具有复数极限的定积分

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具有复数极限的定积分

Mathcad 使用轮廓积分在复平面中沿轮廓线上方路径计算某些积分。

• 可以符号形式计算具有复数极限的积分。

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  <math resultRef="7">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">exp</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
          </apply>
        </lambda>
        <lowerBound>
          <imag symbol="i">1</imag>
        </lowerBound>
        <upperBound>
          <apply>
            <plus />
            <imag symbol="i">2</imag>
            <real>1</real>
          </apply>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <minus />
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT">e</id>
            <apply>
              <plus />
              <real>1</real>
              <imag symbol="i">2</imag>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve" labels="CONSTANT">e</id>
            <imag symbol="i">1</imag>
          </apply>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

• 使用关键字 float 对积分值进行数值运算。

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  <math resultRef="8">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">exp</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
          </apply>
        </lambda>
        <lowerBound>
          <imag symbol="i">1</imag>
        </lowerBound>
        <upperBound>
          <apply>
            <plus />
            <imag symbol="i">2</imag>
            <real>1</real>
          </apply>
        </upperBound>
      </apply>
      <command>
        <id xml:space="preserve">float</id>
      </command>
      <symResult>
        <apply>
          <plus />
          <real>-1.6715066896249533558</real>
          <imag symbol="i">1.630255687196922421</imag>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• 十进制符号可用于数字结果。比较：

<region id="ID0ECS2LB" actualWidth="380.72" actualHeight="57.800000000000004" top="249.60000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="9">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">cos</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
          </apply>
        </lambda>
        <lowerBound>
          <imag symbol="i">1</imag>
        </lowerBound>
        <upperBound>
          <apply>
            <plus />
            <imag symbol="i">2</imag>
            <real>1</real>
          </apply>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <div />
            <apply>
              <mult />
              <imag symbol="i">1</imag>
              <apply>
                <pow />
                <id xml:space="preserve" labels="CONSTANT">e</id>
                <apply>
                  <plus />
                  <imag symbol="i">-1</imag>
                  <real>2</real>
                </apply>
              </apply>
            </apply>
            <real>2</real>
          </apply>
          <parens>
            <apply>
              <minus />
              <apply>
                <div />
                <apply>
                  <mult />
                  <imag symbol="i">1</imag>
                  <apply>
                    <pow />
                    <id xml:space="preserve" labels="CONSTANT">e</id>
                    <real>-1</real>
                  </apply>
                </apply>
                <real>2</real>
              </apply>
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <div />
                    <apply>
                      <mult />
                      <imag symbol="i">1</imag>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="CONSTANT">e</id>
                        <apply>
                          <minus />
                          <imag symbol="i">1</imag>
                          <real>2</real>
                        </apply>
                      </apply>
                    </apply>
                    <real>2</real>
                  </apply>
                  <apply>
                    <div />
                    <apply>
                      <mult />
                      <id xml:space="preserve" labels="CONSTANT">e</id>
                      <imag symbol="i">1</imag>
                    </apply>
                    <real>2</real>
                  </apply>
                </apply>
              </parens>
            </apply>
          </parens>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

与

<region id="ID0ERS2LB" actualWidth="462.25666666666666" actualHeight="57.800000000000004" top="307.20000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="10">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">cos</id>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
          </apply>
        </lambda>
        <lowerBound>
          <imag symbol="i">1.0</imag>
        </lowerBound>
        <upperBound>
          <apply>
            <plus />
            <imag symbol="i">2</imag>
            <real>1</real>
          </apply>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <plus />
          <real>3.1657785132161681467</real>
          <imag symbol="i">0.78439984777780444019</imag>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

• 可使用无穷大或负无穷大作为积分极限。

单击可复制此表达式

<region id="ID0EBU2LB" actualWidth="101.84" actualHeight="81.552" top="422.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="11">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <apply>
              <pow />
              <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
              <real>2</real>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
          <apply>
            <neg />
            <id xml:space="preserve" labels="CONSTANT" label-is-contextual="true">∞</id>
          </apply>
        </lowerBound>
        <upperBound>
          <imag symbol="i">2</imag>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <apply>
          <div />
          <imag symbol="i">1</imag>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

• 若积分不收敛，则无法得到明确结果：

单击可复制此表达式

<region id="ID0ERV2LB" actualWidth="147.14" actualHeight="79.4" top="518.40000000000009" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="12">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
          </apply>
        </lambda>
        <lowerBound>
          <imag symbol="i">-1</imag>
        </lowerBound>
        <upperBound>
          <imag symbol="i">2</imag>
        </upperBound>
      </apply>
      <command>
        <placeholder />
      </command>
      <symResult>
        <id xml:space="preserve" labels="*">undefined</id>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

要在函数具有奇点时计算积分，请使用柯西主值。

单击可复制此表达式

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  <math resultRef="13">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <integral />
        <lambda>
          <boundVars>
            <id xml:space="preserve">x</id>
          </boundVars>
          <apply>
            <div />
            <real>1</real>
            <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">x</id>
          </apply>
        </lambda>
        <lowerBound>
          <imag symbol="i">-1</imag>
        </lowerBound>
        <upperBound>
          <imag symbol="i">2</imag>
        </upperBound>
      </apply>
      <command>
        <id xml:space="preserve">柯西</id>
      </command>
      <symResult>
        <apply>
          <id xml:space="preserve" labels="FUNCTION">ln</id>
          <real>2</real>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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