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author | Prefetch | 2022-12-20 20:11:25 +0100 |
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committer | Prefetch | 2022-12-20 20:11:25 +0100 |
commit | 1d700ab734aa9b6711eb31796beb25cb7659d8e0 (patch) | |
tree | efdd26b83be1d350d7c6c01baef11a54fa2c5b36 /source/know/concept/residue-theorem | |
parent | a39bb3b8aab1aeb4fceaedc54c756703819776c3 (diff) |
More improvements to knowledge base
Diffstat (limited to 'source/know/concept/residue-theorem')
-rw-r--r-- | source/know/concept/residue-theorem/index.md | 11 |
1 files changed, 7 insertions, 4 deletions
diff --git a/source/know/concept/residue-theorem/index.md b/source/know/concept/residue-theorem/index.md index a0f515e..49a6f7a 100644 --- a/source/know/concept/residue-theorem/index.md +++ b/source/know/concept/residue-theorem/index.md @@ -12,7 +12,7 @@ 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 +but where the product $$(z - z_p) f(z)$$ is nonzero and still holomorphic close to $$z_p$$. In other words, $$f(z)$$ can be approximated close to $$z_p$$: @@ -26,7 +26,8 @@ represents the rate at which $$f(z)$$ diverges close to $$z_p$$: $$\begin{aligned} \boxed{ - R_p = \lim_{z \to z_p} (z - z_p) f(z) + R_p + \equiv \lim_{z \to z_p} (z - z_p) f(z) } \end{aligned}$$ @@ -37,7 +38,8 @@ purely depends on the simple poles $$z_p$$ enclosed by $$C$$: $$\begin{aligned} \boxed{ - \oint_C f(z) \dd{z} = i 2 \pi \sum_{z_p} R_p + \oint_C f(z) \dd{z} + = i 2 \pi \sum_{z_p} R_p } \end{aligned}$$ @@ -48,7 +50,8 @@ we know that we can decompose $$f(z)$$ like so, where $$h(z)$$ is holomorphic and $$z_p$$ are all its poles: $$\begin{aligned} - f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p} + f(z) + = h(z) + \sum_{z_p} \frac{R_p}{z - z_p} \end{aligned}$$ We integrate this over a contour $$C$$ which contains all poles, and apply |