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author | Prefetch | 2022-10-27 20:40:09 +0200 |
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committer | Prefetch | 2022-10-27 20:40:09 +0200 |
commit | 6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch) | |
tree | a8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/holomorphic-function | |
parent | bcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (diff) |
Optimize last images, add proof template, improve CSS
Diffstat (limited to 'source/know/concept/holomorphic-function')
-rw-r--r-- | source/know/concept/holomorphic-function/index.md | 32 |
1 files changed, 11 insertions, 21 deletions
diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md index 5dde240..cf252c0 100644 --- a/source/know/concept/holomorphic-function/index.md +++ b/source/know/concept/holomorphic-function/index.md @@ -61,6 +61,7 @@ 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 @@ -73,11 +74,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-int-theorem"/> -<label for="proof-int-theorem">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-int-theorem">Proof.</label> + +{% include proof/start.html id="proof-int-theorem" -%} Just like before, we decompose $$f(z)$$ into its real and imaginary parts: $$\begin{aligned} @@ -97,8 +95,8 @@ $$\begin{aligned} Since $$f(z)$$ is holomorphic, $$u$$ and $$v$$ satisfy the Cauchy-Riemann equations, such that the integrands disappear and the final result is zero. -</div> -</div> +{% include proof/end.html id="proof-int-theorem" %} + An interesting consequence is **Cauchy's integral formula**, which states that the value of $$f(z)$$ at an arbitrary point $$z_0$$ is @@ -110,11 +108,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-int-formula"/> -<label for="proof-int-formula">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-int-formula">Proof.</label> + +{% include proof/start.html id="proof-int-formula" -%} Thanks to the integral theorem, we know that the shape and size of $$C$$ is irrelevant. Therefore we choose it to be a circle with radius $$r$$, such that the integration variable becomes $$z = z_0 + r e^{i \theta}$$. Then @@ -133,9 +128,8 @@ $$\begin{aligned} &= \frac{f(z_0)}{2 \pi} \int_0^{2 \pi} \dd{\theta} = f(z_0) \end{aligned}$$ +{% include proof/end.html id="proof-int-formula" %} -</div> -</div> Similarly, **Cauchy's differentiation formula**, or **Cauchy's integral formula for derivatives** @@ -149,11 +143,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-diff-formula"/> -<label for="proof-diff-formula">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-diff-formula">Proof.</label> + +{% include proof/start.html id="proof-dv-formula" -%} By definition, the first derivative $$f'(z)$$ of a holomorphic function exists and is: @@ -186,6 +177,5 @@ $$\begin{aligned} Since the second-order derivative $$f''(z)$$ is simply the derivative of $$f'(z)$$, this proof works inductively for all higher orders $$n$$. -</div> -</div> +{% include proof/end.html id="proof-dv-formula" %} |