指數積分

函數 > 符號函數 > 指數積分

指數積分

• Ei(x) - 傳回 x 的指數積分函數 (科西主值)，其定義如下︰

按一下以複製此運算式

<region id="ID0EBWZIB" top="115.20000000000002" left="57.600000000000009" xml:lang="zh_TW" height="80.552" width="118.27166666666668" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="7" xml:lang="zh_TW">
    <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="zh_TW" />
      <apply xml:lang="zh_TW">
        <id xml:space="preserve" xml:lang="zh_TW">Ei</id>
        <id xml:space="preserve" xml:lang="zh_TW">x</id>
      </apply>
      <apply xml:lang="zh_TW">
        <integral xml:lang="zh_TW" />
        <lambda xml:lang="zh_TW">
          <boundVars xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">t</id>
          </boundVars>
          <apply xml:lang="zh_TW">
            <div xml:lang="zh_TW" />
            <apply xml:lang="zh_TW">
              <pow xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">e</id>
              <id xml:space="preserve" xml:lang="zh_TW">t</id>
            </apply>
            <id xml:space="preserve" xml:lang="zh_TW">t</id>
          </apply>
        </lambda>
        <lowerBound xml:lang="zh_TW">
          <apply xml:lang="zh_TW">
            <neg xml:lang="zh_TW" />
            <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW">∞</id>
          </apply>
        </lowerBound>
        <upperBound xml:lang="zh_TW">
          <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW">π</id>
        </upperBound>
      </apply>
    </apply>
  </math>
</region>

Ei(x) 只會傳回複數指數積分函數的實數部份。

若 x > 0，會將此積分解譯為科西主值。

• Ei(n, x) - 傳回 x 的廣義指數積分函數。

對於任意整數 n，Ei(n, x) 會以下列循環關係定義：

按一下以複製此運算式

<region id="ID0EEZZIB" top="307.20000000000005" left="57.600000000000009" xml:lang="zh_TW" height="27.752000000000002" width="229.48666666666668" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="13" xml:lang="zh_TW">
    <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="zh_TW" />
      <apply xml:lang="zh_TW">
        <mult xml:lang="zh_TW" />
        <id xml:space="preserve" xml:lang="zh_TW">n</id>
        <apply xml:lang="zh_TW">
          <id xml:space="preserve" xml:lang="zh_TW">Ei</id>
          <sequence xml:lang="zh_TW">
            <apply xml:lang="zh_TW">
              <plus xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">n</id>
              <real xml:lang="zh_TW">1</real>
            </apply>
            <id xml:space="preserve" xml:lang="zh_TW">x</id>
          </sequence>
        </apply>
      </apply>
      <apply xml:lang="zh_TW">
        <minus xml:lang="zh_TW" />
        <apply xml:lang="zh_TW">
          <pow xml:lang="zh_TW" />
          <id xml:space="preserve" xml:lang="zh_TW">e</id>
          <apply xml:lang="zh_TW">
            <neg xml:lang="zh_TW" />
            <id xml:space="preserve" xml:lang="zh_TW">x</id>
          </apply>
        </apply>
        <apply xml:lang="zh_TW">
          <mult xml:lang="zh_TW" />
          <id xml:space="preserve" xml:lang="zh_TW">x</id>
          <apply xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">Ei</id>
            <sequence xml:lang="zh_TW">
              <id xml:space="preserve" xml:lang="zh_TW">n</id>
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
            </sequence>
          </apply>
        </apply>
      </apply>
    </apply>
  </math>
</region>

對於 n = 1，Ei(n, x) 的定義如下︰

按一下以複製此運算式

<region id="ID0EG1ZIB" top="249.60000000000002" left="57.600000000000009" xml:lang="zh_TW" height="50.000000000000007" width="242.08666666666667" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="11" xml:lang="zh_TW">
    <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="zh_TW" />
      <apply xml:lang="zh_TW">
        <neg xml:lang="zh_TW" />
        <apply xml:lang="zh_TW">
          <id xml:space="preserve" xml:lang="zh_TW">Ei</id>
          <sequence xml:lang="zh_TW">
            <real xml:lang="zh_TW">1</real>
            <apply xml:lang="zh_TW">
              <neg xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
            </apply>
          </sequence>
        </apply>
      </apply>
      <apply xml:lang="zh_TW">
        <plus xml:lang="zh_TW" />
        <apply xml:lang="zh_TW">
          <plus xml:lang="zh_TW" />
          <id labels="CONSTANT" xml:space="preserve" xml:lang="zh_TW">γ</id>
          <apply xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">ln</id>
            <apply xml:lang="zh_TW">
              <neg xml:lang="zh_TW" />
              <id xml:space="preserve" xml:lang="zh_TW">x</id>
            </apply>
          </apply>
        </apply>
        <apply xml:lang="zh_TW">
          <summation xml:lang="zh_TW" />
          <lambda xml:lang="zh_TW">
            <boundVars xml:lang="zh_TW">
              <id xml:space="preserve" xml:lang="zh_TW">k</id>
            </boundVars>
            <apply xml:lang="zh_TW">
              <div xml:lang="zh_TW" />
              <apply xml:lang="zh_TW">
                <pow xml:lang="zh_TW" />
                <id xml:space="preserve" xml:lang="zh_TW">x</id>
                <id xml:space="preserve" xml:lang="zh_TW">k</id>
              </apply>
              <apply xml:lang="zh_TW">
                <mult xml:lang="zh_TW" />
                <id xml:space="preserve" xml:lang="zh_TW">k</id>
                <apply xml:lang="zh_TW">
                  <factorial xml:lang="zh_TW" />
                  <id xml:space="preserve" xml:lang="zh_TW">k</id>
                </apply>
              </apply>
            </apply>
          </lambda>
          <lowerBound xml:lang="zh_TW">
            <real xml:lang="zh_TW">1</real>
          </lowerBound>
          <upperBound xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">∞</id>
          </upperBound>
        </apply>
      </apply>
    </apply>
  </math>
</region>

其中 γ 是歐拉常數。

針對實數 x，Ei(x) 與 Ei(n, x) 會彼此相關，如下所示︰

按一下以複製此運算式

<region id="ID0EY2ZIB" top="192.00000000000003" left="57.600000000000009" xml:lang="zh_TW" height="25.6" width="171.34333333333336" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="9" xml:lang="zh_TW">
    <apply xml:lang="zh_TW" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="zh_TW" />
      <apply xml:lang="zh_TW">
        <id xml:space="preserve" xml:lang="zh_TW">Ei</id>
        <id xml:space="preserve" xml:lang="zh_TW">x</id>
      </apply>
      <apply xml:lang="zh_TW">
        <id xml:space="preserve" xml:lang="zh_TW">Re</id>
        <apply xml:lang="zh_TW">
          <neg xml:lang="zh_TW" />
          <apply xml:lang="zh_TW">
            <id xml:space="preserve" xml:lang="zh_TW">Ei</id>
            <sequence xml:lang="zh_TW">
              <real xml:lang="zh_TW">1</real>
              <apply xml:lang="zh_TW">
                <neg xml:lang="zh_TW" />
                <id xml:space="preserve" xml:lang="zh_TW">x</id>
              </apply>
            </sequence>
          </apply>
        </apply>
      </apply>
    </apply>
  </math>
</region>

引數

• x 可以是純量、向量或方形矩陣。當使用廣義指數積分函數時，x 也可以是複數純量。

• n 是實數或複數純量。