Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

learn more… | top users | synonyms

1
vote
1answer
141 views

Integrating $\sin(x)/x$, how to treat the pole at the origin? [duplicate]

I want to use residue theory to integrate $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ What would be a good contour to use? I plan to take the imaginary part of this integral: $$\int \frac ...
0
votes
1answer
46 views

residue theorem to evaluate an integral

I have encountered a problem; Use Residue theorem to evaluate $\displaystyle\int_{-\infty}^{\infty} \frac{\sin z}{(z^2+4)(z^2 -2z +2)} \, dz$ How is this done?
0
votes
1answer
74 views

Determine poles and residues of contour integral using Laurent series

I want to find the residues of the integral $F = \int_{-\infty}^{\infty} \dfrac{1}{x+(a-ib)} \dfrac{1}{\exp(-x/c)-1} dx$ I know that $x=-(a-ib)$ is a simple pole which contributes a non-zero ...
4
votes
0answers
68 views

inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since ...
2
votes
2answers
68 views

Inverse Laplace transform of an exponential function

What is the inverse Laplace transform of $$\frac{e^{\frac{-2}{s}}}{s}$$ I have seen an answer using Maclaurin series expansion of this function. This function is not analytic at $0$, so, is such ...
1
vote
1answer
60 views

how to integrate the definite integral using residue theorem? [duplicate]

How to evaluate $\int_{0}^{\infty}\dfrac{1}{x^a+1}dx$, where $a>1$. I don't know where to start since $x^a+1$ could have infinitely many roots, then it is impossible(?) to evaluate its residues. ...
2
votes
2answers
46 views

$\int_{C_N} \frac{dz}{z^2\sin(z)}$ complex integral, problem with residues

Let $C_n$ be the rectangle, positively oriented, which sides are in the lines $$x=\pm(N+\dfrac{1}{2})\pi~~~y=\pm(N+\dfrac{1}{2})\pi$$ with $N\in\mathbb{N}$. Prove that $$ ...
1
vote
1answer
74 views

Integration $\int_0^{2\pi} \frac{\cos^2 3\theta d\theta}{5-4\cos2\theta}$ by using residues

$$\int_0^{2\pi} \frac{\cos^2 3\theta d\theta}{5-4\cos2\theta}$$ By substituting $\cos m\theta$ to $\frac{z^m+z^{-m}}{2}$ and $d\theta$ to $\frac{-i}{z}dz$,I get $$\int_0^{2\pi} \frac{\cos^2 ...
3
votes
1answer
53 views

Find $ \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$.

Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$. My Try: So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq ...
0
votes
1answer
26 views

Calculating $\int \limits _{\gamma_r}\frac{e^{iz}-1}{z^2}dz$

I don't understand the following example. The second term on the right-hand side is $\pi$, since $$\lim \limits _{r \to 0} \int \limits _{\gamma _r} \frac {\Bbb e ^{\Bbb i z} - 1} {z^2} \Bbb d z = ...
1
vote
1answer
67 views

Singularities of $\sin(z)/(1-\cos(\sqrt{z}\,))$

$\displaystyle f(z) = \frac{\sin(z)}{1-\cos(\sqrt{z}\,)}$. The assignment is to find all the singularities of $f$, determine the type of them and the residue. It is clear that the singularities are ...
1
vote
0answers
30 views

How to compute this integral on contour

How to compute this following integral? $$\int \limits_{C}^{}\frac{e^{az^2}\,dz}{z^4+1}$$ Given $ a>0$ and $C:=\{z: |z+1| = 1\}$ is positively oriented. Where should I start? What would ...
0
votes
0answers
49 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + ...
0
votes
3answers
67 views

Integrate $\oint\frac{z}{\cos z-1}dz$ with residue theorem

$$\oint\limits_{|z-3|=4}^{}\frac{z}{\cos z-1}dz$$ My attempt: $$\cos z=1$$ $$z=2\pi k$$ The set includes only $z=0$ and $z=2\pi$. What next?
-1
votes
1answer
25 views

the residue at the singular point

We need to find residue $\frac{1}{\cos^2z}$ $\cos z=0$ $z=\frac{\pi}{2}+\pi k$ - order 2 poles as the next?
0
votes
0answers
20 views

rational function of complex polynomials can be uniquely written as: $R(z)=P(z)+\sum\limits_{i=1}^n\sum\limits_{j=1}^{r_i}\frac{a_{ij}}{(z-z_i)^j},$

Let $R(z)$ be a rational function of complex polynomials, i.e. $R(z)=\frac{f(z)}{g(z)}$ with $f(z),g(z)\in\mathbb{C}[z]$. Claim: $R$ can be uniquely written ...
0
votes
0answers
23 views

Residue of $\frac{\text{cot}(\pi z)}{z^6}$ at $0$

I am trying to compute $\zeta(6)$ = $\sum_1^{\infty} \frac{1}{n^6}$; I generally know how to do this using a residue-based proof, but I am stuck at the last bit, namely calculating the residue of ...
0
votes
3answers
118 views

Very tricky complex integral, with poles on both sides of the real line,

I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...
1
vote
0answers
17 views

Evaluate the integral $(x+1)/(x^2+2)^2$ by choosing an appropriate contour in the upper half plane

How do you solve this question? Evaluate the integral $(x+1)/(x^2+2)^2$ by choosing an appropriate contour in the upper half plane How would the answer change if this question was evaluated with the ...
1
vote
1answer
32 views

finding residues for poles

I'm struggling to find the residues of the equation $$ \frac{-z\ln(z)}{(z^2+a^2)(2-z)} $$ with poles at $z=\pm ai$ and $z=2$ I have the residue for $z=2$ as $$ \frac{-2\ln2}{4+a^2} $$ but I am ...
1
vote
1answer
51 views

Inverse Laplace transform seems to be always vanishing but it couldn't!

Let's consider $x\in (0,1)$ and the distribution $p(x)=\lambda x^\lambda$, $\lambda>0$. I would like to find the pdf of the sum. The characteristic function of the $N$ sum reads: \begin{equation} ...
0
votes
1answer
17 views

Calculate $\int_{C}\frac{e^{z+\frac{1}{z}}}{1-z^2}$

Calculate $$\int_{C}\frac{e^{z+\frac{1}{z}}}{1-z^2}$$ Where $C=\{|z|=2\}$ Ok so if I write $f(z)=\frac{e^{z+\frac{1}{z}}}{1-z^2}=\frac{e^z}{1-z^2}\cdot e^{\frac{1}{z}}$ Then $f(z)$ has an ...
2
votes
0answers
32 views

Integral of the principal value of a hypergeometric function

I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where $z=-\alpha/\beta$ and ...
0
votes
1answer
32 views

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one.

Show that the polynomial $p(z)=z^5+7z-1$ (in $\Bbb{C}[z]$) has one real root with absolute value smaller than one and that the rest of the roots are in $\{1<|z|<2\}$. Ok, so this exercise ...
0
votes
1answer
36 views

finding residue for complex analysis

I am having a tough time finding the residue for a function, suppose my test function is $$\frac{z^2}{{(z^2+a^2)}^2}$$ while I could determine the poles to be $+-ai$ and I know the formula to find ...
0
votes
0answers
49 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
1
vote
1answer
67 views

contour integral branch cut

I need some help to solve the following integral by contour integration. $$\int_{0}^{1} x^a (1-x)^{1-a}\,\mathrm{d}x$$ I attached my ideas and a picture of the paths to fix the labels. Kind ...
4
votes
1answer
152 views

Evaluating the integral $ \int_{-1}^{1} \frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$

I've been trying to find a way to integrate $\int_{-1}^{1}\frac{1}{(1+x^{2})(1-x^{2})^{1/4}}dx$ using contour integration, but I'm having a hard time coming up with a contour to use. Since I have a ...
0
votes
2answers
34 views

Using the residue theorem

Is it possible to evaluate $$\int_{-\infty}^\infty \frac{x^2}{(x^2+1)^2} \, dx $$ using the residue theorem, as opposed to Calc 1 methods? How can I get started using the residue theorem? What ...
0
votes
1answer
57 views

how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
1
vote
1answer
47 views

Evaluate this integral using residue theorem

So we have $\int_{0}^{+\infty}\dfrac{x^2-a^2}{x^2+a^2}\cdot\dfrac{\sin x}{x}dx$. ($a>0$). I considered that we can just calculate the half of the imaginary part of ...
0
votes
0answers
40 views

logarithmic singularities in contour integration

How to evaluate the contour integral using the residue theorem if there is a logarithmic derivative? For example this: $$\int_C \log\zeta(s)\frac{x^s}{s} ds$$ or even this: $$ \int_C \frac{\log ...
0
votes
1answer
36 views

Integration involving complex singular function

I am stuck with the following integral that came up during my research and I am not sure how to correctly evaluate this expression. $\int_{-\infty}^{\infty} dk\left[\Im[ ...
1
vote
1answer
60 views

Evaluate the following integral $\int_{-1/2}^{1/2}\big(\frac{\sin(n\pi f)}{\sin(\pi f)}\big)^4 df$

There are similar questions out there, but I was hoping someone could show how to would evaluate the following integral $$\int_{-1/2}^{1/2}\bigg(\frac{\sin(n\pi f)}{\sin(\pi f)}\bigg)^4 df$$ I've ...
0
votes
0answers
30 views

Limit inside countour integral that depends on limit

Suppose we have a sequence of functions such that: $\lim_{r \to \infty} f_r(x) = f(x)$ uniformly. Now, my question: is it possible to take the limit inside a countour integral (or a sum) which is ...
2
votes
1answer
65 views

Another combinatorial identity of McKay

Suppose $v\ge 2$ and $s\ge 1$ are integers. I'm stuck trying to show that $$ v\sum_{k=0}^{s-1} \binom{2s}{k} \frac{s-k}{s}(v-1)^k = \sum_{k=0}^s \binom{2s}{k} \frac{2s-2k+1}{2s-k+1}(v-1)^k $$ I've ...
1
vote
1answer
47 views

Would a keyhole contour be advisable to use for this integration?

The integral is $$\int_0^{\infty}\frac {1}{\sqrt{x}(1+x^2)}dx$$ which is to be evaluated by contour integration. So, the integrand clearly has simple poles at $+/- i$. But what kind of pole ...
0
votes
0answers
49 views

Integrate $\int_{0}^{\infty}\frac{x^a\ln(x)}{(x+b)}dx$ by the method of residues

How to evaluate $$\int_{0}^{\infty}\frac{x^a\ln(x)}{(x+b)}dx$$ where $b > 0$ and $-1 < a < 0$ using the method of residues, but I have done only problems of simple poles, but this is much ...
0
votes
1answer
141 views

Evaluating $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ with a rectangular contour

I need to try to evaluate $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ and it seems like this is supposed to be done using some sort of rectangular contour based on looking at other questions. My ...
2
votes
3answers
100 views

Evaluate the integral of function involving $\cosh$

Evaluate the integral $$ \int_0^{\infty} \frac{\cosh(ax)}{\cosh(x)}\,dx, $$ where $|a|<1$. Consider the closed loop integral of $\displaystyle\frac{e^{az}}{\cosh(z)}$ where the contour $C$ is ...
3
votes
1answer
49 views

Calculate infinite sum with residues

I'm trying to use the residue theorem to calculate $$\sum_{k=1}^{\infty} \dfrac{1}{(2k-1)^2}. $$ I came up with $\operatorname{Res}\left(\dfrac{\pi \cot(\pi z)}{(2z-1)^2},\frac12\right)=-\pi^2$ and ...
0
votes
0answers
41 views

Cauchy Residue Theorem Question

I am finding difficulty with the following couple of problems a) show that $$\int_{-\infty}^\infty \frac{\cos(\pi x)}{2x-1} \ne\frac{\pi}{2}$$ I've been trying to do this over a semi-circle, but ...
0
votes
1answer
32 views

Reversing the direction of a contour integral

If $$\int_{C} f(z) dz$$ is some contour integral over a closed curve $C$, and $-C$ is the contour taken in the opposite direction, can $$ \int_{-C} f(z) dz$$ be treated as a closed curve around the ...
0
votes
1answer
22 views

Cauchy's residue application problem

I would like to know what did I do wrong. There's my problem : I= $\frac{1}{2\pi i}$ $\int_a \frac{1}{z^4+1}~dz$ Where a $x^2 + y^2 = 2x$ I already know: there're 4 poles, but only 2 fits for me ...
0
votes
2answers
53 views

How to evaluate the contour integral $\int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz$ over the unit circle?

Let $C(0, 1)$ be the unit circle centered at the origin with radius $1$. Then I need to evaluate the following complex contour integral: $$ \int_{C(0,1)} \frac{z e^z }{\tan^2 z}dz$$ I know the ...
2
votes
1answer
89 views

Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
0
votes
0answers
25 views

Find $\int_0^{2\pi} \frac {dx}{a + \sin^2(x)}$ using method of residues.

I want to find $\int_0^{2\pi} \frac {dx}{a + \sin^2(x)}, a > 0$ using method of residues. My Attempt: First I deduced that $$\int_0^{\pi/2}\frac {dx}{a + sin^2(x)} = \int_0^{\pi}\frac {dt}{(2a ...
0
votes
0answers
69 views

Contour integral of $\int_{0}^\infty \frac{\sinh(kx)}{\sinh(x)}dx = \frac{1}{2}\tan{\frac{a}{2}}$

From Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ In the case of zero $\omega$ and integral starts as 0, how do I prove that using contour integral $\int_{0}^\infty ...
1
vote
1answer
38 views

Bounding a function

I'm trying to solve for $\sum_{0}^{\infty} \dfrac{1}{n^2}$ using the residue theorem. The integral in question is $\int_C f(z)\pi \cot(\pi z) dz$ where $f(z)=\dfrac{1}{z^2}$. I am bounding over the ...
1
vote
0answers
37 views

Complex residue at infinity of $f(z)=\frac{z^5}{\sin\left(\frac{1}{z^2}\right)}$

I'm having trouble finding residue of the function $$f(z)=\frac{z^5}{\sin\left(\frac{1}{\large{z^2}}\right)}$$ at infinity. Wolfram kindly informs that it is equal to $-\frac{7}{360}$ (and gives ...