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Diffstat (limited to 'source/know/concept/residue-theorem/index.md')
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1 files changed, 14 insertions, 13 deletions
diff --git a/source/know/concept/residue-theorem/index.md b/source/know/concept/residue-theorem/index.md index bcbd6bd..b58e3c2 100644 --- a/source/know/concept/residue-theorem/index.md +++ b/source/know/concept/residue-theorem/index.md @@ -8,21 +8,21 @@ categories: layout: "concept" --- -A function $f(z)$ is **meromorphic** if it is +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 $z_p$: +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 $z_p$: +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 $$z_p$$: $$\begin{aligned} \boxed{ @@ -32,8 +32,8 @@ $$\begin{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 $C$: +and states that the integral on a contour $$C$$ +purely depends on the simple poles $$z_p$$ enclosed by $$C$$: $$\begin{aligned} \boxed{ @@ -47,14 +47,14 @@ $$\begin{aligned} <div class="hidden" markdown="1"> <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: +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 +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: $$\begin{aligned} @@ -62,11 +62,12 @@ $$\begin{aligned} &= \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}$$ + </div> </div> This theorem might not seem very useful, -but in fact, by cleverly choosing the contour $C$, +but in fact, by cleverly choosing the contour $$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). |