2
votes
1answer
83 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
0
votes
0answers
41 views

Proper way to set up “Pac-Man” contour integral

I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$ using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only ...
5
votes
3answers
128 views

Contour Integral: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$

I want to compute: $\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the ...
2
votes
1answer
117 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
0
votes
0answers
56 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
2
votes
1answer
53 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
1
vote
2answers
36 views

Show elementarily that $\lim_{R\to\infty}\int_{\Gamma_1} \frac{e^{iz}}{z} = 0$

Context: I am trying to show that $\int_0^\infty x^{-1}\sin x dx = \frac{\pi}{2}$ using complex analysis, by first integrating $\oint_{\Gamma} z^{-1}e^{iz}$, where $\Gamma$ is a closed contour ...
2
votes
1answer
59 views

Computing a very messy contour integral

I'm hoping that someone might be able to help me with the following problem. I'll walk through my current work and indicate where I'm stuck. Compute the contour integral: $$ \oint_{|z-1-i| = ...
2
votes
1answer
57 views

Contour integral method

Let $f(z)=z^5-3iz^2+2z-1+i$. Evaluate the integral of $\frac{f'(z)}{f(z)}$ around a contour $C$ where $C$ encloses all the zeroes of $f$. I'm not sure what to do here. It seems unlikely I should be ...
1
vote
0answers
17 views

Complex contour integral properties

Do I understand correctly, that for complex line integrals the properties of common integrals (e.g. Riemann-integrals) cannot be applied? Neither linearity: $$\int_\gamma \beta \;f(z)dz \not = ...
2
votes
1answer
38 views

Computing a contour integral over curve not centered at origin

Consider the integral $$ \int_C \frac{1}{z} \, dz $$ where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ ...
4
votes
0answers
52 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
7
votes
1answer
136 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
0
votes
1answer
25 views

Contour integrals and Cauchy's theorem

1) Let $C$ be a contour beginning and ending at 1. Suppose that $f(z)$ is analytic on $C$. Then is it true that the contour integral of $f$ around $C$ is 0? This looks to be true by Cauchy's theorem ...
0
votes
1answer
29 views

Theoretical question regarding Cauchy integral theorem

As we know, according to the Cauchy integral theorem we can easily evaluate an integral of an analytic complex function along the curve connecting two points in a complex plane. Thus for a curve ...
3
votes
1answer
62 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
0
votes
3answers
56 views

Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial)

Let $p$ be a polynomial of degree $n\geq2$ and has $n$ different roots $z_1,\dots,z_n$. Prove that $\oint\frac{\mathrm{d}z}{p(z)}=0$ where the closed path is large enough so that all roots are in the ...
0
votes
0answers
58 views

Contour integration on a rectangle

Another qualifier problem: Suppose $f(z): \mathbb{C} \mapsto \mathbb{C}$ is entire and $\exists M,A>0$ such that $$ |f(x+iy)| \leq \frac{A}{1+x^2} e^{2 \pi M |y|} \: \forall x,y \in \mathbb{R}. ...
1
vote
0answers
49 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
0
votes
1answer
22 views

The integral along a circle of the inverse linear function is zero

Assume ${\rm C}$ is a circle and $a,b$ are distinct points in the interior of ${\rm C}$. How can we see that the complex integral $$ \frac{1}{b - a} \int_{\rm C}\left(\frac{1}{z - a} - \frac{1}{z - ...
5
votes
2answers
59 views

How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
4
votes
0answers
50 views

Does anyone have a good reference on calculating contour integrals around the unit circle (numerically or otherwise)?

I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I ...
1
vote
0answers
32 views

Conditions for changing the order of integration for contour integral.

I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$ where the function $f(x)$ can be represented as a contour integral in complex plane: $$f(x)=\oint_\Delta ...
2
votes
0answers
62 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
0
votes
0answers
29 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
0
votes
0answers
41 views

Method for evaluating $\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$

I have a problem where I must evaluate $$\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$$ Where $P(z)$ is a polynomial with degree at least four and has exactly four roots in the unit circle. I know ...
1
vote
2answers
41 views

Definite integrals with complex analysis

Can anyone explain me why: $$\int_{C_r}\frac{1}{z} \mathrm{d}z+\int_{C_r}\frac{e^{iz}-1}{z}\mathrm{d}z\stackrel{r \to 0^+}{=}\pi i$$ $C_r$ is half circle from $r$ to $-r$
2
votes
1answer
45 views

Showing integral on contour tends to zero

I'm trying to prove: $$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$ Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients. ...
0
votes
3answers
54 views

Integrating $1/\sqrt{z^{2}-1}$ on some contour

If I wanted to integrate $$\oint \frac{1}{\sqrt{z^{2}-1}}$$ Say around a circular contour radius $2$ centre $0$, how would I do that? Does the function have poles at $\pm 1$ or are they just "branch ...
1
vote
1answer
53 views

Contour Integral of $\int_0^{\infty} \frac{1}{x^4+1} dx $ - Missing a factor of 2

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{1}{x^4+1} dx $$ Consider $$ \oint \frac{1}{z^4+1} dz = \oint \frac{1}{(z - \frac{1-i}{\sqrt 2})(z + \frac{1-i}{\sqrt 2})(z - \frac{1+i}{\sqrt ...
0
votes
0answers
32 views

Line Integrals in complex plane

I recently learned about line integral in the complex plane in my Complex Analysis class but I'm struggling a bit with some exercises, namely: Calculate $\oint_c f(z) dz$, where: a) $f(z)= ...
0
votes
0answers
44 views

Complex Integration: $\int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $$ Attempt Consider $ \int_0^{\infty} \frac{e^{iz}}{x(k^2x^2 +1)} dz $ Simple poles at $z = \pm \frac{i}{k} $, simple ...
1
vote
0answers
53 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
2
votes
1answer
51 views

If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
0
votes
2answers
50 views

Contour integral of $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$

I'm supposed to evaluate $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$ Using a contour of a unit circle, $z=e^{i\theta}$. This is the same as: $$2i \oint \frac{1}{z^2 - Az + 1 } dz $$ The ...
1
vote
1answer
85 views

Calculate $\int_0^\infty\frac{\sin x}xdx$ by integration of a suitable function along given paths [duplicate]

How can I calculate $$\int_0^\infty\frac{\sin x}xdx$$ by integration of a suitable function along the following paths: where $R$ and $\varepsilon$ are the radius of the shown outer and inner ...
0
votes
1answer
21 views

Show that $f(a)=\frac{1} {2\pi}{\int_C {\frac{(R^2-a \overline a)f(z)}{(z-a)(R^2-z \overline a)}dz}}$

The function $f(z)$ is regular when $|z|<R'$ Show that if $|a|<R<R'$ then $$f(a)=\frac{1} {2\pi}{\int_C {\frac{(R^2-a \overline a)f(z)}{(z-a)(R^2-z \overline a)}dz}}$$ Where $C$ is the ...
3
votes
1answer
60 views

Contour integration of $\frac{(\ln z)^2}{z^2+1} $

I'm supposed to take the principal branch of $\ln z$ and evaluate this integral: $$ \oint \frac{(\ln z)^2}{z^2+1} $$ Attempt I suppose the integral they are talking about is something like ...
3
votes
1answer
50 views

Complex integral of $\frac{\cos x}{x^2+4} $

I want to evaluate: $$ \int_{-\infty}^{\infty}\frac{\cos x}{x^2 +4} dx $$ Using wolfram alpha, it gave an answer of $\frac{\pi}{2e^2}$. Wolfram Alpha is never wrong. Attempt $$ ...
2
votes
1answer
61 views

Complex Contour Integration - Complex Analysis

I'm just practising for my upcoming exam, and I've come across a question I'm having a bit of difficulty with. I've been asked to show the following; $$\int_{0}^{\infty} \frac{dz}{\cosh(z)} = ...
1
vote
1answer
51 views

Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx $$ Attempt Considering $$ \oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz $$ So first I find the branch points of the function. This ...
0
votes
1answer
21 views

Contour Integration of this path circling branch point

If we let the semi-cricle blow up to infinity and the radius of the tiny circle encircling the branch point at origin go to zero, by residue theorem we have: $$\int_\gamma + \int_{AB} + \int_{BC} ...
0
votes
1answer
61 views

Contour Integration: What is the function?

I know that the integral around a closed path = 0 since there are no poles. Why is the integral along the slanted path $\int_0^R e^{-x^2w^2} w dx$? If $w$ is defined to be along the slope, ...
1
vote
1answer
38 views

Help in understanding contour integration

I would like help in understanding the process of contour integration. As an (hopefully straightforward) example, I have chosen the calculation of Bernoulli number $B_2$. I should be very grateful ...
1
vote
1answer
47 views

Finding definite integral using contour integration

Wanting to find the value of the integral $\int_{0}^{\infty} \dfrac{1}{\cosh (x)} dx$ and know I have to find the residue at $\dfrac{\pi i}{2}$ which I find to be $-i$. So far so good. So then, I know ...
0
votes
1answer
42 views

Compute $\int_{\gamma} x dz$

Let the perimeter of the square formed by the points $0$, $1$, $1 + i$, $i$ and $z = x + iy$. How can i compute $$\int_{\gamma} x dz$$. Some help to compute this complex integral please.
2
votes
1answer
100 views

Evaluation of tricky integral

I want to evaluate the integral $$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$, where $b$ and $s$ are positive real numbers. I thought of ...
3
votes
2answers
55 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
1
vote
1answer
55 views

How would Cauchy calculate $\int_{\partial B_1(2i)}\frac{e^{z^2}}{2i-z}dz$?

Please consider the following curve integral: $$I:=\int_{\partial B_1(2i)}\frac{e^{z^2}}{2i-z}dz$$ where $$B_r(z_0):=\left\{z\in\mathbb{C}:|z-z_0|<r\right\}$$ Let $\gamma :[a,b]\to\Omega$ denote ...
0
votes
1answer
30 views

Evaluate $ \int_{C(0,5)} \frac{1}{i-\cos z}dz $.

How do I evaluate $$ \int_{C(0,5)} \frac{1}{i-\cos z}dz? $$ Do I have to find the poles first, and then use residue theorem, or find where the function is holomorphic and then integrate using ...