3
votes
2answers
80 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
0
votes
0answers
54 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
2
votes
1answer
23 views

Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
0
votes
0answers
28 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
2
votes
1answer
28 views

$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ where $f(z) = z^{1/2}$ with branch such that $\Re f(z) \geq 0$

As the title states, the definite integral in question is $$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz,$$ where $f(z) = z^{1/2}$ with branch cut such that $\Re f(z) \geq 0$, i.e., the cut is the ...
0
votes
2answers
55 views

How should I calculate $\displaystyle\int_{-\infty}^\infty\exp\left\{-\frac{1}{2}(x-it)^2\right\}dx$?

I've read that the residue theorem would help to calculate $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\exp\left\{-\frac{1}{2}(x-it)^2\right\}}_{=:f(x)}dx$$ Since $f$ is an entire function ...
1
vote
3answers
83 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
2
votes
2answers
54 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
0
votes
0answers
18 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] ...
0
votes
1answer
25 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
3
votes
1answer
25 views

Contour Integrals evaluation verification

$$ \begin{align} \gamma(t)=2cost + isint, t\in[0,2\pi] \end{align} $$ Could someone verify my thinking and my results for the following Integrals and if I have properly justified my thoughts: $$ ...
4
votes
3answers
64 views

Can $\frac1{z^2}$ be integrated on $|z+i|=\frac32$ using Cauchy's theorem?

$$ \begin{align} \int_{|z+i|=\frac{3}{2}}\frac{1}{z^2}dz=0 \end{align} $$ Is it safe to say the Integral is $0$ due to cauchy's Theorem? Does this apply for any $z_0$ that lies inside the circle ...
1
vote
1answer
46 views

Is this Integral Calculation correct? $\int_{|z|=1}(z^2+2z)^{-1}dz=\pi i$

$$ \begin{align} \int_{|z|=1}(z^2+2z)^{-1}dz=\int_{|z|=1}\frac{1}{z(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+\int_{|z|=1}\frac{1}{2(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+ 0= \\ ...
1
vote
1answer
21 views

Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be ...
3
votes
1answer
46 views

Calculation of a Contour Integral

$$ \begin{align} \int_{|z|=1}(4-z^2)^{-1/2}dz \end{align} $$ The exercise hints at the usage of $e^{2Logz}$ , although any solution or methodology for such integrals would be welcome.
0
votes
1answer
29 views

Geometric explanation of a contour's image

$$\gamma(t)= t^2 + i\, t^4 , \quad t\in [ -1, 1]$$ What is the geometric explanation of the image of the above contour? Intuitively , I think it's ellipsoid-like, but I don't know how to put it in a ...
0
votes
1answer
34 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
1
vote
1answer
30 views

Contour Integration Confusion

I am trying to find the value of $\displaystyle\int_0^\infty\frac{(\log x)^2}{1 + x^2}\,dx$ using contour integration. My approach: I have calculated the residue at z = $i$ and have shown that ...
1
vote
2answers
43 views

Contour expression explanation

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$ contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and ...
3
votes
1answer
82 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
5
votes
2answers
76 views

Complex Numbers - Finding Roots

Hi there I was wondering if someone could help me? I am struggling to find the roots of the polynomial $z^4+2z+3=0$ It is not a quadratic so can't use the quadratic formula so am not quite sure ...
2
votes
1answer
17 views

Estimate of line integral of O(x^n) function

Let $f$ be an analytic function in some sector in the complex plane behaving as $$f(z)=\mathcal O(z^n)$$ for some $n$ as $z\to\infty$. Can one prove in general that line integrals of $f$ (in this ...
0
votes
0answers
28 views

integration with 4 branch point

I come across a problem of contour integration with 4 branch point. The problem came down to be equivalent to integrate $\sqrt{(x^2-1)^n(x^2-2)^m}$ over the imaginary axis. So, there are two branch ...
4
votes
0answers
24 views

$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $ on a closed curve of index $N$ with respect to the point $1$.

Let $\gamma$ be a closed curve in the right half plane that has index $N$ with respect to the point $1$. Find $$ \int_\gamma e^{\frac{1}{z^2-1}}\sin{(\pi z)}dz $$ This is a problem from an old ...
2
votes
1answer
124 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
49 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 ...
10
votes
4answers
255 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
135 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
60 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
56 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
37 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
63 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
59 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
24 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
39 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 $z(\theta) = z_0 + Re^{i\theta}$ ...
5
votes
0answers
57 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
137 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
27 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
33 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
67 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
60 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
67 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
51 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
64 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
53 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
35 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
64 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 ...