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author | Prefetch | 2022-10-14 23:25:28 +0200 |
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committer | Prefetch | 2022-10-14 23:25:28 +0200 |
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diff --git a/source/know/concept/residue-theorem/index.md b/source/know/concept/residue-theorem/index.md new file mode 100644 index 0000000..1880ade --- /dev/null +++ b/source/know/concept/residue-theorem/index.md @@ -0,0 +1,71 @@ +--- +title: "Residue theorem" +date: 2021-11-13 +categories: +- Mathematics +- Complex analysis +layout: "concept" +--- + +A function $f(z)$ is **meromorphic** if it is +[holomorphic](/know/concept/holomorphic-function/) +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 $z_p$. +In other words, $f(z)$ can be approximated close to $z_p$: + +$$\begin{aligned} + f(z) + \approx \frac{R_p}{z - z_p} +\end{aligned}$$ + +Where 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** for meromorphic functions +is a generalization of Cauchy's integral theorem for holomorphic functions, +and states that the integral on a contour $C$ +purely depends on the simple poles $z_p$ enclosed by $C$: + +$$\begin{aligned} + \boxed{ + \oint_C f(z) \dd{z} = i 2 \pi \sum_{z_p} R_p + } +\end{aligned}$$ + +<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 $z_p$ are all its poles: + +$$\begin{aligned} + f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p} +\end{aligned}$$ + +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, by cleverly choosing the contour $C$, +it lets us 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). |