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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-...
Dennis Marx's user avatar
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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 }} \...
Elliot's user avatar
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1 answer
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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 ...
Very Interesting's user avatar
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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=...
deomanu01's user avatar
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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!(...
Isllier's user avatar
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1 answer
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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}\...
J P's user avatar
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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: ...
Yvonne's user avatar
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2 answers
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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 ...
Anacardium's user avatar
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5 votes
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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{...
Conreu's user avatar
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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} \...
number8's user avatar
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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 \...
robert lewison's user avatar
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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 ...
user760's user avatar
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How to use Cauchy integral theorem to evaluate this real integral $\int_0^\infty x^\alpha e^{ix}dx$?

I recently learnt that to evaluate the integral $$\int_0^\infty x^\alpha e^{ix}dx\quad\text{with }-1<\alpha<0, $$ we have to apply Cauchy's Integral Theorem to deform the integral from the real ...
Chang's user avatar
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How to Show that Taylor Series of Holmorphic Function Centred at the Origin Converges Everywhere on the Open Unit Disk

I am interested in how one can show that the Taylor series of a holomorphic function defined within the open unit disk converges everywhere within the open unit disk. To put this more clearly, suppose ...
Liam Elias's user avatar
2 votes
2 answers
103 views

Find $\int_0^\pi \frac{8 \, d \theta}{5+2 \cos \theta}$

Find $\int_0^\pi \frac{8 \, d \theta}{5+2 \cos \theta}$ Let $z = e^{i \theta}$. Then $dz = \frac{d \theta}{iz}$ \begin{align} \begin{split} 8 \int_0^\pi \frac{\, d \theta}{5+2 \cos \theta}...
Grigor Hakobyan's user avatar
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3 answers
70 views

Evaluate $\int_0^{2 \pi} e^{\sin(e^{i \theta})} \hspace{0.1cm} d \theta$

Evaluate $\int_0^{2 \pi} e^{\sin(e^{i \theta})} \hspace{0.1cm} d \theta$ Here's what I have: \begin{align} \begin{split} \int_0^{2 \pi} e^{\sin(e^{i \theta})} \hspace{0.1cm} d \theta &...
Grigor Hakobyan's user avatar
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Intersection of Simple Closed Curves with finite point

Suppose $C_1$ and $C_2$ are two Rectifiable curve simple closed Curves, $C_1 \cap C_2$ ={$x_1,x_2,...,x_n$}, Are the intersection of the interiors of two curves be expressed as the union of the ...
wxw030910's user avatar
1 vote
2 answers
122 views

Let $\xi \in \mathbb{R}$. Use Cauchy's Theorem to prove that $\int_{-\infty}^{\infty}e^{-\pi x^2}e^{-2\pi ix \xi} dx = e^{-\pi \xi^2}$.

It wants us to prove that $\int_{-\infty}^{\infty}e^{-\pi x^2}e^{-2\pi ix \xi} dx = e^{-\pi \xi^2}$ by integrating $f(z) = e^{-\pi z^2}$ over the rectangle with vertices $\pm n, \pm n + i \xi$ and ...
robert lewison's user avatar
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Cauchy's Theorem Application Example

My professor went over an example in class and I am kind of confused. We are giving a proof of the theorem that $\forall a \gt 0, \int_{0}^{\infty} e^{-(a+ib)t} \, dt \, \colon= \lim_{R \to \infty} \...
robert lewison's user avatar
1 vote
1 answer
114 views

integration of$ f(z)=\frac{1}{z^4+1}$ using Cauchy's Integral Formula vs. Partial Fractions

I get different answers when I try to evaluate the following integral when I use partial fraction decomposition and Cauchy's Integral Formula. The integral is, $$\color{blue}{\int _{-2}^{2}\frac{1}{z^...
Travis Miller's user avatar
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37 views

Residue calculation for $z^{-2}\sin(z^{-2})$

I am trying to calculate the residue of $f(z) = \frac{1}{z^2}\sin\left(\frac{1}{z^2}\right)$ at $z=0$. Looking at the Taylor expansion of $f$ around $0$, as $c_{-n}$ (the coefficients of the powers of ...
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Combining Cauchy Integral Formula and Taylor Series

We know that if $f(z)$ is analytic at $a \in \mathbb{C}$, then by Taylor, $$f(z) =\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(z-a)^n$$ By Cauchy, we also know that $$f^{(n)}(a) = \frac{n!}{2 \pi i } \...
Grigor Hakobyan's user avatar
2 votes
0 answers
48 views

Would the following not be a correct proof of Cauchy's Integral Theorem?

I'm somewhat confused as to how the general version of Cauchy's Theorem does not follow (almost) immediately from its version in a disk. At least Ahlfors as well as Stewart & Tall prove the ...
Sam's user avatar
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1 vote
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Cauchy's integral theorem for rectifiable simple closed curve. [closed]

Γ is a rectifiable simple closed curve and Ω is its interior. f is holomorphic on Ω and continuous on Ω∪Γ. Prove the integral of f along Γ is 0. Since we already know the Cauchy's integral theorem for ...
xtwxtw's user avatar
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1 vote
2 answers
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Given f holomorphic on $B_1(0)$ and $max_{z \in C_r(0)} |f(z)| \to 0$ as $r \to 1$. Show f is identically 0.

I have tried to use cauchy integral formula and the deformation theorem but that got nowhere: I got $f(z_0) = max_{z \in C_r'(z_0)}|f(z)|$ where $r' > 0$ such that $B_r'(z_0) \subset B_1(0)$. Note: ...
Kh Nguyen's user avatar
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2 answers
71 views

How is Ahlfors applying Cauchy's Integral Formula when deriving Taylor's Theorem?

From Ahlfors' Complex Analysis (by 'analytic' he means 'complex-differentiable' or 'holomorphic'; he does not mean that the function can be expanded as a power series, as that is what he sets out to ...
Sam's user avatar
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1 vote
0 answers
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Cauchy Integral Formula and the Dirichlet Problem

Does the Cauchy Integral Formula by itself proves that the Dirichlet Problem is solvable for any simply connected region? I was looking for the proof of the Poisson Integral Formula and one uses the ...
underfilho's user avatar
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1 answer
83 views

Evaluate the integral $\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}dz$

Evaluate the integral $$\int_{\gamma} \frac{z^2+1}{(z+1)(z+4)}dz$$ if $\gamma = \beta + [4 \pi ,0]$ and $\beta(t) = te^{it}$ for $0 \le t \le 4 \pi$. My attempt By Cauchy's integral formula. \begin{...
Confused's user avatar
1 vote
0 answers
87 views

Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration. [duplicate]

Question: Understanding how to compute $\int_{0}^{+\infty} \frac{\sin t}{t}dt $ via complex integration. Let us first define $ I(r)=\int_{(|z|=r)} \frac{e^{iz}}{z}dz. $ (a) Show that $I(r) \to 0 $ as $...
Confused's user avatar
3 votes
1 answer
52 views

Understanding the proof of Theorem 10.1 in Montgomery & Vaughan's Multiplicative Number Theory

In the last step of the proof of Theorem 10.1 in the book Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan I couldn't understand what exactly "turn the ...
Ali's user avatar
  • 281
2 votes
1 answer
230 views

Mistakes in proving $\int_{\gamma}\frac{dz}{z-a}=2k\pi i$ in Ahlfors' Complex Analysis

Thanks again for Martin's new extension What a great extension! But I still need to check $\int_{\alpha}^{\beta}w(s)ds=\int_{\gamma}\frac{1}{z-a}dz$ . The integrands are different only at "...
studyhard's user avatar
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0 votes
1 answer
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An improper integral of an inverse of a square root of a higher degree polynomial.

Let $1 \le n_1 < n $ and $ n\ge 3$ be integers and let ${\bf \lambda}= \left( \lambda_j \right)_{j=1}^n \in {\mathbb R}$ such that $\lambda_j > 0 $ for $j=1,\cdots, n_1$ and $\lambda_j <0 $...
Przemo's user avatar
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1 vote
2 answers
137 views

Proof of Cauchy's theorem for punctured domains.

I am unable to prove the following lemma of Cauchy's integral theorem for simple closed curve. Lemma: Let $R$ be a simply connected region. If $f(z)$ be analytic in $R-\{a\}$ and is continuous on $R$ ...
General Mathematics's user avatar
1 vote
1 answer
52 views

expansion of real asymmetric integral on complex plane

I would like to solve the following integral for $f(x)$ $$\int_{0}^{\infty} \frac{1}{(a+ix)(b+ix)(c+ix)} dx $$ by expanding it to the complex and then using a contourlike the half circle, i.e. if $C$ ...
Jannis Erhard's user avatar
4 votes
1 answer
265 views

The proof about Cauchy's Integral Formula in Ahlfors' Complex Analysis

On the third edition of Ahlfors' Complex Analysis, page 122 Lemma 3 it states: Now, if we divide the identity by $z-z_0$ and let $z$ tend to $z_0$, the quotient in the first term tends to a derivative ...
studyhard's user avatar
  • 174
1 vote
1 answer
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Resolvent integral formula for defective matrices

As far as I understand, given a matrix $\mathbf M$, the resolvent integral $$ P = -\frac{1}{2i\pi}\oint_{\partial C} (\mathbf M-\lambda)^{-1} d\lambda,$$ should be equal to a projection on the sum of ...
Cyril Soler's user avatar
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0 answers
45 views

Integration using Cauchy's theorem vs numerical method

It is not a homework question. I just want to learn complex integrations. I want to evaluate the following integral $$ I = \int \frac{dE}{2\pi} E^2 \left(\frac{1}{(E-E_n+iη)^2 (E-E_m+iη) }-\frac{1}{(E-...
Luqman Saleem's user avatar
2 votes
1 answer
50 views

Small Question regarding contour integration of $\int_{1=|z-i|} \frac {1}{z^2+1}dz$

I am not certain if my Process of using Cauchy's Theorem is sound. $$ \int_{1=|z-i|} \frac {1}{z^2+1}dz = \int_{1=|z-i|} \frac {1}{(z-i)(z+i)}dz = \int_{1=|z-i|} \frac {\frac {1}{z-1}}{(z+i)}dz $$ ...
Pascal's user avatar
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3 votes
1 answer
297 views

Does Cauchy's integral formula generalize to non-analytic functions?

Cauchy's integral formula states that if the complex function $f(z)$ is analytic on a closed domain $D$ of the complex plane and $a$ is in the interior of $D$, then $$f(a) = \frac{1}{2 \pi i}\oint_{\...
tparker's user avatar
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0 answers
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Proving Integral of Analytic Function over Unit Circle Equals Zero

I'm trying to prove the following result: if f is an analytic function defined for all complex numbers, and f(0) = 0 , how can I show that the integral over the unit circle |z| = 1 of z^-n * g(z) ...
VALENTIN NEME GABRIEL's user avatar
1 vote
0 answers
62 views

Show that $\widehat {f}$ is holomorphic in each variable.

I am going through Theorem $1.1$ from the book Holomorphic Functions and Integral Representations in Several Complex Variables by Michael Range (Page no. $43$) which states the following $:$ Theorem $...
Anacardium's user avatar
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1 vote
1 answer
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Confusion in working out $\mathcal{P} \int_{-\infty}^{\infty} \frac{dz}{z-i}$.

I'm trying to work out $$I=\mathcal{P} \int_{-\infty}^{\infty} \frac{dz}{z-i}$$ where $\mathcal{P}$ denotes the fact we are taking the Cauchy principal value. Let $\gamma$ be the contour such that $\...
Robin's user avatar
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0 votes
1 answer
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Evaluating $\int_\gamma \frac{dz}{z^2 - \frac{1}{4}}$ using Cauchy's Integral Formula

Cauchy's Integral Formula states that for $a \in \mathbb{C}$, $r > 0$, if $f:B_r(a) \to \mathbb{C}$ is holomorphic, then for every $w \in B_r(a)$ such that $|w-a| < \rho < r$ we have $$ \int_{...
Ryderr's user avatar
  • 133
3 votes
1 answer
68 views

Solve Cauchy Integral using residues

I have an exam tomorrow, and we were given like an "example test" without answers, and one question is to solve this Cauchy integral: $$ \oint_c = \frac {2z+1} {(z+1)^2(z-3)} $$ with circle ...
Michal Gally's user avatar
-2 votes
1 answer
88 views

A complex integral using Cauchy's Formula

Let $m \in \mathbb{Z}$, compute $\oint_{|z|=1} \frac{e^z}{z^m}dz$. This exercise is supposed to be solve using Cauchy's Formula. The case when $m\leq 0$ could be solved by setting the auxiliar ...
clopenset's user avatar
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0 votes
1 answer
65 views

Calculating the integral of analytic functions inside and outside the unit circle

Calculate the integral: $$I = {1 \over {2\pi i}}\int_r {{{\omega (t) \cdot \overline {\varphi(t)} } \over {t - {z_0}}}} dt$$ where $r$ is a curve on the positive unit circle, ${{z_0}}$ is a point ...
Elliot's user avatar
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1 vote
1 answer
73 views

Cauchy Integral Theorem, Send Circle Radius to 0.

I can’t seem to wrap my head around this old qual problem, despite it seeming rather straightforward. We are given a function $f$ that is continuous in a neighborhood around a point a. We are asked to ...
Pancho_Squancho's user avatar

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