From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/holomorphic-function/index.md | 189 ++++++++++++++++++++++ 1 file changed, 189 insertions(+) create mode 100644 source/know/concept/holomorphic-function/index.md (limited to 'source/know/concept/holomorphic-function') diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md new file mode 100644 index 0000000..17bd5a6 --- /dev/null +++ b/source/know/concept/holomorphic-function/index.md @@ -0,0 +1,189 @@ +--- +title: "Holomorphic function" +date: 2021-02-25 +categories: +- Mathematics +- Complex analysis +layout: "concept" +--- + +In complex analysis, a complex function $f(z)$ of a complex variable $z$ +is called **holomorphic** or **analytic** if it is complex differentiable in the +neighbourhood of every point of its domain. +This is a very strong condition. + +As a result, holomorphic functions are infinitely differentiable and +equal their Taylor expansion at every point. In physicists' terms, +they are extremely "well-behaved" throughout their domain. + +More formally, a given function $f(z)$ is holomorphic in a certain region +if the following limit exists for all $z$ in that region, +and for all directions of $\Delta z$: + +$$\begin{aligned} + \boxed{ + f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} + } +\end{aligned}$$ + +We decompose $f$ into the real functions $u$ and $v$ of real variables $x$ and $y$: + +$$\begin{aligned} + f(z) = f(x + i y) = u(x, y) + i v(x, y) +\end{aligned}$$ + +Since we are free to choose the direction of $\Delta z$, we choose $\Delta x$ and $\Delta y$: + +$$\begin{aligned} + f'(z) + &= \lim_{\Delta x \to 0} \frac{f(z + \Delta x) - f(z)}{\Delta x} + = \pdv{u}{x} + i \pdv{v}{x} + \\ + &= \lim_{\Delta y \to 0} \frac{f(z + i \Delta y) - f(z)}{i \Delta y} + = \pdv{v}{y} - i \pdv{u}{y} +\end{aligned}$$ + +For $f(z)$ to be holomorphic, these two results must be equivalent. +Because $u$ and $v$ are real by definition, +we thus arrive at the **Cauchy-Riemann equations**: + +$$\begin{aligned} + \boxed{ + \pdv{u}{x} = \pdv{v}{y} + \qquad + \pdv{v}{x} = - \pdv{u}{y} + } +\end{aligned}$$ + +Therefore, a given function $f(z)$ is holomorphic if and only if its real +and imaginary parts satisfy these equations. This gives an idea of how +strict the criteria are to qualify as holomorphic. + + +## Integration formulas + +Holomorphic functions satisfy **Cauchy's integral theorem**, which states +that the integral of $f(z)$ over any closed curve $C$ in the complex plane is zero, +provided that $f(z)$ is holomorphic for all $z$ in the area enclosed by $C$: + +$$\begin{aligned} + \boxed{ + \oint_C f(z) \dd{z} = 0 + } +\end{aligned}$$ + +
+ + + +
+ +An interesting consequence is **Cauchy's integral formula**, which +states that the value of $f(z)$ at an arbitrary point $z_0$ is +determined by its values on an arbitrary contour $C$ around $z_0$: + +$$\begin{aligned} + \boxed{ + f(z_0) = \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z - z_0} \dd{z} + } +\end{aligned}$$ + +
+ + + +
+ +Similarly, **Cauchy's differentiation formula**, +or **Cauchy's integral formula for derivatives** +gives all derivatives of a holomorphic function as follows, +and also guarantees their existence: + +$$\begin{aligned} + \boxed{ + f^{(n)}(z_0) + = \frac{n!}{2 \pi i} \oint_C \frac{f(z)}{(z - z_0)^{n + 1}} \dd{z} + } +\end{aligned}$$ + +
+ + + +
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