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view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <polar item-idref="1"/> <xy item-idref="2"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="176" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="2" left="24" top="12.75" width="346.5" height="31.5" align-x="24" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Найти все разложения функции в ряд Лорана по степеням</f> <region region-id="3" left="24" top="27" width="27.75" height="17.25" align-x="34.5" align-y="38.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и установить области сходимости полученных разложений</inlineAttr> </f> </text> </region> <region region-id="4" left="24" top="60.75" width="117" height="13.5" align-x="24" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Задача 12.352 </f> <f family="Arial Cyr" charset="204" size="12">. Дано:</f> </text> </region> <region region-id="6" left="30" top="96.75" width="35.25" height="15.75" align-x="48.75" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z0</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="7" left="90" top="88.5" width="84" height="33.75" align-x="115.5" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">z</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="8" left="36" top="138" width="427.5" height="32.25" align-x="36" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Функция </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">) имеет особые точки: z=0 и<sp count="2"/>z=z0=1. Это полюсы первого порядка. </f> <f family="Arial Cyr" charset="204" size="12">1. Будем раскладывать </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">) </f> <f family="Arial Cyr" charset="204" size="12">в ряд Лорана </f> <f family="Arial Cyr" charset="204" size="12">по степеням </f> <region region-id="10" left="335.25" top="153" width="27.75" height="17.25" align-x="345.75" align-y="164.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="6"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="13" left="102" top="177" width="141.75" height="51.75" align-x="126.75" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> <ml:id xml:space="preserve">∞</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="7"/> </region> <region region-id="14" left="396" top="192.75" width="15.75" height="15.75" align-x="402.75" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="8"/> </region> <region region-id="15" left="36" top="246" width="441" height="62.25" align-x="36" align-y="258" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">вначале в кольце </f> <region 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xml:space="preserve">z0</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z0</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>1.6</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">R</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z0</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="27" left="156" top="420" width="99.75" height="16.5" align-x="188.25" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">r</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="28" left="66" top="468" width="234" height="210.75" align-x="66" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="29" left="42" top="708.75" width="423.75" height="41.25" align-x="42" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В соответствии с тем как изложено в книге "Компьютерная математика", гл.4,</f> <f family="Times New Roman" charset="204" size="12"> <c val="#000">§</c> </f> <f family="Arial Cyr" charset="204" size="12"> 5, (см. также www.bmstu.ru/ps/~kvtitov, папку "Решение задач из Ефимова") вычисляем коэффициенты по формуле: </f> </text> </region> <region region-id="32" left="108" top="768" width="249" height="51" align-x="108" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="17"/> <rendering item-idref="18"/> </region> <region region-id="37" left="420" top="786.75" width="15.75" height="15.75" align-x="426.75" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="19"/> </region> <region region-id="33" left="48" top="840" width="442.5" height="47.25" align-x="48" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Из (2) следует, что для всех </f> <region region-id="34" left="202.5" top="840.75" width="34.5" height="15.75" align-x="215.25" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:id xml:space="preserve">n</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="20"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> имеем </inlineAttr> </f> <region region-id="35" left="279.75" top="840.75" width="42" height="17.25" align-x="306.75" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C1</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="21"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Для остальных </inlineAttr> </f> <region region-id="36" left="413.25" top="840.75" width="35.25" height="15.75" align-x="426" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:id xml:space="preserve">n</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="22"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, и</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">спользуя интегральную формулу Коши (</f> <f family="Times New Roman" charset="204" size="12"> <c val="#000">§</c> </f> <f family="Arial Cyr" charset="204" size="12"> 5, гл.4</f> <f family="Arial Cyr" charset="204" size="12">) и вводя обозначение </f> <f family="Arial Cyr" charset="204" size="14">m=n+1</f> <f family="Arial Cyr" charset="204" size="12">, находим:</f> </text> </region> <region region-id="130" left="102" top="918" width="284.25" height="38.25" align-x="102" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="23"/> <rendering item-idref="24"/> </region> <region region-id="131" left="438" top="930.75" width="15.75" height="15.75" align-x="444.75" align-y="942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="25"/> </region> <region region-id="132" left="48" top="972.75" width="317.25" height="13.5" align-x="48" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Алгоритмически это можно записать следующим образом:</f> </text> </region> <region region-id="103" left="54" top="1014.75" width="35.25" height="15.75" align-x="72.75" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="89" left="318" top="1014.75" width="141.75" height="13.5" align-x="318" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Альтернативный вариант:</f> </text> </region> <region region-id="52" left="54" top="1038.75" width="181.5" height="86.25" align-x="90.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">C1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:factorial/> <ml:parens> <ml:id xml:space="preserve">m</ml:id> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:degree> <ml:id xml:space="preserve">m</ml:id> </ml:degree> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> <ml:id xml:space="preserve">F</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="80" left="312" top="1062.75" width="182.25" height="86.25" align-x="342.75" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="bad_variable mc_z mc_z value_zone " optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">CC1</ml:id> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">FF</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:factorial/> <ml:parens> <ml:id xml:space="preserve">m</ml:id> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:degree> <ml:id xml:space="preserve">m</ml:id> </ml:degree> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> <ml:id xml:space="preserve">FF</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="108" left="60" top="1168.5" width="169.5" height="39" align-x="103.5" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:transpose/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="1" cols="5"> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="29"/> </region> <region region-id="109" left="66" top="1218.75" width="144" height="21.75" align-x="111" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:transpose/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="1" cols="5"> <ml:real>1</ml:real> <ml:real>-1</ml:real> <ml:real>1</ml:real> <ml:real>-1</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="30"/> </region> <region region-id="110" left="282" top="1224.75" width="66.75" height="17.25" align-x="325.5" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:symResult> <ml:real>-1</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="31"/> </region> <region region-id="93" left="54" top="1278.75" width="140.25" height="59.25" align-x="77.25" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">K1</ml:id> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:localDefine> </ml:for> <ml:id xml:space="preserve">K1</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="94" left="54" top="1350.75" width="56.25" height="17.25" align-x="81.75" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-1.000000000000004</ml:real> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="102" left="144" top="1350.75" width="49.5" height="17.25" align-x="171.75" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="95" left="60" top="1410.75" width="423.75" height="27" align-x="60" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Теперь можно записать ряд Лорана в кольце r1 с наперёд выбранным чисом</f> <f family="Arial Cyr" charset="204" size="12">его членов, так как бесконечный ряд мы записать не всегда можем</f> </text> </region> <region region-id="105" left="66" top="1461.75" width="169.5" height="50.25" align-x="97.5" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="104" left="264" top="1468.5" width="206.25" height="33.75" align-x="295.5" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="36"/> </region> <region region-id="111" left="48" top="1542.75" width="441" height="27" align-x="48" align-y="1554" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Так как символьная математика в Mathcad работает плохо, то записать получен-</f> <f family="Arial Cyr" charset="204" size="12">ный ряд Лорана в общем виде можно только вручную<sp count="2"/> </f> </text> </region> <region region-id="152" left="72" top="1587.75" width="203.25" height="50.25" align-x="116.25" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">S</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">M</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="151" left="318" top="1602.75" width="78" height="15.75" align-x="381" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="38"/> </region> <region region-id="150" left="438" top="1602.75" width="15.75" height="15.75" align-x="444.75" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="39"/> </region> <region region-id="116" left="72" top="1648.5" width="222.75" height="33.75" align-x="120" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:sequence> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">M</ml:id> </ml:sequence> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="40"/> </region> <region region-id="118" left="36" top="1704" width="447" height="66" align-x="36" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">2. Будем раскладывать </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">) </f> <f family="Arial Cyr" charset="204" size="12">в ряд Лорана </f> <f family="Arial Cyr" charset="204" size="12">по степеням </f> <region region-id="119" left="335.25" top="1704.75" width="27.75" height="17.25" align-x="345.75" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="41"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">в кольце</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="120" left="39" top="1722.75" width="85.5" height="16.5" align-x="106.5" align-y="1734.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>1</ml:real> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> </math> <rendering item-idref="42"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="6"/>(см. рис.1)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">, в котором находятся уже два полюса.</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">В соответствии с методикой<sp count="2"/>гл.4, </f> <c val="#000"> <f family="Times New Roman" charset="204" size="12"> § </f> </c> <f family="Arial Cyr" charset="204" size="12"> 5 нам необходимо вычислить коэффициент</f> <f family="Arial Cyr" charset="204" size="12">C2</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">или контурный интеграл относительно второго полюса по формуле</f> </text> </region> <region region-id="121" left="54" top="1794" width="249.75" height="53.25" align-x="54" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="43"/> <rendering item-idref="44"/> </region> <region region-id="123" left="420" top="1818.75" width="15.75" height="15.75" align-x="426.75" align-y="1830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="45"/> </region> <region region-id="125" left="36" top="1872.75" width="456" height="57.75" align-x="36" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Здесь контур </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">g</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">имеет малый радиус такой, что внутри него содержится только одна</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">особая точка (полюс z=0).<sp count="2"/>Поэтому в числителе (5) стоит регулярная функция в </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">области, охваченной этим контуром, и вычислять C2</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">n </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">согласно интегральной фор-</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">муле Коши надо так</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="128" left="150" top="1956" width="167.25" height="42.75" align-x="150" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="46"/> <rendering item-idref="47"/> </region> <region region-id="129" left="438" top="1968.75" width="15.75" height="15.75" align-x="444.75" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="48"/> </region> <region region-id="137" left="120" top="2034" width="209.25" height="36.75" align-x="120" align-y="2034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="49"/> <rendering item-idref="50"/> </region> <region region-id="138" left="36" top="2046.75" width="42" height="13.5" align-x="36" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Так как </f> </text> </region> <region region-id="140" left="36" top="2124.75" width="459" height="31.5" align-x="36" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">то возвращаясь к методике<sp count="2"/>гл.4, </f> <f family="Times New Roman" charset="204" size="12"> <c val="#000">§</c> </f> <f family="Arial Cyr" charset="204" size="12"> 5, </f> <f family="Arial Cyr" charset="204" size="12"> запишем формулу вычисления коэффициента C</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">= C1</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">+ C2</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">ряда Лорана в кольце </f> <f family="Arial Cyr" charset="204" size="12"> r </f> <f family="Arial Cyr" charset="204" size="12">2 </f> </text> </region> <region region-id="141" left="78" top="2190" width="218.25" height="44.25" align-x="78" align-y="2190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="51"/> <rendering item-idref="52"/> </region> <region region-id="142" left="420" top="2202.75" width="15.75" height="15.75" align-x="426.75" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="53"/> </region> <region region-id="144" left="36" top="2262" width="444.75" height="35.25" align-x="36" align-y="2274" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Таким образом, разложение функции </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12"> в ряд Лорана по степеням </f> <region region-id="145" left="407.25" top="2262.75" width="27.75" height="17.25" align-x="417.75" align-y="2274" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="54"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в кольце </inlineAttr> </f> <region region-id="153" left="77.25" top="2280.75" width="85.5" height="16.5" align-x="144.75" align-y="2292.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>1</ml:real> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> </math> <rendering item-idref="55"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> будет иметь вид (j = -(n+2))</inlineAttr> </f> </text> </region> <region 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top="2311.5" width="54.75" height="22.5" align-x="193.5" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="172" left="54" top="2343.75" width="150.75" height="50.25" align-x="85.5" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">j</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">J</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="171" left="300" top="2358" width="81" height="16.5" align-x="363" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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<currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <polar item-idref="1"/> <xy item-idref="2"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="176" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="2" left="24" top="12.75" width="346.5" height="31.5" align-x="24" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Найти все разложения функции в ряд Лорана по степеням</f> <region region-id="3" left="24" top="27" width="27.75" height="17.25" align-x="34.5" align-y="38.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="3"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> и установить области сходимости полученных разложений</inlineAttr> </f> </text> </region> <region region-id="4" left="24" top="60.75" width="117" height="13.5" align-x="24" align-y="72" show-border="false" 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Дано:</f> </text> </region> <region region-id="6" left="30" top="96.75" width="35.25" height="15.75" align-x="48.75" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">z0</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="7" left="90" top="88.5" width="84" height="33.75" align-x="115.5" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">z</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="8" left="36" top="138" width="427.5" height="32.25" align-x="36" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Функция </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">) имеет особые точки: z=0 и<sp count="2"/>z=z0=1. Это полюсы первого порядка. </f> <f family="Arial Cyr" charset="204" size="12">1. Будем раскладывать </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">) </f> <f family="Arial Cyr" charset="204" size="12">в ряд Лорана </f> <f family="Arial Cyr" charset="204" size="12">по степеням </f> <region region-id="10" left="335.25" top="153" width="27.75" height="17.25" align-x="345.75" align-y="164.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="6"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="13" left="102" top="177" width="141.75" height="51.75" align-x="126.75" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> <ml:id xml:space="preserve">∞</ml:id> </ml:bounds> </ml:apply> </ml:apply> </math> <rendering item-idref="7"/> </region> <region region-id="14" left="396" top="192.75" width="15.75" height="15.75" align-x="402.75" align-y="204" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="8"/> </region> <region region-id="15" left="36" top="246" width="441" height="62.25" align-x="36" align-y="258" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">вначале в кольце </f> <region 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xml:space="preserve">z0</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z0</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>1.6</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">R</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z0</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="27" left="156" top="420" width="99.75" height="16.5" align-x="188.25" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">r</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">R</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="28" left="66" top="468" width="234" height="210.75" align-x="66" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="29" left="42" top="708.75" width="423.75" height="41.25" align-x="42" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">В соответствии с тем как изложено в книге "Компьютерная математика", гл.4,</f> <f family="Times New Roman" charset="204" size="12"> <c val="#000">§</c> </f> <f family="Arial Cyr" charset="204" size="12"> 5, (см. также www.bmstu.ru/ps/~kvtitov, папку "Решение задач из Ефимова") вычисляем коэффициенты по формуле: </f> </text> </region> <region region-id="32" left="108" top="768" width="249" height="51" align-x="108" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="17"/> <rendering item-idref="18"/> </region> <region region-id="37" left="420" top="786.75" width="15.75" height="15.75" align-x="426.75" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="19"/> </region> <region region-id="33" left="48" top="840" width="442.5" height="47.25" align-x="48" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Из (2) следует, что для всех </f> <region region-id="34" left="202.5" top="840.75" width="34.5" height="15.75" align-x="215.25" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:id xml:space="preserve">n</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="20"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> имеем </inlineAttr> </f> <region region-id="35" left="279.75" top="840.75" width="42" height="17.25" align-x="306.75" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C1</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="21"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Для остальных </inlineAttr> </f> <region region-id="36" left="413.25" top="840.75" width="35.25" height="15.75" align-x="426" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:id xml:space="preserve">n</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="22"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, и</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">спользуя интегральную формулу Коши (</f> <f family="Times New Roman" charset="204" size="12"> <c val="#000">§</c> </f> <f family="Arial Cyr" charset="204" size="12"> 5, гл.4</f> <f family="Arial Cyr" charset="204" size="12">) и вводя обозначение </f> <f family="Arial Cyr" charset="204" size="14">m=n+1</f> <f family="Arial Cyr" charset="204" size="12">, находим:</f> </text> </region> <region region-id="130" left="102" top="918" width="284.25" height="38.25" align-x="102" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="23"/> <rendering item-idref="24"/> </region> <region region-id="131" left="438" top="930.75" width="15.75" height="15.75" align-x="444.75" align-y="942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="25"/> </region> <region region-id="132" left="48" top="972.75" width="317.25" height="13.5" align-x="48" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Алгоритмически это можно записать следующим образом:</f> </text> </region> <region region-id="103" left="54" top="1014.75" width="35.25" height="15.75" align-x="72.75" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="89" left="318" top="1014.75" width="141.75" height="13.5" align-x="318" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Альтернативный вариант:</f> </text> </region> <region region-id="52" left="54" top="1038.75" width="181.5" height="86.25" align-x="90.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">C1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:factorial/> <ml:parens> <ml:id xml:space="preserve">m</ml:id> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:degree> <ml:id xml:space="preserve">m</ml:id> </ml:degree> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> <ml:id xml:space="preserve">F</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="80" left="312" top="1062.75" width="182.25" height="86.25" align-x="342.75" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="bad_variable mc_z mc_z value_zone " optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">CC1</ml:id> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">FF</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:factorial/> <ml:parens> <ml:id xml:space="preserve">m</ml:id> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:degree> <ml:id xml:space="preserve">m</ml:id> </ml:degree> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> <ml:id xml:space="preserve">FF</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="108" left="60" top="1168.5" width="169.5" height="39" align-x="103.5" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:transpose/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="1" cols="5"> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="29"/> </region> <region region-id="109" left="66" top="1218.75" width="144" height="21.75" align-x="111" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:transpose/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:symResult> <ml:matrix rows="1" cols="5"> <ml:real>1</ml:real> <ml:real>-1</ml:real> <ml:real>1</ml:real> <ml:real>-1</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="30"/> </region> <region region-id="110" left="282" top="1224.75" width="66.75" height="17.25" align-x="325.5" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:symResult> <ml:real>-1</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="31"/> </region> <region region-id="93" left="54" top="1278.75" width="140.25" height="59.25" align-x="77.25" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">K1</ml:id> <ml:program> <ml:for> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">C1</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:localDefine> </ml:for> <ml:id xml:space="preserve">K1</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="94" left="54" top="1350.75" width="56.25" height="17.25" align-x="81.75" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-1.000000000000004</ml:real> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="102" left="144" top="1350.75" width="49.5" height="17.25" align-x="171.75" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="95" left="60" top="1410.75" width="423.75" height="27" align-x="60" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Теперь можно записать ряд Лорана в кольце r1 с наперёд выбранным чисом</f> <f family="Arial Cyr" charset="204" size="12">его членов, так как бесконечный ряд мы записать не всегда можем</f> </text> </region> <region region-id="105" left="66" top="1461.75" width="169.5" height="50.25" align-x="97.5" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">K1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">m</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">M</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="104" left="264" top="1468.5" width="206.25" height="33.75" align-x="295.5" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="36"/> </region> <region region-id="111" left="48" top="1542.75" width="441" height="27" align-x="48" align-y="1554" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Так как символьная математика в Mathcad работает плохо, то записать получен-</f> <f family="Arial Cyr" charset="204" size="12">ный ряд Лорана в общем виде можно только вручную<sp count="2"/> </f> </text> </region> <region region-id="152" left="72" top="1587.75" width="203.25" height="50.25" align-x="116.25" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">S</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">M</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="151" left="318" top="1602.75" width="78" height="15.75" align-x="381" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="38"/> </region> <region region-id="150" left="438" top="1602.75" width="15.75" height="15.75" align-x="444.75" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="39"/> </region> <region region-id="116" left="72" top="1648.5" width="222.75" height="33.75" align-x="120" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:sequence> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">M</ml:id> </ml:sequence> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="40"/> </region> <region region-id="118" left="36" top="1704" width="447" height="66" align-x="36" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">2. Будем раскладывать </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">) </f> <f family="Arial Cyr" charset="204" size="12">в ряд Лорана </f> <f family="Arial Cyr" charset="204" size="12">по степеням </f> <region region-id="119" left="335.25" top="1704.75" width="27.75" height="17.25" align-x="345.75" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="41"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">в кольце</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <region region-id="120" left="39" top="1722.75" width="85.5" height="16.5" align-x="106.5" align-y="1734.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>1</ml:real> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> </math> <rendering item-idref="42"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="6"/>(см. рис.1)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">, в котором находятся уже два полюса.</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">В соответствии с методикой<sp count="2"/>гл.4, </f> <c val="#000"> <f family="Times New Roman" charset="204" size="12"> § </f> </c> <f family="Arial Cyr" charset="204" size="12"> 5 нам необходимо вычислить коэффициент</f> <f family="Arial Cyr" charset="204" size="12">C2</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">или контурный интеграл относительно второго полюса по формуле</f> </text> </region> <region region-id="121" left="54" top="1794" width="249.75" height="53.25" align-x="54" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="43"/> <rendering item-idref="44"/> </region> <region region-id="123" left="420" top="1818.75" width="15.75" height="15.75" align-x="426.75" align-y="1830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="45"/> </region> <region region-id="125" left="36" top="1872.75" width="456" height="57.75" align-x="36" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Здесь контур </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">g</inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">имеет малый радиус такой, что внутри него содержится только одна</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">особая точка (полюс z=0).<sp count="2"/>Поэтому в числителе (5) стоит регулярная функция в </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">области, охваченной этим контуром, и вычислять C2</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">n </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">согласно интегральной фор-</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">муле Коши надо так</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="128" left="150" top="1956" width="167.25" height="42.75" align-x="150" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="46"/> <rendering item-idref="47"/> </region> <region region-id="129" left="438" top="1968.75" width="15.75" height="15.75" align-x="444.75" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="48"/> </region> <region region-id="137" left="120" top="2034" width="209.25" height="36.75" align-x="120" align-y="2034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="49"/> <rendering item-idref="50"/> </region> <region region-id="138" left="36" top="2046.75" width="42" height="13.5" align-x="36" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Так как </f> </text> </region> <region region-id="140" left="36" top="2124.75" width="459" height="31.5" align-x="36" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">то возвращаясь к методике<sp count="2"/>гл.4, </f> <f family="Times New Roman" charset="204" size="12"> <c val="#000">§</c> </f> <f family="Arial Cyr" charset="204" size="12"> 5, </f> <f family="Arial Cyr" charset="204" size="12"> запишем формулу вычисления коэффициента C</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">= C1</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">+ C2</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">ряда Лорана в кольце </f> <f family="Arial Cyr" charset="204" size="12"> r </f> <f family="Arial Cyr" charset="204" size="12">2 </f> </text> </region> <region region-id="141" left="78" top="2190" width="218.25" height="44.25" align-x="78" align-y="2190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="51"/> <rendering item-idref="52"/> </region> <region region-id="142" left="420" top="2202.75" width="15.75" height="15.75" align-x="426.75" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="53"/> </region> <region region-id="144" left="36" top="2262" width="444.75" height="35.25" align-x="36" align-y="2274" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Таким образом, разложение функции </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12"> в ряд Лорана по степеням </f> <region region-id="145" left="407.25" top="2262.75" width="27.75" height="17.25" align-x="417.75" align-y="2274" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">z</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="54"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в кольце </inlineAttr> </f> <region region-id="153" left="77.25" top="2280.75" width="85.5" height="16.5" align-x="144.75" align-y="2292.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>1</ml:real> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">z0</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> </math> <rendering item-idref="55"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> будет иметь вид (j = -(n+2))</inlineAttr> </f> </text> </region> <region 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top="2311.5" width="54.75" height="22.5" align-x="193.5" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="172" left="54" top="2343.75" width="150.75" height="50.25" align-x="85.5" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">j</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">J</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="171" left="300" top="2358" width="81" height="16.5" align-x="363" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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