查看“︁調和共軛”︁的源代码

在數學中，'''調和共軛'''（Harmonic conjugate）是針對[[函數]]的概念。定義在開集<math>\Omega\subset\R^2</math>中的函數<math>u(x,\,y)</math>，另一個函數<math>v(x,\,y)</math>為其共軛函數的充分必要條件是<math>u(x,\,y)</math>和<math>v(x,\,y)</math>需要是[[全純函數]]<math> f(z)</math>（<math>z:=x+iy\in\Omega</math>）的實部及虛部。

因此，若<math>f(z):=u(x,y)+iv(x,y)</math>在<math>\Omega</math>中為全純函數，<math>v</math>就為<math>u</math>的共軛函數。而<math>v</math>和<math>u</math>也是<math>\Omega</math>中的[[调和函数]]。<math>v</math>為<math>u</math>的共軛函數，若且唯若<math>u</math>為<math>-v</math>的共軛函數。

在<math>\Omega</math>區間內，<math> v</math>是<math>u</math>共軛函數的充分必要條件是<math>u</math>和<math>v</math>滿足[[柯西－黎曼方程]]。

:{{quad}} <math>{ \partial u \over \partial x } = { \partial v \over \partial y } </math>
:{{quad}}<math>{ \partial u \over \partial y } = -{ \partial v \over \partial x } . </math>
==舉例==
例如，考慮函數<math display="block">u(x,y) = e^x \sin y. </math>

因為
<math display="block">{\partial u \over \partial x } = e^x \sin y, \quad {\partial^2 u \over \partial x^2} = e^x \sin y</math>
且
<math display="block">{\partial u \over \partial y} = e^x \cos y, \quad {\partial^2 u \over \partial y^2} = - e^x \sin y,</math>
會滿足
<math display="block"> \Delta u = \nabla^2 u = 0</math>
（<math>\Delta</math>是[[拉普拉斯算子]]），因此是调和函数。現在假設存在<math>v(x,y)</math>，可以滿足柯西－黎曼方程：

<math display="block">{\partial u \over \partial x} = {\partial v \over \partial y} = e^x \sin y</math>
and
<math display="block">{\partial u \over \partial y} = -{\partial v \over \partial x} =  e^x \cos y.</math>

化簡後可得
<math display="block">{\partial v \over \partial y} = e^x \sin y</math>
且
<math display="block">{\partial v \over \partial x} = -e^x \cos y</math>
因此可得
<math display="block"> v = -e^x \cos y + C.</math>

若{{math|''u''}}和{{math|''v''}}的關係對調，函數就不是調和共軛函數了，因為柯西－黎曼方程中的負號，讓此關係是非對稱的關係。
==參考資料==
*{{cite book |last1=Brown |first1=James Ward |last2=Churchill |first2=Ruel V. |title=Complex variables and applications |url=https://archive.org/details/complexvariables00brow_0 |year=1996 |publisher=McGraw-Hill |location=New York |isbn=0-07-912147-0 |edition=6th |page=[https://archive.org/details/complexvariables00brow_0/page/61 61] |quote=If two given functions ''u'' and ''v'' are harmonic in a domain ''D'' and their first-order partial derivatives satisfy the Cauchy-Riemann equations (2) throughout ''D'', ''v'' is said to be a ''harmonic conjugate'' of ''u''.}}

==外部連結==
* [http://www.cut-the-knot.org/pythagoras/HarmonicRatio.shtml Harmonic Ratio] {{Wayback|url=http://www.cut-the-knot.org/pythagoras/HarmonicRatio.shtml |date=20210426011432 }}
* {{springer|title=Conjugate harmonic functions|id=p/c025040}}

[[Category:调和函数]]
[[Category:偏微分方程]]