Questions tagged [cauchy-integral-formula]
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration"
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Complex integrals that look like they agree, differ by sign (according to Mathematica)
Consider the integral
$$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$
I would assume it to agree with the integral
$$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$
However, according to Mathematica the ...
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Proof of zeta functional equation in Edwards
I'm trying to understand the first proof of the zeta functional equation given in Riemann's Zeta Function by H M Edwards. Referring to the excerpt from page 13 below, I'm stuck on how he derives the ...
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Application of Cauchy integral formula / residue theorem for evaluation of real-valued integrals
I try to evaluate integrals of the form
\begin{equation}
I(x) = \int_{0}^{x}\frac{f(t)}{t^{2}}\,\mathrm{d}t
\end{equation}
with $f(t)$ being a differentiable and real-valued function of the real-...
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Using the Cauchy Integral Theorem to Solve a Contour Integral
Calculate the contour integral of the given function on the unit circle:
$$f\left( t \right) = 2\left( {t + {1 \over t }} \right) - t \sqrt {{1 \over {{t ^2}}}{{\left( {2\left( {t + {1 \over t }} \...
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Applying the generalized version of Cauchy's Integral Formula
I am having some issues with the following exercise:
Let $\Gamma$ be a chain in $G=\mathbb{C}^*$ , $f$ be a function that is holomorphic in $G$ and bounded on $\mathbb{C} \setminus K_1(0)$. Show that
...
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How to expand a complex function around the point at infinity?
I came across a problem that asked to expand the function
$$
f(z) = \frac{1-e^{2iz}}{z^2}
$$
both around the point $z=0$ and $z=\infty$. The correct expansion around $z=0$ should be
$$
f(z) = -\sum_{k=...
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Estimate $f^{(n)}(a)$ with L1 norm for f holomorphic
$\Omega\subset \mathbb{C}$ is a region, $\overline{B(a,r)}=\{z\in\mathbb{C}||z-a|\leqslant r\}\subset\Omega$ $(r>0)$. Suppose $f\in H(\Omega)$, prove the inequality:
$$|f^{(n)}(a)|\leqslant\frac{n!(...
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Complex integral $\frac{\sin(z)}{z(z+1)^3}$
I have the following integral $\int_{|z|=2}\frac{\sin(z)}{z(z+1)^3}$ taking $f(z)=\frac{\sin(z)}{z}$ and considering the analytic continuation $F$ at $z=0$ with $F(0)=1$
, I'd obtain:
$$ \int_{|z|=2}\...
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Integral over the real line of a function with a second-order pole $\int_{-\infty}^\infty \frac{e^{-(A\omega+iB)^2+C}}{\omega^2} d\omega$
I am trying to solve an integral of the form
\begin{equation}
\int_{-\infty}^\infty \frac{e^{-(A\omega+iB)^2+C}}{\omega^2} d\omega,
\end{equation}
where $A,B,C\in \mathbb{R}$, $A>0$.
Attempt 1: ...
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Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property. [duplicate]
Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
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Cycles around compacta and the global Cauchy theorem
Recall the global Cauchy theorem/formula:
Let $U$ be open and $\Gamma\subset U$ a cycle. If $\Gamma$ is homologous to zero in $U$, then for all $f:U\to\Bbb C$ holomorphic and $w\in U\setminus\Gamma$ ...
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How to prove $\displaystyle f(n,k;p)=\frac{1}{2\pi i}\oint_{|z|=1}\dfrac{(pz+1-p)^n}{z^{k+1}}\,\mathrm dz={n\choose k}p^k(1-p)^{n-k}$?
How do you prove that
$$f(n,k;p)=\frac{1}{2\pi i}\oint_{|z|=1}\dfrac{(pz+1-p)^n}{z^{k+1}}\,\mathrm dz={n\choose k}p^k(1-p)^{n-k}$$
if $n\in\mathbf Z^+$, $k=0,1,...,n$ and $0<p<1$?
$\mathbf{...
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Calculate $\oint_{S} \frac{e^{\pi z}}{4z^3 + z} dz$ where $S = [2, 2i,−2,−2i, 2]$
Calculate $\oint_{S} \frac{e^{\pi z}}{4z^3 + z} dz$ where $S = [2, 2i,−2,−2i, 2]$
Using Partial fractions, the integral can be wrote as:
$$\oint_{\gamma_1} \frac{e^{\pi z}}{4z} dz - \oint_{\gamma_2} \...
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Trying to calculate $\int_{\vert z \vert = 2} \frac z{\cos z}dz$ [duplicate]
Trying to calculate $$\int\limits_{\vert z \vert = 2} \frac{z}{\cos z}dz$$ but running into a lot of issues. I decided to try and use $$\int\limits_{\vert z \vert = 2} \frac{z}{\cos{z}}dz = 2 \pi i \...
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Proof of Cauchy Integral formulas for the derivatives and taking differetiation under the integral sign.
I'm currently reviewing complex analysis, and reading two texts atm.
In Stein&Shakarchi, their proof of Corollary 4.2 is a quite straightforward computation. But they don't apply any lemma/theorem ...
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