0
votes
1answer
41 views

Is there an analytic function defined on $\Bbb C$ except for Gaussian integers where it has poles of order 1 and residue 1?

I need a function defined for all complex variables $z$, except for at all the Gaussian integers, where it has poles of order 1 and residue 1. The function has to be complex-analytic. Can anyone ...
0
votes
1answer
36 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
5
votes
3answers
344 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
0
votes
0answers
56 views

Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0 $ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
4
votes
3answers
74 views

Integral $I(a,b)= P\int_{0}^{\pi}\frac{d\theta}{a-b\cos\theta}$

Hi I am trying to calculate this integral $$ I(a,b)= P\int_{0}^{\pi}\frac{d\theta}{a-b\cos\theta},\quad 0 <a<b,\quad a,b\in \mathbb{R}. $$ We can first write $$ I(a,b)=\frac{1}{2} ...
3
votes
1answer
46 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
4
votes
0answers
90 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
1
vote
1answer
44 views

The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
15
votes
3answers
322 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
0
votes
1answer
46 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
16
votes
2answers
420 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
6
votes
3answers
103 views

calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...
1
vote
2answers
79 views

question on integrals

Let $\displaystyle A=\int_0^1 \frac{dx}{1+x^8}$. Then which of the following are true: 1) $A\lt 1$, 2) $A\gt 1$, ...
0
votes
1answer
97 views

real integrals using residues

How to evaluating this integral using residues where $a>0$: $$\int _0^{\infty }\frac{x^3dx}{x^5-a^5}$$ Any help is appreciated
1
vote
1answer
102 views

Summing a series by using residues

For the same series $$\sum_{n=0}^{\infty}\binom{3n}{2n} x^n$$ I am trying to calculate te sum by using residue theory. At the last line, I need to find the roots of $z^2-(z+1)^3x=0$ and one of ...
3
votes
3answers
116 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
10
votes
3answers
420 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,dx$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ...
4
votes
2answers
395 views

Can you solve this integral with residue theorem

How do you compute $$\int_{-\infty}^\infty \frac{\cos x }{e^x + e^{-x}} dx$$
12
votes
3answers
747 views

A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$

The following question comes from Some integral with sine post $$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$ but now I'd be curious to know how to deal with it by methods of ...
26
votes
4answers
890 views

Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$

I plan to evaluate $$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$ and I need a starting point for both real and complex methods. Thanks ! Sis.
10
votes
3answers
684 views

Evaluation by methods of complex analysis $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$

How would we evaluate the below integral by methods of complex analysis? $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$$ I asked the question a while ago, but at that time I didn't specify this ...
5
votes
3answers
522 views

Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$

Evaluate by complex methods $$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$ Sis.
0
votes
1answer
88 views

A improper integral with series

Show that $$\begin{align} & \int_{-\infty }^{+\infty }\frac{\cos \alpha x}{( \beta^2+x^2 )( ( \beta +1 )^2+x^2 )\cdots ( ( \beta +n )^2+x^2 )} \, \text{d}x = 2\pi \sum\limits_{k=0}^n ( -1 )^k ...
7
votes
5answers
416 views

Definite integral, quotient of logarithm and polynomial

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
9
votes
3answers
905 views

Applications of Residue Theorem in complex analysis?

Does anyone know the applications of Residue Theorem in complex analysis? I would like to do a quick paper on the matter, but am not sure where to start. The residue theorem The residue ...
0
votes
4answers
553 views

Graphing $r =2\sin(2\theta)$

I am calculating some residue calculus stuff, where I need to know if the prescribed poles are inside the curve given above, namely $2\sin(2\theta)$ for $0\leq \theta<2\pi$. I actually need to know ...
1
vote
2answers
151 views

Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?

How do I use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
2
votes
1answer
294 views

Singularities of $ \ \frac{z-1}{z^2 \sin z} \ $

Find all singularities of $ \ \frac{z-1}{z^2 \sin z} \ $ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ...
4
votes
1answer
507 views

Evaluating Integral with Residue Theorem

The integral in question is $$\int_{_C} \frac{z}{z^2+1}\,dz,$$ where $C$ is the path $|z-1| = 3.$ The two pole of $f(x)$ where $f(x)=\frac{z}{z^2+1}$ is $-j$ and $j$ $${\rm ...