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"

Filter by
Sorted by
Tagged with
2
votes
0answers
42 views

Show $\frac{2}{\pi} \int_{0}^{2\pi} f(e^{i\theta})\cos^2(\frac{\theta}{2})d\theta = 2f(0)+f'(0)$ when $f\in A(|z|<1+\varepsilon)$. [duplicate]

Saw this question in a test, but got stuck, here is my attempt: \begin{align} \frac{2}{\pi} \int_{0}^{2\pi} f(e^{i\theta})\cos^2\bigg(\frac{\theta}{2}\bigg)\,d\theta &= \frac{2}{\pi} \int_{0}^{2\...
1
vote
0answers
29 views

Derivation of the retarded Green function: Problem with a Cauchy integral

I'm currently following a derivation of the retarded Greens function in my Electrodynamics course. We arrived at a integral of the following form: $$\int_{-\infty}^{\infty}\mathrm d\omega\frac{1}{c^2k^...
1
vote
1answer
23 views

Cauchy's inequalities in horizontal strip

Let $f:B\to\mathbb{C}$ a holomorphic function in the horizontal strip $B:=\{z\in \mathbb{C} : -1 < \mathrm{Im}\:{z} < 1\}$. Assume that we have $\beta > 0$ such that for every $z \in B$, it ...
0
votes
2answers
29 views

Cauchy's integral Theorem for functions with removable singularities

If a function $f$ has a removable singularity at $z_0$ then there exists an analytic function g such that $f(z)=g(z)$ for all z in some deleted neighborhood of $z_0$. My question is: if $g$ is ...
0
votes
0answers
36 views

A general Cauchy integral formula for open sets with holes

Suppose $f \in H(\Omega_{2})$ is a holomorphic functions defined on the open set $\Omega_{2} \subseteq \mathbb{C}$ which possibly has infinitely many holes. Furthermore, assume $\Omega_{1} \subsetneq \...
4
votes
1answer
95 views

When can functions of a complex variable be integrated 'normally'?

In our complex functions lecture notes, the lecturer integrates the complex function $f(z) = \frac{1}{z}$ around a circle of abitrary radius (i.e. over $z = r e^{i \theta}$). He first does a normal ...
0
votes
0answers
20 views

Use of Stone-Weierstrass Theorem for $1/z$ on complex circle

Let $C:=\{x\in\mathbb{C}:|x|=1\}$ be the complex circle and $f:C\rightarrow\mathbb{C}$ the mapping $x\mapsto 1/x$. From the Stone-Weierstrass theorem I get $\oint fdx =0$, but by Cauchy's theorem $\...
1
vote
1answer
58 views

Finding the coefficient of the Laurent Series using the integral formula

This question has been asked multiple times, but I can't seem to find one that does it using the formula. My book skips the steps and says that you can find the coefficients with the formula, but I am ...
0
votes
0answers
43 views

Proving integral with Poisson kernel

I am trying to prove that for a holomorphic function $f$ on the disk around $0$ with radius $r>1$ the following identity holds for any $z$ in the unit disk: $$f(z)=\frac{1}{2\pi}\int_{0}^{2\pi} \...
1
vote
1answer
58 views

Prove $\int_{|z|=1}\frac{P_n(z)}{z^{n+1}(z-a)}dz=0$

I've been solving problems from my Complex Analysis course, and I want to make sure if what I think what I think may be the path to solution is correct. The problem says: Prove that $$\int_{|z|=r}\...
-1
votes
2answers
63 views

Compute $\int_{\{|z|=4\}}\frac{z^2+2}{(z-3)^2}dz$ [duplicate]

I want to compute the value of the complex line integral $$\int_{\{|z|=4\}}\frac{z^2+2}{(z-3)^2}dz$$ I don't think I can apply Cauchy's integral theorem nor Cauchy's integral formula here. So far, I ...
0
votes
2answers
73 views

Study the improper integral for convergence and absolute convergence [closed]

$$\int_1^\infty \cos( x^2\ln(x) )\ dx $$ here is an integration which I have to study for convergence and absolute convergence. I cannot find f(x) and g(x) to use Dirichlet test. (Hint: Use Dirichlet ...
1
vote
2answers
44 views

Complex line integral: $\int_{|z|=1}\frac{z^2+2}{z(z-3)}dz$

I am trying to compute the line integral $$\int_{|z|=1}\frac{z^2+2}{z(z-3)}dz$$ where the circumference $\{|z|=1\}$ is positively orientated. I have used two different methods and I have obtained two ...
-1
votes
2answers
44 views

Cauchy's differentiation formula consequences

I have just started complex analysis. I understand Cauchy's integral formula and its differentiation formula, but I do not understand this consequence my professor listed: "The Cauchy ...
2
votes
1answer
79 views

Solve the integral $\int^{+\infty}_0\frac{x\sin x}{1+x^4}dx$ using complex analysis

I am trying to solve the following integral using only complex analysis: $$\int^{+\infty}_0\frac{x\sin x}{1+x^4}dx$$ So, as the function $f(x)$ inside the integral is even, the integral can be ...
0
votes
0answers
30 views

Applying Cauchy-Goursat Theorem

Apply the Cauchy–Goursat theorem to show that $\int_{C}^{}f(z)dz=0$ when the contour C is the unit circle $\left | z \right |=1$ , in either direction, and when $f(z)=sechz$. here is solution: Note ...
1
vote
1answer
45 views

Bounded holomorphic function over unit disk uniquely determines a bounded function on the boundary by taking the limit

Now given $f \in H^\infty(D)$, I want to show by functional analysis that there exists $f^\star \in L^\infty(\partial D)$ such that $f$ is determined by $f^\star$ via cauchy integral over the boundary ...
11
votes
1answer
188 views

$\int\limits_{-\infty}^\infty \left(f_T(\frac{x-\mu}{1+\Psi/2})-f_T(\frac{x+\mu}{1-\Psi/2})\right)\frac{x\gamma}{(x-x_{0})^{2}+\gamma^2/4}dx$

EDIT: I realized from numerical implementation that the step from \begin{align} \mathcal{I}_2=&\frac{\gamma}{2}\int\limits_{-\infty}^\infty \left(f_T(\frac{x-\mu}{1+\Psi/2})-f_T(\frac{x+\mu}{1-\...
2
votes
1answer
36 views

In the proof of Cauchy Integral Theorem, transforming $dz$ to polar coordinates drops $dr$ term, why?

The proof on Wikipedia and the textbook both parameterize $z=x+iy$ in polar coordinates around the singularity as, $z=z_0 + re^{i\theta}$, and so I believe that $dz=e^{i\theta}dr + ire^{i\theta}d\...
3
votes
0answers
41 views

Cauchy Integral Question

Evaluate $$\int_{\gamma}\frac{z^2+z^7}{(z-\frac{e}{3})^3}$$ here $\gamma$ is a rectangular path traversed in the counter clockwise direction with vertices $1,2i,−1,−2i$. So my thoughts for this ...
0
votes
0answers
5 views

How to find multiple connected domain using Cauchy integral formula?

According to the theorem as we know Cauchy integral says the summation of all integral is equal to zero . So now if i use the concept then i get : $2+3+x=0$ $x=-5$ where x is $\int_{c}f(z)dz$ . I ...
1
vote
3answers
47 views

Complex Analysis: Show that $\int_{C_\epsilon} \frac{1}{(z-x_0)^{2n+1}}dz = 0$

We have to show that: $$\int_{C_\epsilon} \frac{1}{(z-x_0)^{2n+1}}dz = 0$$ Where $x_0 \in \mathbb{R}$, $n$ is a positive integer and $C_\epsilon$ is the path that follows counterclockwise, in the ...
0
votes
1answer
23 views

Cauchy integral formula for a point outside the simple closed curve

Show that if $f$ is analytic inside and on a simple closed curve $C$ and $z_0$ not in $C$ then $$ (n-1)!\int_C \frac{f^{(m)}(z)}{(z-z_0)^n}dz=(m+n-1)!\int_C \frac{f(z)}{(z-z_0)^{m+n}}dz$$ Since it is ...
0
votes
1answer
29 views

Numerical integration of an integral with singularity

I am trying to solve this integral numerically using Mathematica. Here is my integral $$\int_0^{\infty} dx\;\frac{\Gamma(\delta-4ix)}{(i(x-1)+\epsilon)^{1-4ix}}\;, $$ where $0<\delta,\ll 1$ and $0&...
2
votes
1answer
49 views

Possible proof of Cauchy's Integral Formula for derivatives - completion and verification

First, let me state Cauchy's Integral formula: Let $U$ be an open region in the complex plane and $D = \{z : |z-z_0| \leq R\}$ a disk in $U$. If $f : U \to \mathbb C$ is holomorphic and $\gamma$ is ...
0
votes
0answers
72 views

Find $\int_{-\infty}^{\infty}\frac{sin^2(t)}{t^2}dt$ [duplicate]

Let $$f(z) = \frac{1-e^{2iz}}{z^2}$$ and let R > r > $0$ Find $$\int_{-\infty}^{\infty}\frac{sin^2(t)}{t^2}dt$$ using $$\int_{\gamma}f(z)dz = 0$$ where $\gamma=L(-R,-r)\oplus - S(0,r)\oplus L(r,...
0
votes
1answer
87 views

Find the integral $\int_{c(0,1)}\frac{c_0+c_1z+…+c_{2n}z^{2n}}{z^{n+1}}dz$

By considering the case when p = −1 separately find $$\int_{c(0,1)}z^p dz$$ $c(0,1)$ means a circle around the origin of radius 1 where p ∈ $\mathbb{Z}$ and hence if n ∈ $\mathbb{N}$ ∪ {0} and $c_0,...
0
votes
1answer
46 views

Attitude for solving $\int_{|z+1|=1} \frac{1}{z^{3}-i} d z$

I am asked to calculate $$\int_{|z+1|=1} \frac{1}{z^{3}-i} d z$$ I was thinking to apply the Cauchy integral theorem but I am not sure how to express the set $|z+1|=1$ as a boundary of a disk in $\...
0
votes
1answer
56 views

Apply Cauchy integral formula to product

Given that $$ \frac{d^nf(z)}{dz^n} \bigg|_{z=z_0}=\frac{n!}{2\pi i}\oint \frac{f(z)}{(z-z_0)^{n+1}}dz $$ Can we apply the same formula to a product of the form $g(z) = z^mf(z)$? For instance $$ \frac{...
0
votes
1answer
48 views

Does the analytic continuation of $\zeta(s)$ for $s\in\mathbb{C} \ \backslash \ \{1\}$ require Cauchy's integral formula?

I've seen Cauchy's integral formula used in proofs for the analytic continuation of the Riemann zeta function to the entire complex plane ($s\neq1$). My question is whether a counterexample can be ...
0
votes
1answer
38 views

Cauchy residue formula question

The Cauchy formula can be used to find the $n$th derivative of an analytical function $f(z)$ as $$ \frac{d^nf}{dz^n}\bigg|_{z=z_0}=\frac{n!}{2\pi i }\oint \frac{f(z)}{(z-z_0)^{n+1}}dz $$ which happens ...
0
votes
1answer
31 views

how to show this integral is $\in \mathbb{R}$ (attempt added)

Let $f: \Omega \rightarrow \mathbb{C}$ holomorphic such that $\Omega \subseteq \mathbb{C}$. Let $z_0 \in \Omega$ such that $\overline{D_{R}}\left(z_{0}\right) \subseteq \Omega$. Given that for every ...
1
vote
1answer
53 views

Derivative question and nested Cauchy formula

Consider $F(z)=\sum_i a_iz^i$ to be a formal power series with coefficients $a_i$. It is known that the coefficients of the series can be recovered from the $n$th terms of the associated Taylor series ...
2
votes
1answer
52 views

Find $\int_{|z|=1} f(z) d z$, when $f(z)=(z \sin z) /(z+2)+\bar{z}$

Find $\int_{|z|=1} f(z) d z$, when $f(z)=(z \sin z) /(z+2)+\bar{z}$ $f(z)$ looks like analytic function. If it is analytic, then by Cauchy-Theorem, integration will be zero. I tried to show Cauchy-...
0
votes
0answers
24 views

1) $\oint_C \frac{\cos(z)}{z} dz$, 2) $\oint_C \frac{\sin(z)}{z} dz$ and 3 ) $\oint_C \frac{\cos^2(z)}{z} dz$ where C is a unit circle.

first -> Sorry for my bad english Can I just use cauchy theorem? $\oint_C \frac{f(z)}{z - z_0} = 2i\pi f(z_0)$ So: 1) $f(z) = \cos(z)$ and $z_0 = 0 \Rightarrow$ $f(z_0) = 1.$ so $\oint_C \frac{\...
-1
votes
2answers
43 views

Calculate using Cauchy formula $\oint\limits_{\gamma} \frac{d z}{z^{2}+4}$, where $\gamma = |z-2i|=1$ – a closed loop counterclockwise. [closed]

How can I calculate that using Cauchy formula $\oint\limits_{\gamma} \frac{d z}{z^{2}+4}$, where $\gamma = |z-2i|=1$ – a closed loop counterclockwise? However, for Cauchy formula I need the ...
1
vote
1answer
49 views

Show that there is a strict inequality $|\int_{|z|=R} \frac{z^n}{z^m-1}| < \frac{2\pi R^{n+1}}{R^m-1}$

I need to prove this strict inequality holds. $\Big \lvert \int_{|z|=R} \frac{z^n}{z^m-1} \Big \rvert < \frac{2\pi R^{n+1}}{R^m-1}$ where $R>1, m \geq 1, n \geq 0$ Letting $f(z)= \frac{z^n}{z^m-...
0
votes
1answer
30 views

A condition of linearity for a holomorphic function [duplicate]

Consider the following exercise from the book Complex Analysis Stein and Shakarchi I already proved that $\varphi(z)=z+a_nz^n+O(z^{n+1})$ and $\varphi_k(z)=z+ka_nz^n+O(z^{n+1})$, but I do not see how ...
0
votes
0answers
33 views

holomorphic function on unit disk having zero of order $n$ at $0$

Assume that $f: \mathbb{D} \rightarrow \mathbb{D}$ is holomorphic function with a zero at $0$ of order $n$, where $n \geq 2$. What can we say about $f^{(n)}(0)$? and what can we say about $f^{(n)}(0)$...
1
vote
1answer
38 views

two disjoint compact sets, holomorphic function there exists a decomposition $f=f_1+f_2$

Let $D_1$ and $D_2$ be two compact sets in $\Bbb C$, $D_1\cap D_2=\emptyset$, and $ f\colon \Bbb C\setminus(D_1\cup D_2)\to\Bbb C$ be a holomorphic function. Show that there exist two holomorphic ...
1
vote
0answers
15 views

What techniques can be used to find fused global-local estimators of fractional derivatives?

The history of fractional calculus is a long one. A question on this topic appeared earlier today. This is a nice example of a minimal version of a global fractional derivative in a Fourier sense. But ...
0
votes
0answers
20 views

Numbers of the solution in a Cauchy problem

I am looking for an example for a real first order ODE which has $0$, $\infty$ or exactly $1$ solution depending on the initial $x(t_{0})$ value. The solution doesn't necessary has to have the domain ...
0
votes
0answers
21 views

Cauchy integral calculation outside the region

I have an integral $\oint \frac{2z}{z^2-9}dz$. It's said that the integral taken around the circle |z|=2 Here, the roots of z are $3,-3$ so it's outside the region. Does it mean the integral is 0?
1
vote
0answers
47 views

Sharpness of bound on derivative of bounded holomorphic function on a strip

Let $U$ be the open set of all $z \in \mathbb C$ such that $-1< \text{Im } z <1$. Let $f: U \to \mathbb C$ be a bounded homomorphic function and let $A :=\sup_{z\in U}|f(z)| $. Then using ...
0
votes
0answers
42 views

Derivative of an entire function of finite order

Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function of order $\rho$, that is, $\rho$ is the least positive real so that there are constants $x,y$ with $|f(z)| \leq xe^{y|z|^\rho}$ for every $z \in ...
1
vote
1answer
50 views

computing $\int_{-\infty}^{\infty} x e^{itx} dx$ using Cauchy Theorem and gamma function rapresentation

I want to solve the above integral with $t \in \mathbb{R}$; using a software (Wolfram) to verify the result I obtain the following result: $I(t) \equiv \int_{-\infty}^{\infty} x e^{itx} dx = -i \frac{...
1
vote
1answer
78 views

Verify integral using Residue Theorem

Verify the following equation: $$ \int_{0}^{2\pi}{\log{\left(\sin^{2}{2\theta}\right)}\,\mathrm{d}\theta} = 4\int_{0}^{\pi}{\log{\left(\sin{\theta}\right)}\,\mathrm{d}\theta} = -4\pi \log{2}$$ I am ...
2
votes
1answer
50 views

Understanding the Cauchy Integral Formula

This is the statement of the Cauchy Integral Formual. I have seen its proof, and I have used this formula many times to compute integrals. However, I feel like I do not understand this formula ...
0
votes
2answers
31 views

Can Cauchy's integral formula be used on the boundary point?

Can Cauchy's integral formula be used if the point a is on the boundary of the circle ? The theorem is given below.
0
votes
1answer
75 views

Prove that $f(z)=g(z)$ on $D$ considering Taylor expansions

Let $f$ and $g$ be differentiable on the strip $D = \{z \in \mathbb{C}: -2 < \operatorname{Im}z <2$. Suppose that $ f(z) =g(z)$ for all $z$ such that $|z|<0.01$. By considering Taylor ...

1
2 3 4 5
16