Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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"

2
votes
1answer
66 views

Is it possible to evaulate $I = \frac{2i}{\pi} \int_\gamma \ln(z) dz$ explicitly even though $\ln(z)$ isn't holomorphic at $z=0$?

Is it possible to calculate the following integral explicitly: $$ I = \frac{2i}{\pi} \int_\gamma \ln(z) dz, $$ where $\gamma$ can be a disk at the origin of $\mathbb{C}$? Unfortunately $\ln$ isn't ...
0
votes
1answer
37 views

Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
2
votes
2answers
93 views

Solving a complex integral $\oint_L\frac{e^{1/(z-a)}}zdz$ using Cauchy's formula

I am practicing complex integration using Cauchy's formula, and I ran into a problem. The following integral: $$\oint_{L}{\frac{e^{\frac{1}{z-a}}}{z}}dz$$ where $$L=\{z\in\mathbb{C}:|z|=r\}$$ for ...
9
votes
2answers
605 views

Integral with two different answers using real and complex analysis

The integral is$$\int_0^{2\pi}\frac{\mathrm dθ}{2-\cosθ}.$$Just to skip time, the answer of the indefinite integral is $\dfrac2{\sqrt{3}}\tan^{-1}\left(\sqrt3\tan\left(\dfracθ2\right)\right)$. ...
3
votes
2answers
192 views

Real integrals with two poles in the complex plane

I'm studying the Cauchy Integral Theorem / Formula, but realised I have a misunderstanding. Consider an integral over the function $f: \mathbb{R} \to \mathbb{C}$ $$ I = \int^\infty_{-\infty} f(x) \, ...
0
votes
1answer
37 views

When would I use Cauchy's Integral Formula over Residue

Just a quick question I've been wondering about. When would I use Cauchy's Integral Formula over the Residue Theorem to solve complex integration problems with poles? To me it seems that Residue ...
0
votes
1answer
24 views

When to resolve into partial fractions for applying Cauchy's integral formula?

Suppose I have to calculate $\int \frac{f(z)}{(z-a)(z-b)}dz$ around a curve in which both $a$ and $b$ lie inside. Should I apply Residue theorem like this: Residue at $a$= $\frac{f(a)}{a-b}$ ...
5
votes
0answers
55 views

$L_f(z) = \frac 1 {2 \pi i}\int_{ \mathbb{T} } \frac{ \zeta+z}{ \zeta ( \zeta -z)} f( \zeta ) d\zeta$

I'm trying to prove that for any harmonic function $u$, we have : let $ \Omega \subset \mathbb{R}^2$ and $ \overline B(0,R) \subset \Omega $ $$ u \colon \Omega \to \mathbb R $$ $$\forall z \in ...
0
votes
0answers
28 views

How to get correct residue of complicated function with exponentials and associated contour integral?

I am trying to calculate the improper integral: $$ I=\int_{-\infty}^{+\infty} f(x) dx $$ with $$ f(x)=\frac1{8\pi^3}\frac{x^2 \sqrt{1+x^2}}{1+e^\sqrt{1+x^2}}.$$ The function $f(x)$ has poles at $x_0=\...
1
vote
2answers
31 views

Using Cauchy’s Theorem/ Cauchy’s Integral Formula for Higher Derivatives or otherwise, evaluate, with justification, the following integrals:

$$\int \frac{5 \cos(\pi z)}{(z+3i)(z-7i)}dz$$ i) where γ is the circle centre 0 and radius 4; ii)γ is the circle centre 0 and radius 10 For part i) I have calculated using CIF that because z=-3i is ...
2
votes
2answers
73 views

Stuck on proof using Cauchy's integral formula

I posted my attempted proof to this question here but I realized that I was wrong in taking the limit, and that the proof did not make sense. So I am still stuck on this problem let $f: \Omega \...
2
votes
2answers
69 views

Prove inequalities with Cauchy's integral theorem [closed]

Let $$f:\overline{B(0,1)}\rightarrow\mathbb{C}$$ be continuous and holomorphic on $B(0,1)$. Consider the function $$z\mapsto F(z):=f(z)\overline{f(\overline{z})}.$$ Show (i) $\int_{\gamma}F(z)dz=0$ ...
1
vote
0answers
47 views

Proof using Cauchy's Integral Formula

let $f: \Omega \rightarrow \mathbb{C}$ be analytic and $z_0 \in \mathbb{C}$. Define $$g(z) = \begin{cases} \frac{f(z)-f(z_0)}{z- z_0} & z \not = z_0 \\ f'(z_0) & z = z_0 \...
3
votes
1answer
37 views

Cauchy Integral Theorem with $f(z)=e^{z^2}$

I have $z(t)=t(1-t)e^t + \cos(2 \pi \cdot t^3)i$ with $0 \le t \le 1$ and need to evaluate the line integral of $e^{z^2}$. I know that the endpoints are $z(0)=z(1)=0+i$, so the line is a closed ...
0
votes
1answer
33 views

Proving a certain holomorphic function is polinomic

Suppose we have an holomorphic function $f$ on the open unit disk $D(0,1)$ s.t.: $$\forall r\in (0,1) \exists n\in \mathbb{N}| \max_{C(0,r)}|f|=r^n$$ prove that $f$ is polinomic. Honestly I don't ...
1
vote
0answers
15 views

Show that for a function $f$ bounded by $M$ on a disk $D_{r}$ show that $|f^{n}(z)|\leq n!M/\delta^{n}$ for $D_{r-\delta}$

Suppose that a function $f$ is analytic in the open disk $$D_{r}=\{z\in \mathbb{C}:|z|<r\}$$ where $r>0$, and there is a number $M\in\mathbb{R} $ such that $|f(z)| \leq M$ for all $z \...
0
votes
1answer
34 views

Questions about Cauchy's thm. on complex integration

I have the following integral $$I = \int_{\gamma_{1}} \frac{e^{z^2}}{(z-1)^2}dz,$$ where $$\gamma_{1} : [0,2Pi] \rightarrow \mathbb{C}, \\ \quad t \quad \mapsto \quad 2e^{i t} .$$ I want to show that ...
2
votes
1answer
56 views

Integral by Residue Theorem

I'm working through a Complex Analysis text and am working through the Residue chapter. I am not sure if I am approaching this question correctly. $$ \int_\gamma \frac{1}{(z-1)^2(z^2+1)}$$ Such that ...
1
vote
1answer
61 views

How to evaluate integrals like $ \int_C \frac{z}{e^z-1} dz$ without Residue Theorem?

I'm trying to evaluate two integrals: $$ I_1 = \int_{C(0, 1)} \frac{z}{e^z-1} dz,\quad I_2 = \int_{C(1, 3)} \frac{\sin(iz)}{e^{iz}-1} dz $$ $ C(a, r) $ is a circular contour with radius $r$ ...
0
votes
1answer
54 views

Proving $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$

using the following result $\int _{\gamma}(z+ \frac{1}{z})^{2n}\frac{dz}{z}= {2n \choose {n}}2\pi i $ Prove $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$ I cant see how ...
0
votes
1answer
63 views

Cauchy's integral formula with special contour 4

suppose $\gamma: [a,b] \rightarrow \mathbb{C} $ is a path of integration with $ \gamma(a)=0, \gamma(b)=1 \ and \ \pm i \notin\gamma([a,b]) $ Show that, $$ \int_{\gamma} \frac{1}{1+z^2} = \frac{\pi}{...
0
votes
1answer
48 views

Holomorphic integrals

I am struggling to understand how the center and radius effect a certain circular contour. e.g. $ \int _{\gamma} \frac {z^{2}+1} {e^{z}(z-1)(z+1)^{2}} dz $ can anyone explain this?
0
votes
1answer
31 views

Complicated Cauchy complex integral.

I have the function $f(z)$ that defined on the closed disc $ \bar D_3(0)$ by the integeral on the boundry $C_3(0)$: $f(z) = \int_{C_3(0)} \frac{3w^2+7w+1}{w-z}dw $ I need to find $f'(1+i)$ Caushy ...
1
vote
1answer
42 views

Integral Cauchy formula

How do I put: $\int_\gamma (z + \frac{1}{z})^{2n} \frac{dz} {z} $ In the form: $\int_\gamma \frac{f(z)}{(z-a)^{2n}} $ Initially I have gone for $f(z)=\frac {1}{z} $ but I cannot get the rest in ...
1
vote
1answer
11 views

Prove simple closed curves $f$'s exist, so $\Gamma = C-\sum_{i=1}^{k}{f_i}$ satisfies $ \int_{\Gamma}{\frac{z^3e^{1/z}}{(z^2 + z + 1)(z^2 + 1)}dz}=0$

Let $C$ be the circle $C(0,2)$ traversed one time counter-clockwise. Prove that there exist $k\in \mathbb {Z}_+$ and $C^1$ simple closed cuves $f_1, \dots ,f_k$ such that the cycle $\Gamma = C-\sum_{i=...
0
votes
3answers
54 views

If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

If $f$ is an entire function so that $|\,f(z)|\le|\operatorname{Re}(z)|$ for all $\mathbb{C}$, then $f\equiv0$ on $\mathbb{C}$. There are various ways showing the above property. For example, using ...
1
vote
0answers
20 views

Undestand the proof of Cauchy Integral Formula using homotopy of curves.

https://math.berkeley.edu/~vvdatar/m185f16/notes/Lecture-16_CIF_Analyticity.pdf I'm trying to understand the proof of Theorem 0.1. Only the following part is missing: "Consider the contour in the ...
0
votes
0answers
20 views

Showing a Result Using Cauchy's Generalised Integral Formula

Using Cauchy's generalised integral formula, show that if $f$ is an entire function and $|f(z)|\leq |z|+1$, then $f^{(k)}(0)=0$. Hence deduce that there are complex constants $a,b$ such that $f(...
0
votes
1answer
17 views

Interating when one singularity is inside C and the other is not

I tried to evaluate $\frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz$ where C is the square with vertices $\pm2$, $\pm2+4i$ thusly: $$\frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz=\frac{1}{2\pi i}\oint_C \...
1
vote
2answers
75 views

How to show an entire function bounded from above is identically zero

Suppose that $f$ is entire. By writing $f$ as a Taylor Series, prove that if $\lvert f(z)\rvert<m\lvert z\rvert^\alpha$ for $m>0$ and $0<\alpha<1$ then $f$ must be identically zero in $\...
0
votes
0answers
47 views

Evaluating the integral $\int_c \frac{e^z}{z(z-4)^3}dz$

I would like to evaluate \begin{align} \int_c \frac{e^z}{z(z-4)^3}dz \end{align} where $C$ is the square centered at the origin with sides length 6 units and let $C_1$ be the unit circle centered at ...
1
vote
1answer
110 views

$\int_0^∞ \dfrac{x^a}{(x^2 + 1)^2} dx$ for $-1<a<3$

I tried to evaluate $A=\int_0^∞ \dfrac{x^a}{(x^2 + 1)^2} dx$ for $-1<a<3$, several times. I followed lengthy examples of the book which supposes real positive axis as branch cut. Every time I am ...
0
votes
1answer
27 views

Evaluating a complex integral with two singularities

I am trying to evaluate $\frac{1}{2\pi i}\oint_C\frac{e^{et}}{(z^2+1)^2}dz$, where $C$ is $|z|=3$, knowing that it should be $\frac{1}{2}(\sin t-t \cos t)$. Here is what I've done: $$f^{(n)}(a)=\...
1
vote
1answer
24 views

Application of Cauchy theorem On Integral

I'm taking my first course in complex analysis and I've come across a question I haven't encountered before. $$\int_C \frac{z dz}{(z+2)(z-1)}$$ Such that C is a circle where $|Z| = 4$. So my ...
1
vote
1answer
45 views

Evaluating the integral $\frac{1}{2\pi i} \oint_C\frac{e^{zt}}{z^2+1}dz$ with Cauchy's theorem.

I would like to show that $$\frac{1}{2\pi i} \oint_C\frac{e^{zt}}{z^2+1}dz=\sin t$$ if $t>0$ and $C$ is the circle $|z|=3$. I am pretty sure that I need to use Cauchy to do this because it ...
0
votes
1answer
37 views

Calculate complex curve integral along rectangle

Determine the line/contour integral of: $$\int_{\gamma}\frac {z}{z^3+1}dz$$ where $\gamma$ is the boundary of a rectangle defined for $0\leq x\leq 2$ and $-2\leq y\leq 2$. I am almost certain we ...
1
vote
1answer
65 views

Let $f$ be the disk $D[0, 1]$ be a holomorphic function. Show the following

I was able to prove a) using the limit of $g$. I know that b) has something to do with Cauchy's formula, and I tried to use the RHS to get the LHS, but got stuck after expanding it. I also tried ...
2
votes
2answers
43 views

Complex line integral of $\frac{e^z}z$

I have an integral $$\oint_{|z|=1}\frac{e^z}{z}\,\mathrm dz.$$ I defined $g: [0,2\pi]$, $g(t)=e^{it}$. The integral then becomes. $$\int_0^{2\pi}e^{e^{it}}\mathrm dt.$$ I used the property that ...
1
vote
0answers
26 views

Integrating a Modulus Under the Complex Path Integral on a Circle

I have an integral that involves a modulus term under the integral sign, that I'm not too sure how to deal with. $$\int_{|z|=R} \dfrac{|dz|}{|z-c|^4}, R > 0, |c| \neq R$$ OK, so now suppose I ...
0
votes
1answer
32 views

$|f(z_1)-f(z_2)| \leq \frac{4M}{R} |z_1-z_2|$

Let $f: G \to \mathbb{C}$ be holomorphic, where $G$ is a domain subset of $\mathbb{C}$. Let $|z-a| \leq R \subset G$. Show that if $|f(z)| \leq M$ for all $z \in |z-a|=R$ then for any $z_1,z_2 \in \{...
0
votes
0answers
23 views

when to apply cauchy's-lemma 1 or cauchy's-lemma 2 or jordan's-lemma?

I know the procedure of computing contour integration of a given complex function using cauchy's lemma's rule.But,I am confused about when to use cauchy's-lemma 1 and when caushy's-lemma 2 or jordan's-...
0
votes
2answers
77 views

complex analysis integral with $z^8$ function [closed]

can anyone help with this integral I'm trying to do. Will a semicircle contour work for this function? $$\int_{0}^{\infty}\frac{1}{1+x^{8}} dx$$
1
vote
0answers
42 views

Hyperbolic function in complex analysis integral

I'm trying to solve this integral for my complex analysis course however I can't seem to figure out how to get started on it. I think since there's exponentials in the function you want to maybe use a ...
1
vote
1answer
41 views

Complex Analysis Question - Cosine and quadratic combined

i was doing some questions in my complex analysis booklet and have came across the following question that i don't seem to be able to get the answer for. Hoping someone on here can help! So it's the ...
0
votes
0answers
39 views

Evaluate $\int_{|z|=3} \frac{cos(\pi z)}{(z-2)^2(z+5)(z+1)} \ dz$

I am trying to solve $$I=\int_{|z|=3} \frac{cos(\pi z)}{(z-2)^2(z+5)(z+1)} \ dz.$$ My attempt: Residue Theorem: Let $$f(z)=\frac{cos(\pi z)}{(z-2)^2(z+5)(z+1)}.$$ Now, $$\text{Res}(f,2)=\lim_{z\to ...
1
vote
1answer
28 views

Integrating with Cauchy Goursat Theorem with exponential

How do I apply the cauchy goursat theorem with exponentials? I know i should be using partial fractions but not sure how to break down the numerator here. Any help is greatly appreciated.
2
votes
2answers
215 views

Find $\int_{\gamma} e^zz^n dz$ where $\gamma$ is the unit circle, using Cauchy's Integral Formula

I'm been banging my head against the wall trying to solve the following question which ask to solve the following integral using the Cauchy integral formula, and hence evaluating the corresponding ...
2
votes
1answer
81 views

Cauchy's Integral formula real part confusion

Suppose that $f$ is holomorphic in $D_{R_0}(0) = \{|z| < R_0\}$. Show that whenever $0 < R < R_0$ and $|z| < R$, then $f(z) = \frac{1}{2\pi}\int_0^{2\pi} f(Re^{i\theta})\text{Re}(\frac{Re^{...
4
votes
0answers
55 views

Find $I_n=\int_{\gamma} e^zz^n \ dz, \ n\in\mathbb{Z}$

Let $\gamma$ be the unit circle $\{e^{i\theta}:-\pi\leq\theta\leq\pi\}$. Using the Cauchy integral formula to find, $$I_n=\int_{\gamma} e^zz^n \ dz, \ n\in\mathbb{Z}.$$ Hence evaluate the ...
0
votes
0answers
45 views

Is it possible to generalize Cauchy Integral Inequality for any point other than center?

I know that Cauchy Inequality in complex analysis which gives bound for nth derivative of the function at $z_0$ which holomorphic on disk centred at $z_0$ and Radius R by $$f^{(n)}(z_0)\leq \frac{n!|...