Harmonic conjugate

In mathematics, a function $u(x,\,y)$ defined on some open domain $\Omega\subset\R^2$ is said to have as a conjugate a function $v(x,\,y)$ if and only if they are respectively real and imaginary parts of a holomorphic function $f(z)$ of the complex variable $z:=x+iy\in\Omega.$ That is, $v$ is conjugate to $u$ if $f(z):=u(x,y)+iv(x,y)$ is holomorphic on $\Omega.$ As a first consequence of the definition, they are both harmonic real-valued functions on $\Omega$ . Moreover, the conjugate of $u,$ if it exists, is unique up to an additive constant. Also, $u$ is conjugate to $v$ if and only if $v$ is conjugate to $-u$ .

单词	Harmonic conjugate
释义	Harmonic conjugate 英语百科 Harmonic conjugate In mathematics, a function $u(x,\,y)$ defined on some open domain $\Omega\subset\R^2$ is said to have as a conjugate a function $v(x,\,y)$ if and only if they are respectively real and imaginary parts of a holomorphic function $f(z)$ of the complex variable $z:=x+iy\in\Omega.$ That is, $v$ is conjugate to $u$ if $f(z):=u(x,y)+iv(x,y)$ is holomorphic on $\Omega.$ As a first consequence of the definition, they are both harmonic real-valued functions on $\Omega$ . Moreover, the conjugate of $u,$ if it exists, is unique up to an additive constant. Also, $u$ is conjugate to $v$ if and only if $v$ is conjugate to $-u$ .
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