For questions about integration methods that use results from complex analysis and their applications

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21
votes
2answers
685 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 ...
15
votes
7answers
2k views

Integration by means of complex analysis

Dear all: this time I have the integral $$\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$$and we must try to solve it using complex integration, residues, Cauchy's Theorem and the whole lot. (BTW, does ...
10
votes
4answers
315 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 ...
9
votes
5answers
497 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 $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
8
votes
1answer
99 views

Analytic function $f$ in $\overline{\mathbb{D}}$ satisfying $\left\lvert\,f'(\tfrac{1}{2})\,\right \rvert\leq 8.$

Let $f$ be an analytic function on the closed unit disk $\overline{\mathbb{D}}$. On its boundary $\partial \mathbb{D}$ it holds that $\vert\,f(z) -z\rvert < \lvert z\rvert$. I now have to show ...
7
votes
2answers
145 views

Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
7
votes
1answer
287 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
7
votes
0answers
63 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
6
votes
3answers
151 views

Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
6
votes
2answers
98 views

The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$

How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$. I can easily take the residue in ...
5
votes
3answers
317 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.
5
votes
3answers
149 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
5
votes
2answers
148 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 ...
5
votes
2answers
118 views

Why do we need a branch cut for $\int_0^{\infty} \frac{x^{\frac{1}{2}}}{{(1 + x)^2}}dx$?

What is the significance of the $x^{\frac{1}{2}}$ in the numerator of this integral. I have read this kind of integral requires taking a branch cut. Why do we need a branch cut, what does it enable us ...
5
votes
1answer
81 views

Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $ \deg Q - \deg P > 2$. Is it possible to have a general formula for ...
5
votes
1answer
239 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
5
votes
1answer
50 views

Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$

I tried to evaluate $$\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$$ when $r>0$. Clearly $\log z$ is not continuous at $z=-r$. So is this integral meaningful then? Since the function is continuous and ...
5
votes
2answers
67 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
5
votes
1answer
119 views

How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
4
votes
4answers
182 views

Intuitive reason for why many complex integrals vanish when the path is “blown-up”?

It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along ...
4
votes
2answers
151 views

Complex integration, any ideas?

I need to solve the following integral of a function analytic on an open environmemt of the unit circle, but i dont know how to get that answers $$\frac{1}{2\pi ...
4
votes
2answers
46 views

Show that $\int_{|z|=1} \frac{e^z}{z^k} dz = \frac{2\pi i}{(k-1)!}$

I'm supposed to show that $$\int_{|z|=1} \frac{e^z}{z^k} dz = \frac{2\pi i}{(k-1)!}$$ where $|z|=1$ is traversed counterclockwise and $k>0$. We can parametrize this path as $\gamma(t)=e^{it}$ for ...
4
votes
2answers
396 views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
4
votes
1answer
117 views

The substitution $y = ix$

Given $\int_0^\infty f(x) dx$, under what conditions would the substitution $y = ix$ not change the limits of integration, so to speak? Since setting $y = ix$ changes the range of integration to a ...
4
votes
2answers
306 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}. ...
4
votes
1answer
149 views

How do you integrate $\int_0^\infty \exp(it^k)\,\mathrm dt$ for $k \in \Bbb N$?

My problem is with the integral $$\int^\infty_0 e^{it^k}\,\mathrm dt$$ with $k\in\mathbb{N}$. Somehow it can be evaluated by use of Cauchy's theorem. But I don't see how. The best thing I can ...
4
votes
2answers
407 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 $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
4
votes
4answers
342 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. ...
4
votes
2answers
281 views

Cauchy Integral formula question

$$\int_{\gamma=(i,1)} \frac{z^3}{(z-i)^n} dz$$ for any $n\in\mathbb{N}$. Can someone please help me answer this question as I cannot seem to get the right answer! Please note that the Cauchy ...
4
votes
1answer
87 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
3
votes
2answers
150 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
72 views

Integrating $I(\alpha)=\int^{\infty}_{0} \frac{x^{\alpha}}{x^4+1}dx$

Here is the question: Let $P(x)$ be a polynomial of degree $d>1$ with $P(x)>0$ for all $x>0$. For what values of $\alpha \in \mathbb{R}$ does the integral $I(\alpha)=\int^{\infty}_{0} ...
3
votes
2answers
150 views

Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
3
votes
1answer
122 views

Using argument principle to compute an integral

Let $f(z)=z^4-2z^3+z^2-12z+20$. Then evaluate the integral by using the argument principle $$\oint_C \frac{zf'(z)}{f(z)} \,ds$$ Where $C$ is the circle $|z|=5$. What I've tried: I tried using the ...
3
votes
2answers
53 views

doubt in complex integration

I was doing problems on complex integration and got stuck at one question. The question is $$ \int_{\gamma}{{\rm e}^{{\rm i}\pi z}\left(z + i\right)^{2}\cos\left(nz\right) ...
3
votes
1answer
580 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 ...
3
votes
1answer
252 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
368 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
43 views

Is this Integral calculation correct?

Can someone confirm if my solution is right or if I have done something that is not permitted $$ \begin{align} & \int_\gamma e^{\pi z}=\int_\gamma \left( \frac{ e^{\pi z}}{\pi}\right)' \, dz ...
3
votes
1answer
181 views

How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over ...
3
votes
1answer
27 views

Analytic function with values at specific points

Problem: Show that there is no function $f$, holomorphic on a neighbourhood of $z=0$, such that its value on $z=\frac{1}{n}$ is $(-1)^n (\frac{1}{n})$. To contrast, find a function holomorphic ...
3
votes
1answer
172 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
3
votes
2answers
46 views

How to integrate $\int_{0}^{\infty} \frac{dx}{\sqrt{x}(x^{2}+1)}$ using the residue theorem.

He was doing this integral using the formula $$\int_{0}^{\infty} \frac{dx}{\sqrt{x}(x^{2}+1)}= \frac{2\pi i}{1-e^{-2\pi i\alpha}}(\sum(Res(\frac{F(z)}{z^{\alpha}};z_{k})))$$ where ...
3
votes
3answers
91 views

How can an improper integral have multiple values?

Integrals like this are said to dependend on the contour of integration: $$\int^{\infty}_{-\infty}\frac{x\sin x}{x^2-\sigma^2}dx=\pi e^{i\sigma}\space \mathrm{or}\quad \pi \cos\sigma $$ How is it ...
3
votes
1answer
95 views

problem about complex integration

The question is to find $\displaystyle\int \frac{z^2-z+1}{z-1}dz$ over $|z|=1$. My solution is : Using cauchy's integral formula we have $\displaystyle f(1) = \frac{1}{2\pi i}\int ...
3
votes
1answer
53 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 ...
3
votes
1answer
80 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
3
votes
1answer
60 views

Compute $\int_{\gamma} \frac{z}{z^3-1} dz$ where $\gamma$ is circle centered at origin of radius 2

Compute $$\int_{\gamma} \frac{z}{z^3-1} dz$$ where $\gamma(t)=2e^{it}$, $t\in[0,2\pi]$. The first part of the problem had me compute the same integral over the path $\gamma(t)=\frac{1}{2} ...
3
votes
2answers
79 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
3
votes
1answer
72 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 $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...