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\PMlinkescapeword{iff} |
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A complex function \,$f\!: D \to \mathbb{C}$,\, where $D$ is a domain of the complex plane, having the derivative
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A complex function \,$f\!: D \to \mathbb{C}$,\, where $D$ is a domain of the complex plane, is {\em antiholomorphic} in $D$ if the derivative
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| $$\frac{df}{d \overline{z}}$$ |
$$\frac{df}{d \overline{z}}$$ |
| in each point $z$ of $D$, is said to be {\em antiholomorphic} in $D$.\\ |
exists in each point $z$ of $D$.\, The real part \,$u(x,\,y)$\, and the imaginary part \,$v(x,\,y)$\, of an antiholomorphic function $f$ satisfy the equations |
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| The following conditions are \PMlinkname{equivalent}{Equivalent3}: |
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| \begin{itemize} |
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| \item $f(z)$ is antiholomorphic in $D$. |
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| \item \, $\overline{f(z)}$\, is holomorphic in $D$. |
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| \item $f(\overline{z})$ is holomorphic in\, $\overline{D} \,:=\, \{\overline{z}\;\vdots\;\, z \in D\}$. |
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| \item $f(z)$ may be \PMlinkescapetext{expanded} to a power series $\sum_{n=0}^\infty a_n(\overline{z}-u)^n$ at each\, $u \in D$. |
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| \item The real part \,$u(x,\,y)$\, and the imaginary part \,$v(x,\,y)$\, of the function $f$ satisfy the equations |
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| $$\frac{\partial u}{\partial x} \;=\; -\frac{\partial v}{\partial y}, \qquad |
$$\frac{\partial u}{\partial x} \;=\; -\frac{\partial v}{\partial y}, \qquad |
| \frac{\partial u}{\partial y} \;=\; \frac{\partial v}{\partial x}.$$ |
\frac{\partial u}{\partial y} \;=\; \frac{\partial v}{\partial x}.$$ |
| N.B. the \PMlinkescapetext{place} of minus; cf. the \PMlinkname{Cauchy--Riemann equations}{CauchyRiemannEquations}.\\ |
N.B. the \PMlinkescapetext{place} of minus; cf. the \PMlinkname{Cauchy--Riemann equations}{CauchyRiemannEquations}.\\ |
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| \end{itemize} |
The function $f(z)$ is antiholomorphic in $D$ iff\, $\overline{f(z)}$\, is holomorphic in $D$ iff $f(\overline{z})$ is holomorphic in\, $\overline{D} := \{\overline{z}\,\vdots\;\, z \in D\}$.\\ |
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| \textbf{Example.}\, The function\, $\displaystyle z \mapsto \frac{1}{\overline{z}}$ is antiholomorphic in\, |
\textbf{Example.}\, The function\, $\displaystyle z \mapsto \frac{1}{\overline{z}}$ is antiholomorphic in\, |
| $\mathbb{C}\!\smallsetminus\!\{0\}$.\, One has |
$\mathbb{C}\!\smallsetminus\!\{0\}$. |
| $$f(z) \,=\, \frac{z}{|z|^2} \,=\, \underbrace{\frac{x}{x^2+y^2}}_{u}+i\underbrace{\frac{y}{x^2+y^2}}_{v},$$ |
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| $$\frac{\partial u}{\partial x} \;=\; \frac{y^2\!-\!x^2}{(x^2\!+\!y^2)^2}, \quad |
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| \frac{\partial v}{\partial y} \;=\; \frac{x^2\!-\!y^2}{(x^2\!+\!y^2)^2}, \quad |
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| \frac{\partial u}{\partial y} \;=\; -\frac{2xy}{(x^2\!+\!y^2)^2}, \quad |
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| \frac{\partial v}{\partial x} \;=\; -\frac{2xy}{(x^2\!+\!y^2)^2}.$$ |
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