The complex-integration tag has no wiki summary.
15
votes
2answers
280 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 ...
8
votes
4answers
185 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 ...
7
votes
5answers
269 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 $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
5
votes
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 ...
4
votes
2answers
151 views
Show the existence of a complex differentiable function defined outside $|z|=4$ with derivative $\frac{z}{(z-1)(z-2)(z-3)}$
My attempt
I wrote the given function as a sum of rational functions (via partial fraction decomposition), namely
$$
\frac{z}{(z-1)(z-2)(z-3)} = \frac{1/2}{z-1} + \frac{-2}{z-2} + \frac{3/2}{z-3}.
...
3
votes
2answers
112 views
Computation of a certain integral
I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
3
votes
3answers
162 views
Improper integration involving complex analytic arguments
I am trying to evaluate the following:
$\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/
Any help will be much appreciated.
3
votes
4answers
110 views
What is the value of $\int_{\gamma} \bar{z} dz$?
I could use some help in calculating $$\int_{\gamma} \bar{z} \; dz,$$ where $\gamma$ may or may not be a closed curve. Of course, if $\gamma$ is known then this process can be done quite directly (eg. ...
3
votes
1answer
180 views
Complex analysis integration with residues.
I have to show that $$\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}}$$ where $a,b>0$. I have tried using double angle formulas ...
3
votes
1answer
199 views
Complex antiderivative
I am confused on a couple things:
1.) Why is it that an integral of a complex valued function of a complex variable exists if f(z(t)) is piecewise continuous (and/or piecewise continuous on ...
3
votes
1answer
46 views
Did I calculate this (simple) integral correctly?
Given the contour $C$:
we are asked to calculate $\displaystyle\frac{1}{2\pi i}\oint \frac{ze^{z^2-4z}}{z^2-1}dz$. I wrote it as such:
$$\frac{1}{2}\left(\frac{1}{2\pi i}\oint ...
2
votes
2answers
163 views
Evaluate $\int_{C}\frac{e^{\alpha z}}{z}dz$ where $\alpha \in \mathbb R$ and C is the circle $\gamma(t)=e^{it}$…
Let $\alpha \in \mathbb R$ and C be the circle $\gamma(t)=e^\alpha t$, $-\pi\le t \le \pi$
Evaluate $$\int_{C}\frac{e^{\alpha z}}{z}dz.$$
Use the above, to show that $$\int_{0}^{\pi}e^{\alpha ...
2
votes
2answers
84 views
Is an integral in the complex plane an integral over a single number?
A recent question from Juan Saloman reminded me of something that has nagged me for years, and I have never understood and never heard explained. (or maybe I just don't remember, but anyway ...) In ...
2
votes
1answer
47 views
Change of variables in a complex integral
I want to evaluate this integral using Residue Theorem
$$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$
$$ C : |z| = 1 $$
so I substitute letting $$\ W = z ^ {2 } $$
$$ dw = 2z dz $$
and the ...
2
votes
1answer
162 views
Complex integral over circle using Cauchy's formula
I have to integrate the complex function
$$
\frac{e^z-1}{z^5}
$$
over the curve $\gamma(t)=1+re^{-5it}$ where $t \in [0,2\pi]$. The curve has winding number -5 with respect to a point inside the disc ...
2
votes
1answer
49 views
Complex Integral of a meromorphic function
Please help with the following prelim problem. Thanks!
Express the integral as a complex integral of a meromorphic function, where $\rho>0$ and $a$ is complex valued
$$
\int_{|z|=\rho} ...
2
votes
1answer
196 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 ...
2
votes
1answer
92 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 $\approx 1.11072$; ...
2
votes
1answer
222 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
$$
...
2
votes
1answer
100 views
Does anyone know this functional integral equation?
$$\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]$$
I'd like to know if there is a solution for $f\colon(-1,1) \to\mathbb{C}$.
Of course if it ...
2
votes
1answer
104 views
Definite integral involving hyperbolic cosine
I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
2
votes
0answers
115 views
Show Smoothness by Morera
I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function:
$$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$
The function is ...
2
votes
2answers
137 views
Complex Analysis Help
Let $γ\colon[-1,1]\to\mathbb{C}$ , $γ(t)= z_0 + itc$ , $z_0$ fixed and c>0
Prove for x>0 $$\lim_{x\to0} \frac{1}{2πi} \int_γ \left(\frac{1}{z-w} - \frac{1} {z-w'}\right)dz = -1$$
Where $w=z_0 + x$ ...
2
votes
1answer
221 views
integral of complex logarithm
Consider the integral
$$I=\int_0^{2\pi}\log\left|re^{it}-a\right|\,dt$$
where $a$ is a complex number and $0<r<|a|$. We have
...
2
votes
1answer
137 views
Riemann Lebesgue Lemma for polynomial?
I was asked to prove that
$$\lim_{n\to\infty} \int_{0}^{1} \exp(i\cdot n\cdot p(x))\;dx =0 $$
for nonconstant real polynomial $p(x)$.
if $p(x)$ is of degree $1$... It reduces to Riemann-Lebesgue ...
1
vote
4answers
199 views
Integrate $\int_0^\infty \frac{\sqrt{x}}{x^{2}+1}\, \mbox{d} x$
I've been trying to integrate the following
$$\int_{0}^{\infty} \frac{\sqrt{x}}{x^{2}+1} \mbox{d} x$$
on half an annulus in the upper half plane. I keep getting $\frac{\pi}{\sqrt{2}}\ i$, which ...
1
vote
3answers
165 views
Equality of absolute values of complex integrals
It was pretty hard finding a short and precise title, heres my problem:
The equation $$\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$$holds true if f is integratable (where ...
1
vote
1answer
58 views
Need help integrating $\tan x$ and $\tan^n x$ using reduction
I have tried to use integration by parts taking $u$ as $\tan x$ and $v$ as $1$:
$$\int \tan x \,dx = \int \tan x \cdot 1\; dx = \tan x \cdot x - \int \sec^2 x \cdot x\; dx$$
then by taking $u$ as ...
1
vote
1answer
37 views
Complex analysis contour integral
I am working on the integral $\displaystyle\int_0^{\infty}\frac{\log(x)}{x^2-1}$. I see it done here $\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint. but I am wondering if it is possible to ...
1
vote
1answer
67 views
Finding the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis
Trying to find the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis, I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $ d\theta=\dfrac{dz}{iz}$. I am trying ...
1
vote
1answer
131 views
How to solve using Cauchy Integral formula?
Let $C$ be the positively oriented boundary of the square whose sides lie along the lines $x=+/-2$ and $y=+/-2$. I am supposed to use the Cauchy Integral formula to evaluate $$\int_C ...
1
vote
2answers
72 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? $3$?
1
vote
1answer
110 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
$$
...
1
vote
1answer
62 views
Finding $\int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz$.
I'd really like some help with this problem. I'm supposed to find $$ \int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz,$$ where $B_2(0) = \{ z \in \mathbb{C} \; | \; |z|<2 \}$ (ie. the ball of radius ...
1
vote
1answer
87 views
Integrating squared absolute value of a complex sequence
I was reading through my book in complex analysis and i encountered this problem.
Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R.
We are asked to show ...
1
vote
1answer
149 views
line integral versus complex integral
Let $a\in \mathbb C, r>0$ and $\gamma_r=\partial D(0,r)$. I want to evaluate the following line integral
$$I=\int_{\gamma_r}\frac{1}{|z-a|^2}ds.$$
I'm looking for a complex function $g(z)$ such ...
1
vote
0answers
73 views
Laplace transform of a product of functions
While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form:
...
1
vote
0answers
158 views
Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0
The ultimate aim is to solve the following integral:
\begin{equation}
\label{eq:Icos1}
\begin{aligned}
I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
0
votes
2answers
100 views
a question about Cauchy integral formula
I'm new in the complex analysis and I'm stuck with this integral :
$I=\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{(z^2+9)(z+9)} $
the exercise is about Cauchy integral, I don't want the whole ...
0
votes
2answers
102 views
Show the value of a complex integral is independent of R for R > 1
Question: Show that for R > 1 $$\int_{|z|=1} \frac{z^{2011}}{2z^{2012}-1} dz = \int_{|z|=R} \frac{z^{2011}}{2z^{2012}-1} dz$$
Thoughts thus far: (i) I know that we cannot use Cauchy's integral ...
0
votes
6answers
93 views
Integration issue
I am trying to solve $\int^{+\infty}_{-\infty}\frac{1}{x}dx $.
I read that it is a contour integral along the semi-circle of large radius in the lower complex plane. First, is there any justification ...
0
votes
1answer
76 views
How to do complex integration. E.g. $ \int_\frac{\pi}{2}^{\frac{\pi}{2} + i} \cos(2z) \; \mathrm{d}z $
For my homework assignment I've been given a number of complex integrals to solve. I've already asked for help on a specific example here, but I was somewhat dissatisfied with the answers. The answers ...
0
votes
1answer
27 views
Cauchy integral formula and holomorphic functions
I am stuck in a problem about holomorphic functions and using of Cauchy integral formula. I really have no idea how to start, so i would be glad if somebody could help me with it.
Let $C=C(0,1)$ a ...
0
votes
0answers
31 views
Solving an complex Integration with complex exp and other terms
I am trying to solve a partial differential equation and while solving I need to solve the following integral. If anyone could help me solve this integral that would be great.
$$y(x,t) = \int_{c-i ...
0
votes
1answer
59 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 $(\overline{z})^n$ ...
0
votes
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:
$$I=2\pi i \left\{ ...
0
votes
1answer
126 views
Problems in interpreting an integral that should be solved with residue method
Usually, when I solve an integral using residue method, I find real functions as integrands.
I am not able to provide an interpretation for the following complex integral
$$
\int_{-\infty}^{\infty} ...
0
votes
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 ...
0
votes
1answer
85 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 ...
0
votes
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 ...
