# Tagged Questions

The complex-integration tag has no wiki summary.

1answer
43 views

### Change of variables in a complex integral

I want to evaluate this integral using Residue Theorem $$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz =$$ $$C : |z| = 1$$ so I substitute letting $$\ W = z ^ {2 }$$ $$dw = 2z dz$$ and the ...
1answer
34 views

### Complex analysis contour integral

I am working on the integral $\displaystyle\int_0^{\infty}\frac{\log(x)}{x^2-1}$. I see it done here $\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint. but I am wondering if it is possible to ...
1answer
49 views

### Complex Integral of a meromorphic function

Please help with the following prelim problem. Thanks! Express the integral as a complex integral of a meromorphic function, where $\rho>0$ and $a$ is complex valued  \int_{|z|=\rho} ...
1answer
27 views

### Cauchy integral formula and holomorphic functions

I am stuck in a problem about holomorphic functions and using of Cauchy integral formula. I really have no idea how to start, so i would be glad if somebody could help me with it. Let $C=C(0,1)$ a ...
1answer
46 views

### Did I calculate this (simple) integral correctly?

Given the contour $C$: we are asked to calculate $\displaystyle\frac{1}{2\pi i}\oint \frac{ze^{z^2-4z}}{z^2-1}dz$. I wrote it as such: \frac{1}{2}\left(\frac{1}{2\pi i}\oint ...
0answers
71 views

### Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
2answers
66 views

### How to integrate complex exponential??

Consider $$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$ Why do we only look at the real part? What about the imaginary part ...
6answers
92 views

### Integration issue

I am trying to solve $\int^{+\infty}_{-\infty}\frac{1}{x}dx$. I read that it is a contour integral along the semi-circle of large radius in the lower complex plane. First, is there any justification ...
0answers
30 views

### Solving an complex Integration with complex exp and other terms

I am trying to solve a partial differential equation and while solving I need to solve the following integral. If anyone could help me solve this integral that would be great. y(x,t) = \int_{c-i ...
1answer
76 views

### How to do complex integration. E.g. $\int_\frac{\pi}{2}^{\frac{\pi}{2} + i} \cos(2z) \; \mathrm{d}z$

For my homework assignment I've been given a number of complex integrals to solve. I've already asked for help on a specific example here, but I was somewhat dissatisfied with the answers. The answers ...
1answer
121 views

### How to solve using Cauchy Integral formula?

Let $C$ be the positively oriented boundary of the square whose sides lie along the lines $x=+/-2$ and $y=+/-2$. I am supposed to use the Cauchy Integral formula to evaluate \int_C ...
1answer
56 views

### Need help integrating $\tan x$ and $\tan^n x$ using reduction

I have tried to use integration by parts taking $u$ as $\tan x$ and $v$ as $1$: $$\int \tan x \,dx = \int \tan x \cdot 1\; dx = \tan x \cdot x - \int \sec^2 x \cdot x\; dx$$ then by taking $u$ as ...
1answer
57 views

### Integrating $z^n$ and $(\overline{z})^n$ along a line segment in the complex plane

Let $z_1$ and $z_2$ be distinct points of $\mathbb{C}$. Let $[z_1,z_2]$ denote the oriented line segment starting at $z_1$ and ending at $z_2$. Evaluate the integral of $z^n$ and $(\overline{z})^n$ ...
2answers
80 views

### Is an integral in the complex plane an integral over a single number?

A recent question from Juan Saloman reminded me of something that has nagged me for years, and I have never understood and never heard explained. (or maybe I just don't remember, but anyway ...) In ...
1answer
65 views

### Finding the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis

Trying to find the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis, I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $d\theta=\dfrac{dz}{iz}$. I am trying ...
1answer
81 views

### Calculating Residues

I want to calculate this integral $$I:=\int dk^{0}\frac{e^{-ik^{0}(x^{0}-x'^{0})}}{\left(\left(k^{0}\right)^{2}-|\vec{k}|^{2}\right)}$$ for that I recall the Residue Theorem: I=2\pi i \left\{ ...
2answers
273 views

### Summation using residues

In reference to this question about showing that the following interesting series takes on the value $$\sum_{n=0}^\infty \frac{1}{(2n+1)\operatorname{sinh}((2n+1)\pi)}=\frac{\log(2)}{8}$$ I tried ...
1answer
111 views

### Problems in interpreting an integral that should be solved with residue method

Usually, when I solve an integral using residue method, I find real functions as integrands. I am not able to provide an interpretation for the following complex integral  \int_{-\infty}^{\infty} ...
4answers
167 views

### contour integration of logarithm

I must compute the following integral $$\displaystyle\int_{0}^{+\infty}\frac{\log x}{1+x^3}dx$$ Can someone suggest me the right circuit in the complex plane over which to do the integration? I ...
1answer
89 views

### finding $\int_0^\infty \dfrac{dx}{1+x^4}$ through complex analysis

I am trying to find $\int_0^\infty \dfrac{dx}{1+x^4}$ by setting it equal to $\dfrac{1}{2}\oint_C \dfrac{dz}{1+z^4}$ and solving that. By a computer program I've calculated it to be $\approx 1.11072$; ...
2answers
71 views

### Computing with Cauchy Residue theorem

how do I calculate $$\operatorname{Res}\left(\frac{1}{z^2 \cdot \sin(z))}, 0\right)$$ What is the order of the pole? $3$?
2answers
100 views

### a question about Cauchy integral formula

I'm new in the complex analysis and I'm stuck with this integral : $I=\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{(z^2+9)(z+9)}$ the exercise is about Cauchy integral, I don't want the whole ...
1answer
49 views

### Is this OK: $\int_a^b \!\mathrm{d}x \,\,f(x) =^? \int_{\mathrm{i}\,a}^{{\mathrm{i}\,b}} \!\mathrm{d} (\mathrm{-i}y)\,\,f(\mathrm{-i}y).$ Any proof?

This is related to Wick rotation in QFT but it is not exactly it. I'll take a 2-dimensional spacetime to be brief but usually there are more. I've checked with a few functions and with finite ...
1answer
195 views

### How to evaluate this complex integral !?

We have the following complex integral : $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$ Where $x\in\mathbb{R}:x>1$. i tried closing ...
1answer
159 views

### Complex integral over circle using Cauchy's formula

I have to integrate the complex function $$\frac{e^z-1}{z^5}$$ over the curve $\gamma(t)=1+re^{-5it}$ where $t \in [0,2\pi]$. The curve has winding number -5 with respect to a point inside the disc ...
4answers
198 views

### Integrate $\int_0^\infty \frac{\sqrt{x}}{x^{2}+1}\, \mbox{d} x$

I've been trying to integrate the following $$\int_{0}^{\infty} \frac{\sqrt{x}}{x^{2}+1} \mbox{d} x$$ on half an annulus in the upper half plane. I keep getting $\frac{\pi}{\sqrt{2}}\ i$, which ...
1answer
217 views

### Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}.$$ using residue calculus. Its value is $\pi^3/8$, as you can verify (for example) introducing the function  ...
5answers
262 views

### Showing that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left (\sqrt{a^2+1}-1\right)$.

How can I show that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
0answers
115 views

### Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}}$$ The function is ...
1answer
82 views

### Complex form of gauss divergence theorem

Just as complex form of green's theorem $\int {f(z)}dz=i\int\int \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}dxdy$ where $z=x+iy$ , do we have complex form of gauss divergence ...
1answer
105 views

### Evaluating complex integrals involving log (finding bounds)

When evaluating real integrals involving log, I am having trouble with the step that involves finding a bound on circular segments. Let me explain what I mean: If, for example, we have  ...
3answers
157 views

### Equality of absolute values of complex integrals

It was pretty hard finding a short and precise title, heres my problem: The equation $$\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$$holds true if f is integratable (where ...
1answer
99 views

### Does anyone know this functional integral equation?

$$\sqrt{2}f(x) =\lim_{\delta \to 0^{+}}\left[x-i\delta-\int_{-1}^{1} \frac{|f(y)|^2}{y-i\delta-x}dy\right]$$ I'd like to know if there is a solution for $f\colon(-1,1) \to\mathbb{C}$. Of course if it ...
3answers
159 views

### Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/ Any help will be much appreciated.
1answer
62 views

### Finding $\int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz$.

I'd really like some help with this problem. I'm supposed to find $$\int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz,$$ where $B_2(0) = \{ z \in \mathbb{C} \; | \; |z|<2 \}$ (ie. the ball of radius ...
4answers
110 views

### What is the value of $\int_{\gamma} \bar{z} dz$?

I could use some help in calculating $$\int_{\gamma} \bar{z} \; dz,$$ where $\gamma$ may or may not be a closed curve. Of course, if $\gamma$ is known then this process can be done quite directly (eg. ...
1answer
104 views

### Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
2answers
112 views

### Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
2answers
137 views

### Complex Analysis Help

Let $γ\colon[-1,1]\to\mathbb{C}$ , $γ(t)= z_0 + itc$ , $z_0$ fixed and c>0 Prove for x>0 $$\lim_{x\to0} \frac{1}{2πi} \int_γ \left(\frac{1}{z-w} - \frac{1} {z-w'}\right)dz = -1$$ Where $w=z_0 + x$ ...
1answer
176 views

### Complex analysis integration with residues.

I have to show that $$\int_{0}^{2\pi}\frac{d\theta}{(a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta)^{2}} =\frac{ \pi(a^{2}+b^{2})}{a^{3}b^{3}}$$ where $a,b>0$. I have tried using double angle formulas ...
1answer
84 views

### Integrating squared absolute value of a complex sequence

I was reading through my book in complex analysis and i encountered this problem. Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R. We are asked to show ...
1answer
216 views

### integral of complex logarithm

Consider the integral $$I=\int_0^{2\pi}\log\left|re^{it}-a\right|\,dt$$ where $a$ is a complex number and $0<r<|a|$. We have ...
1answer
148 views

### line integral versus complex integral

Let $a\in \mathbb C, r>0$ and $\gamma_r=\partial D(0,r)$. I want to evaluate the following line integral $$I=\int_{\gamma_r}\frac{1}{|z-a|^2}ds.$$ I'm looking for a complex function $g(z)$ such ...
0answers
52 views

### Exercise of Complex Integration

Let $f(z)$ be such that along the path $C_N$ of the following figure If $|f(z)|\leq \frac{M}{|z|^k}$ where $k>1$ and $M$ are constants independent of $N$. How to prove that ...
0answers
155 views

### Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0

The ultimate aim is to solve the following integral: \begin{equation} \label{eq:Icos1} \begin{aligned} I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
2answers
87 views

### $\int_{-1}^{1}\exp(-at^2)/t^2 dt$ using residue theorem

How to calculate $$\int_{-1}^{1}\frac{e^{-at^2}}{t^2}dt$$ where $a>0$, using residue theorem?
1answer
134 views

### Riemann Lebesgue Lemma for polynomial?

I was asked to prove that $$\lim_{n\to\infty} \int_{0}^{1} \exp(i\cdot n\cdot p(x))\;dx =0$$ for nonconstant real polynomial $p(x)$. if $p(x)$ is of degree $1$... It reduces to Riemann-Lebesgue ...
2answers
101 views

### Show the value of a complex integral is independent of R for R > 1

Question: Show that for R > 1 $$\int_{|z|=1} \frac{z^{2011}}{2z^{2012}-1} dz = \int_{|z|=R} \frac{z^{2011}}{2z^{2012}-1} dz$$ Thoughts thus far: (i) I know that we cannot use Cauchy's integral ...
1answer
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### Is $(-1+i)\log(2e^{it}+i)$ same as $\frac{1}{2}\left((2+2i)\tan^{-1}(2e^{it})-(1-i)\log(1+4e^{2it})\right)$?

Is $\displaystyle(-1+i)\log(2e^{it}+i)$ the same as $\displaystyle\frac{1}{2}\left((2+2i)\;\tan^{-1}(2e^{it})-(1-i)\log(1+4e^{2it})\right)$? WolframAlpha shows that they are same, but this page on W|A ...
2answers
121 views

### “Convergent” Integral in Davenport's Multiplicative Number Theory

I am currently learning analytic number theory using Davenport's Multiplicative Number Theory book, and at some point I believe something silly is happening. I have great faith that I am wrong AND ...
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