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style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Задачи из Ефимова [24]. Ряды Фурье. </f> <f family="Arial Cyr" charset="204" size="12"> <b>Задача №12.481</b> </f> </p> </text> </region> <region region-id="39" left="42" top="36" width="451.5" height="27.75" align-x="42" align-y="48" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">В задаче требуется разложить функцию в ряд Фурье. Зададим функцию </f> <f family="Arial Cyr" charset="204" size="12"> <i>f</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>x</i> </f> <f family="Arial Cyr" charset="204" size="12">) следу-</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">ющим образом (см. рис.1) :</f> </p> </text> </region> <region region-id="2" left="36" top="75.75" width="210.75" height="34.5" align-x="62.25" align-y="96" 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">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="123" left="456" top="84.75" width="15.75" height="15.75" align-x="462.75" align-y="96" 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="3"/> </region> <region region-id="40" left="36" top="120" width="471" height="27.75" align-x="36" align-y="132" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Для периодической функции </f> <f family="Arial Cyr" charset="204" size="12"> <i>f</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>x</i> </f> <f family="Arial Cyr" charset="204" size="12">) задан п</f> <f family="Arial Cyr" charset="204" size="12">олупериод </f> <f family="Arial Cyr" charset="204" size="12"> <i>L</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> который можно выбрать, исходя </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">из удобства вычисления (см. [13], стр. 328) коэффициентов ряда Фурье, равный:</f> </p> </text> </region> <region region-id="58" left="48" top="162.75" width="30.75" height="15.75" align-x="62.25" align-y="174" 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">q</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="57" left="108" top="162" width="43.5" height="16.5" align-x="123" align-y="174" 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:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="53" left="42" top="192" width="465.75" height="28.5" align-x="42" align-y="204" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Введём параметр анимации </f> <f family="Arial Cyr" charset="204" size="12"> <i>FRAME </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">так, чтобы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> 20<</f> <f family="Arial Cyr" charset="204" size="12"> <i>FRAME</i> </f> <f family="Arial Cyr" charset="204" size="12">, через который выразим число </f> <f family="Arial Cyr" charset="204" size="12"> <i>N</i> </f> <f family="Arial Cyr" charset="204" size="12"> членов ряда Фурье, например, так:<sp count="2"/> </f> </p> </text> </region> <region region-id="59" left="180" top="228.75" width="87" height="15.75" align-x="197.25" align-y="240" 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:id xml:space="preserve">N</ml:id> <ml:apply> <ml:plus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="61" left="36" top="260.25" width="435" height="12" align-x="36" align-y="270" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="10"> <i>Примечание</i> </f> <f family="Arial Cyr" charset="204" size="10">. Такое ограничение 20<</f> <f family="Arial Cyr" charset="204" size="10"> <i>FRAME</i> </f> <f family="Arial Cyr" charset="204" size="10"> связано со временем записи видеоклипа и только. </f> </p> </text> </region> <region region-id="25" left="36" top="281.25" width="475.5" height="45" align-x="36" align-y="294" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Сумма ряда </f> <f family="Arial Cyr" charset="204" size="12"> <i>S</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>x</i> </f> <f family="Arial Cyr" charset="204" size="12">) приближает функцию </f> <f family="Arial Cyr" charset="204" size="12"> <i>f</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>x</i> </f> <f family="Arial Cyr" charset="204" size="12">) при любом из полупериодов: </f> <region region-id="62" left="443.25" top="282" width="41.25" height="16.5" align-x="457.5" align-y="294" 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:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">q</ml:id> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="7"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i>q</i> </f> <f family="Arial Cyr" charset="204" size="12">=1 или<sp count="2"/>2. Покажем это в режиме </f> <f family="Arial Cyr" charset="204" size="12">анимации на рис.1, предварительно вычислив коэффициенты </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье</f> </p> </text> </region> <region region-id="102" left="42" top="336.75" width="35.25" height="17.25" align-x="60.75" align-y="348" 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:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="101" left="132" top="336.75" width="35.25" height="17.25" align-x="150.75" align-y="348" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define 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<ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region 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<ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">h</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="125" left="492" top="390.75" width="15.75" height="15.75" align-x="498.75" align-y="402" 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="12"/> </region> <region region-id="70" left="36" top="444.75" width="464.25" height="41.25" align-x="36" align-y="456" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Проведём визуализацию численных значений коэффициентов </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье и запишем</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">его </f> <f family="Arial Cyr" charset="204" size="12">сумму<sp count="2"/>S(x), оставив в ней только коэффициенты </f> <f family="Arial Cyr" charset="204" size="12"> <i>b</i> </f> <f family="Arial Cyr" charset="204" size="12">, так как </f> <f family="Arial Cyr" charset="204" size="12">коэффициенты </f> <f family="Arial Cyr" charset="204" size="12"> <i>a</i> </f> <f family="Arial Cyr" charset="204" size="12"> по</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">определению равны нулю, что подтверждается ниже</f> </p> </text> </region> <region region-id="131" left="492" top="504.75" width="15.75" height="15.75" align-x="498.75" align-y="516" 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="13"/> </region> <region region-id="130" left="42" top="510.75" width="44.25" height="15.75" align-x="54" align-y="522" 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">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>3</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="129" left="120" top="510.75" width="36.75" height="81" align-x="129" align-y="522" 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:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix 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<resultFormat> <table item-idref="17"/> </resultFormat> </math> <rendering item-idref="18"/> </region> <region region-id="127" left="288" top="495.75" width="165" height="50.25" align-x="318" align-y="522" 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:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <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:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="114" left="30" top="588.75" width="477.75" height="99.75" align-x="30" align-y="600" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Как указано в </f> <f family="Arial Cyr" charset="204" size="12">[13], с целью</f> <f family="Arial Cyr" charset="204" size="12"> удобства вычисления </f> <f family="Arial Cyr" charset="204" size="12">коэффициентов </f> <f family="Arial Cyr" charset="204" size="12">ряда Фурье, можно выбрать множители<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>1</sub> </f> <f family="Arial Cyr" charset="204" size="12">=0 и<sp count="2"/> </f> <i> <f family="Arial Cyr" charset="204" size="12">h</f> </i> <f family="Arial Cyr" charset="204" size="12"> <sub>2</sub> </f> <f family="Arial Cyr" charset="204" size="12">=2, что упрощает задание функции f(x), оставляя результат прежним. Это также легко проверяется </f> <f family="Arial Cyr" charset="204" size="12">в режиме </f> <f family="Arial Cyr" charset="204" size="12">анимации на рис.1. Заметим, что такой выбор </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>1</sub> </f> <f family="Arial Cyr" charset="204" size="12">=0 и<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>2</sub> </f> <f family="Arial Cyr" charset="204" size="12">=2 равносилен заданию </f> <region region-id="115" left="413.25" top="633.75" width="41.25" height="16.5" align-x="427.5" align-y="645.75" 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:equal/> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </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">, то есть</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> сочетания параметров </inlineAttr> </f> <i> <f family="Arial Cyr" charset="204" size="12">h</f> </i> <f family="Arial Cyr" charset="204" size="12"> <sub>1</sub> </f> <f family="Arial Cyr" charset="204" size="12">=0,<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <sub> <f family="Arial Cyr" charset="204" size="12">2</f> </sub> <f family="Arial Cyr" charset="204" size="12">=2, </f> <f family="Arial Cyr" charset="204" size="12"> <i>q</i> </f> <f family="Arial Cyr" charset="204" size="12">=1<sp count="2"/>и<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>1</sub> </f> <f family="Arial Cyr" charset="204" size="12">=1,<sp count="2"/> </f> <i> <f family="Arial Cyr" charset="204" size="12">h</f> </i> <f family="Arial Cyr" charset="204" size="12"> <sub>2</sub> </f> <f family="Arial Cyr" charset="204" size="12">=1, </f> <f family="Arial Cyr" charset="204" size="12"> <i>q</i> </f> <f family="Arial Cyr" charset="204" size="12">=1 или 2 - равоносильны. </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Вычислим (2) при условии </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>1</sub> </f> <f family="Arial Cyr" charset="204" size="12">=0,<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>h</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sub>2</sub> </f> <f family="Arial Cyr" charset="204" size="12">=2, </f> <f family="Arial Cyr" charset="204" size="12"> <i>q</i> </f> <f family="Arial Cyr" charset="204" size="12">=1:</f> </p> </text> </region> <region region-id="88" left="30" top="720" width="308.25" height="227.25" align-x="30" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="21"/> <rendering item-idref="22"/> </region> <region region-id="140" left="36" top="990" width="206.25" height="56.25" align-x="67.5" align-y="1020" 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:function> <ml:id xml:space="preserve">A</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="141" left="264" top="990" width="201.75" height="56.25" align-x="294.75" align-y="1020" 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">B</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> 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xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>-1.1360953475058735E-16</ml:real> <ml:real>-7.1031258967638857E-18</ml:real> <ml:real>-2.3689580055235426E-17</ml:real> <ml:real>-2.545604000628812E-16</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="25"/> </resultFormat> </math> <rendering item-idref="26"/> </region> <region region-id="148" left="114" top="1062.75" width="49.5" height="81" align-x="123.75" align-y="1074" 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:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0.99999999999999978</ml:real> <ml:real>0.49999999999999978</ml:real> <ml:real>0.33333333333333354</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="27"/> </resultFormat> </math> <rendering item-idref="28"/> </region> <region region-id="133" left="48" top="1158.75" width="456.75" height="54.75" align-x="48" align-y="1170" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Вычисленные коэффициенты A(n) и<sp count="2"/>B(n) полностью совпадают с (2) при том, что</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">подынтегральная функция f(x) проще в её задании.</f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Рассмотрим Функцию Хевисайда </f> <f family="Arial Cyr" charset="204" size="12"> <i>Ф</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>х</i> </f> <f family="Arial Cyr" charset="204" size="12">) и то как с помощью этой функции можно задавать другие функции на примере данной задачи (см. рис.2):</f> </p> </text> </region> <region region-id="112" left="66" top="1236" width="177.75" height="16.5" align-x="97.5" align-y="1248" 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:function> <ml:id xml:space="preserve">H</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="122" left="294" top="1227.75" width="71.25" height="34.5" align-x="321.75" align-y="1248" 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">z</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="121" left="54" top="1284" width="347.25" height="16.5" align-x="85.5" align-y="1296" 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">U</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">z</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Φ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="151" left="30" top="1332" width="470.25" height="31.5" align-x="30" align-y="1344" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">В условиях задачи </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">№12.481 функция задана как </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal">z</inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal">x</inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">), а на рис.2 функцию </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>z</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>x</i> </f> <f family="Arial Cyr" charset="204" size="12">) имитируют</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">функции U(x) и<sp count="2"/>H(x) соответственно на отрезках </f> <region region-id="152" left="355.5" top="1347" width="52.5" height="16.5" align-x="381" align-y="1359" 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:sequence> <ml:apply> <ml:mult style="auto-select"/> <ml:real>-2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:sequence> </ml:parens> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> и<sp count="2"/> </f> <region region-id="153" left="423.75" top="1347" width="40.5" height="16.5" align-x="443.25" align-y="1359" 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:sequence> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:sequence> </ml:parens> </math> <rendering item-idref="33"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> </p> </text> </region> <region region-id="50" left="48" top="1410" width="336.75" height="167.25" align-x="48" align-y="1410" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="34"/> <rendering item-idref="35"/> </region> <region region-id="173" left="42" top="1601.25" width="441" height="30.75" align-x="42" align-y="1614" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Итак, разложим теперь функцию U(x) в ряд Фурье, с полуперидом </f> <region region-id="157" left="402.75" top="1602" width="15" height="16.5" align-x="408" 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:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </math> <rendering item-idref="36"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Тогда</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">будем иметь </inlineAttr> </f> </p> </text> </region> <region region-id="181" left="48" top="1662" width="43.5" height="16.5" align-x="63" align-y="1674" 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="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="182" left="114" top="1644" width="185.25" height="56.25" align-x="150.75" align-y="1674" 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">Ua</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="183" left="318" top="1644" width="182.25" height="56.25" align-x="355.5" align-y="1674" 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">Ub</ml:id> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="167" left="66" top="1716.75" width="38.25" height="81" align-x="78.75" align-y="1728" 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:id xml:space="preserve">Ua</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>-3.5473591401557412E-17</ml:real> <ml:real>0</ml:real> <ml:real>3.72291553710939E-18</ml:real> <ml:real>7.0678992921411486E-17</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="40"/> </resultFormat> </math> <rendering item-idref="41"/> </region> <region region-id="166" left="144" top="1716.75" width="38.25" height="81" align-x="157.5" align-y="1728" 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:id xml:space="preserve">Ub</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>-1.2031328602182185E-17</ml:real> <ml:real>0.99999999999999989</ml:real> <ml:real>-9.7989031717191937E-18</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="42"/> </resultFormat> </math> <rendering item-idref="43"/> </region> <region region-id="169" left="258" top="1731.75" width="181.5" height="50.25" align-x="296.25" align-y="1758" 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">SU</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <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:id xml:space="preserve">Ub</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="170" left="54" top="1806.75" width="335.25" height="13.5" align-x="54" align-y="1818" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Построим график функции z(x) или<sp count="2"/>U(x) и её ряда SU(x) рис.3 </f> </p> </text> </region> <region region-id="172" left="48" top="1842" width="342.75" height="191.25" align-x="48" align-y="1842" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="45"/> <rendering item-idref="46"/> </region> <region region-id="187" left="36" top="2052.75" width="475.5" height="27" align-x="36" align-y="2064" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Полученные ряды Фурье являются неполными, так как один из коэффициентов ряда A(n) равен нулю, что характерно для нечетных функций. </f> </p> </text> </region> </regions> <binaryContent> <item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdzlQwd/GGhMbj9InsCBWJj6CnvifEA1SWPrm5+yCvUf2GGa/ GQZExBiHSVQb3OPjcnep3PO6Oz+q0tURyZljYrVzVzu1aIHplcX2cD8VdWPtmxcFM1ym7Z9K q2r5Stz79lprtVE2w2y4ZZvj8RBn/7u/8m5dueK0zANd2RZMeUJs2Ytitlj6lAO2BtGUDdvQ ixOCIQ8RgfmssOoII6muMhA9YQSOpUfiCRiN98FofABG40P9F8RIGYixMAJ/AQAA//8DAMuS Xhh8AQAA</item> <item item-id="2">iVBORw0KGgoAAAANSUhEUgAAARkAAAAuCAYAAAAcGG22AAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA BYVJREFUeF7tnYFx5CAMRXMdbAlbQkpICVtSOtgStoSUkBJSQkrxRZlzhiOAvxAyxnxmMrm7 gJAe0jfYe/Gf5as9sZEACZCAFwERGbZl+fj4WG632/L29kYcJEACIAGpF6kbqZ9cewJtnbrb 4/FYrtdrEdSpATA4EjAQEIF5fn5epI5SbXqRETCXy8WAmENJgASEgFyoU0Iztch8fn5yB8P6 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