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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Is there a Cauchy integral formula on Riemann surface?
Is there a Cauchy integral formula on Riemann surfaces? It looks like the following formula (it may not be right).
(Cauchy integral formula) Let $M$ be a Riemann surface, $(U,\varphi)$ be a complex ...
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Cauchy integral formula for derivatives — general proof [duplicate]
The Cauchy Integral Formula for Derivatives states that if $f$ is analytic inside and on a simple closed contour $C$, and $z_0$ is a point inside $C$, then for any integer $n \ge 0$,
$$
f^{(n)}(z_0) = ...
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Cauchy Type integral along the real axis
I'm trying to calculate the following Cauchy type integral along the real axis:
$$\int_{-\infty}^{\infty}\frac{\cosh^{-1}\left( \frac{s+r}{a}\right) }{\sqrt{\left( a+r\right) ^{2}-s^{2}}\left( s-...
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Evalution of $I(a) = \operatorname{PV} \int_0^\infty \frac{x \sin bx \, \ln a/x}{a^2 - x^2} \, dx$
$$I(a) = \operatorname{PV} \int_0^\infty \frac{x \sin bx \, \ln a/x}{a^2 - x^2} \, dx$$
I could not find this exact form in the literature or on Math StackExchange, though I found some related ...
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Dispersion relations with modified correlators and multiple branch cuts
In particle physics, we often encounter correlators $F_q(q^2)$ which are functions of the squared momentum transfer $q^2$. These functions are real-valued for some $q^2$ below a threshold $M^2$, and ...
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Cauchy's integral formula and infinity point
I am experiencing a problem with applying Cauchy's integral formula for semi-infinite paths:
$$
f(w) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - w} dz.
$$
I am reading a paper where the following path ...
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Generalisation of Schwarz Integral formula to multiple dimensions
I am looking for $n$-dimensional generalisations of the boundary value solution given by the Schwarz formula. The Schwarz integral formula for 2D (the complex plane) states that:
Given a real valued ...
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Cauchy integral for $n\not\in\mathbb{N}$
Let me start by saying that this is a question of pure curiosity without much context behind it.
Context
The Cauchy integral is
$$f^{(n)}(z)=\frac{n!}{2\pi i}\oint_{\gamma}\frac{f(t)}{(t-z)^{n+1}}\...
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Relation between the residue of $\frac{f(z)}{z^{n+1}}$ at a high-order pole and the series expansion of $f(z)$
Suppose a meromorphic function f(z) at all points in $|z|\le R\space$ ,which is analytic at $z=0, z=R$ and have only one pole(a simple case) $\rho$ with the order of M. Then, by Cauchy's Residue & ...
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Using Cauchy's Integral formula to relate behaviour of $f$ and $f'$
I need to prove that for $1 \le p \le \infty$, $f$ holomorphic in $\mathbb{D}$ and $\beta > 0$:
$$
m_p(f,r) = O\left(\frac{1}{(1-r)^\beta}\right) \iff m_p(f',r) = O\left(\frac{1}{(1-r)^{\beta+1}}\...
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Show $\infty$ is a removable singularity of $f$, analytic in a punctured neighbourhood of $\infty$, given $z^{-1} \text{Re} f(z) \rightarrow 0$ [duplicate]
The following problem is from Ahlfors 4.6.4, Ex.7:
If $f$ is analytic in a neighbourhood of $\infty$ and if $z^{-1} \text{Re} f(z) \rightarrow 0$ when $z \rightarrow \infty$, show that $\lim_{z \...
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Why doesn’t Cauchy’s Theorem (Theorem 8.8 in Stewart's Complex Analysis) imply the boundary version (Theorem 9.6)?
I'm studying from Complex Analysis by Ian Stewart and David Tall, and I came across two theorems that appear closely related, yet the text does not present one as a corollary of the other.
Theorem 8....
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The Cauchy transform of an integrable function
I have some questions regarding the Cauchy transform of an integrable function $f\in L^{1}(C,ds)$ which is given by
$$ K(f)(z)=\frac{1}{2\pi i}\int_{C}\frac{f(\tau)}{\tau-z}d\tau,\quad z\in D,$$
where ...
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Cauchy type integral of Chebyshev weight of third kind $F(z)=\int_{-1}^1 \sqrt{\frac{1+t}{1-t}}\frac{dt}{z-t}$
Let $$F(z)=\int_{-1}^1 \sqrt{\frac{1+t}{1-t}}\frac{dt}{z-t},$$
where $\text{Im}(z)>0$. I know that result is $F(z)=-i\pi\sqrt{\frac{1+z}{1-z}}-\pi$, but don't know how to prove it. In Szego book &...
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The Riesz functional calculus
I'm study the Conway's book and I read Cauchy's theorem (below). I don't understand why in the theorem it states that $\sum_{j=1}^{m} n(\gamma_j;a)=0$. I think it is because when point $a$ is not ...
