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diff --git a/content/know/concept/holomorphic-function/index.pdc b/content/know/concept/holomorphic-function/index.pdc index 4b7221c..3e3984a 100644 --- a/content/know/concept/holomorphic-function/index.pdc +++ b/content/know/concept/holomorphic-function/index.pdc @@ -193,60 +193,3 @@ this proof works inductively for all higher orders $n$. </div> </div> - -## Residue theorem - -A function $f(z)$ is **meromorphic** if it is holomorphic except in -a finite number of **simple poles**, which are points $z_p$ where -$f(z_p)$ diverges, but where the product $(z - z_p) f(z)$ is non-zero and -still holomorphic close to $z_p$. - -The **residue** $R_p$ of a simple pole $z_p$ is defined as follows, and -represents the rate at which $f(z)$ diverges close to $z_p$: - -$$\begin{aligned} - \boxed{ - R_p = \lim_{z \to z_p} (z - z_p) f(z) - } -\end{aligned}$$ - -**Cauchy's residue theorem** generalizes Cauchy's integral theorem -to meromorphic functions, and states that the integral of a contour $C$ -depends on the simple poles $p$ it encloses: - -$$\begin{aligned} - \boxed{ - \oint_C f(z) \dd{z} = i 2 \pi \sum_{p} R_p - } -\end{aligned}$$ - -<div class="accordion"> -<input type="checkbox" id="proof-res-theorem"/> -<label for="proof-res-theorem">Proof</label> -<div class="hidden"> -<label for="proof-res-theorem">Proof.</label> -From the definition of a meromorphic function, -we know that we can decompose $f(z)$ like so, -where $h(z)$ is holomorphic and $p$ are all its poles: - -$$\begin{aligned} - f(z) = h(z) + \sum_{p} \frac{R_p}{z - z_p} -\end{aligned}$$ - -We integrate this over a contour $C$ which contains all poles, and apply -both Cauchy's integral theorem and Cauchy's integral formula to get: - -$$\begin{aligned} - \oint_C f(z) \dd{z} - &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z} - = \sum_{p} R_p \: 2 \pi i -\end{aligned}$$ -</div> -</div> - -This theorem might not seem very useful, -but in fact, thanks to some clever mathematical magic, -it allows us to evaluate many integrals along the real axis, -most notably [Fourier transforms](/know/concept/fourier-transform/). -It can also be used to derive the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations). - |