5
votes
2answers
93 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
1
vote
1answer
43 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
1
vote
1answer
29 views

Contour integral of $\frac{\bar{z}}{z-Z}$ on a square centered at the origin

I am having trouble calculating the following integral: $\oint_C \frac{\bar{z}}{z-Z} dz$ Here, Z is a complex constant and C is the contour of a square of side $2a$ centered at the origin. I ...
0
votes
2answers
13 views

Parametric equations in complex analysis

I am trying to find $\int_C (1+i-2z')dz$ where$z'$ is the conjugate of $z$ and where C is the parabola $y=x^2$ from $z_1=0$ to $z_2=1+i$. How do I write the parametric equations for this?
0
votes
0answers
17 views

Showing $\int_{\gamma}f(z)dz = \int_{\gamma_1}f(z)dz + \int_{\gamma_2}f(z)dz$ with non analytic points.

Suppose $f$ is analytic on the complex plane except at $z_1,z_2$, that $\gamma_1$ and $\gamma_2$ are simple closed curves with $z_1,z_2$ in their interiors and $\gamma_1$ and $\gamma_2$ are in the ...
4
votes
1answer
66 views

Residue theorem with exponential and trig functions

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

How does the integral $\int_{D_C} e^{ia z}P(z)/Q/(z)\,\mathrm{d}z$ blow up.

In my book I have a theorem that goes something like the following Let $P(x)$ be $Q(x)$ polynomials such that $\deg(Q) \geq \deg(P) + 2$. Then \begin{align*} \int_{-\infty}^{\infty} ...
0
votes
2answers
30 views

Complex Analysis - Contour Integration

By considering the integral of the function $f(z) = exp(-az^2)$, with $a > 0$, around an appropriate contour, show that the integral $$ I(p,a) = \int_{\infty+ip}^{\infty+ip} exp(-az^2)dz$$ which ...
0
votes
0answers
47 views

Find the countour integral of $\int_{γ} \sqrt{z} dz$ where $γ=C(2,1)^+$ or $γ=C(1,1)^{+}$ or $γ=C(0,1)^{+}$

Find the countour integral of $\int_{γ} \sqrt{z} dz$ where $γ=C(2,1)^+$ or $γ=C(1,1)^{+}$ or $γ=C(0,1)^{+}$ With $\sqrt{z}$ I mean the branch with the non-positive real axis as branch cut. With ...
2
votes
2answers
186 views

Problem with Cauchy integrals

Hello everybody I need to solve some integral with the help of the Cauchy Integral Formula (CIF). I'll post near each integral the job that I've done and the question that I can't answer. let $\kappa ...
2
votes
1answer
36 views

Three questions concerning holomorphic functions defined by contour integrals

Consider the following situation: a simple, closed, piecewise smooth curve $\gamma$ in the complex plane and $\Omega$ the bounded connected component of the complement of $\gamma$ in $\mathbb{C}$; a ...
4
votes
0answers
74 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 ...
3
votes
3answers
59 views

Contour integration problem

I am to evaluate $\displaystyle\int_0^{\infty} \dfrac{\sin x}{x(x^2+1)}dx$ via contour integration. Now I used an indented semicircular contour, and the parts lying on the real line and the big arc ...
1
vote
1answer
53 views

Calculation of a Residue

Does anyone know of a good way to calculate the residue at zero of the following function? I was able to calculate it with the higher order pole formula for residues and then used Mathematica to find ...
3
votes
1answer
59 views

Integrating around the upper half of $|z|=R$

In a textbook it says that you can show that $ \displaystyle\int_{-\infty}^{\infty} \frac{\cos(x^{2})+\sin(x^{2})-1}{x^{2}} \ dx = 0$ by considering $ \displaystyle f(z) = \frac{e^{iz^{2}}-1}{z^{2}}$ ...
5
votes
4answers
127 views

Integrating $ \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
5
votes
1answer
142 views

Evaluating trigonometric integral using residues

I am trying to evaluate for real positive $\alpha,\beta$ $$\int_{0}^{\infty}\arctan\left(\frac{\alpha}{x}\right)\sin(\beta x)dx$$ using a hint to consider $$\int \log\left(\frac{z+ia}{z}\right) ...
1
vote
2answers
61 views

Infinite sums and integrals using residues

I have no idea how to solve these two, any help? $\mathtt{i)}$ $$\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\frac{e^{tz}}{\sqrt{z+1}}dz$$ $$ a,t\gt0$$ $\mathtt{ii)}$ $$ \sum_{n=1}^\infty ...
3
votes
1answer
70 views

Understanding Dogbone contour example

I am trying to understand example VI on the wikipedia page http://en.wikipedia.org/wiki/Methods_of_contour_integration, but one particular point has mystified me for hours. After it is shown that ...
1
vote
0answers
30 views

Show that $\int_{-\infty}^\infty f(t)dt=0$ where $f\in H^\infty(\mathbb{H})$

The problem is stated as follows: Let $\mathbb{H}$ denote the open upper half plane. Let $f \in H^{\infty}(\mathbb{H})$ Suppose $f$ can be extended to be continuous on $\overline{\mathbb{H}}$ with ...
2
votes
0answers
34 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} ...
0
votes
1answer
31 views

Is there a lower bound for integration of complex functions?

We are given in our book that the upper bound for complex integrations is $|\int_\gamma f(z)\,dz| \leq mL$ where $L$ is the length of $\gamma$ and $m$ is the $\max(|f(z)|: z\in \gamma)$ and were ...
3
votes
2answers
72 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
0
votes
0answers
26 views

Contour Integrals around a square

Compute the following contour integral: $\int_\Gamma\frac{e^{-z}}{z-1}dz$ where $\Gamma$ is the square with sides parallel to the axes, centre 15+$i$25 and side length 5 traversed in the ...
1
vote
1answer
79 views

Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
2
votes
1answer
38 views

Determining asymptotic behavior through generating functions

I need to determine the asymptotic behavior of $$a_n=\sum_{k=2}^{n-2}\frac1{\ln k\ln(n-k)}$$ as $n\to\infty$. I know some elementary methods that might help. For example, split the index $\lvert ...
1
vote
2answers
63 views

Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
0
votes
0answers
32 views

Integration using Cauchy Integral Formula

If $f(z)$ is analytic in $|z-a|<R$ and $0<r<R$, then $f'(a) = \frac{1}{r\pi}\int_{0}^{2pi}F(\theta)e^{-i\theta}\,d\theta$, where F($\theta$) is the real part of $f(a+re^{i\theta})$. I ...
3
votes
2answers
109 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
2
votes
1answer
68 views

Why does the Cauchy-Goursat theorem not apply here?

Let $C$ denote the positively oriented boundary of the half disk $0 \le r \le 1, 0 \le \theta \le \pi$, and let $f(z)$ be a continuous function defined on that half disk by writing $f(0) = 0$ and ...
0
votes
0answers
42 views

Evaluating this complex integral, how?? [duplicate]

Looking through past papers to prepare for my exam and found this: You are asked to integrate this around the following contour: and show that it's equal to: I have found the residue of this ...
0
votes
1answer
38 views

Cauchy Riemann equations, do these satisfy it??

I have this question and am unsure of my approach. I have applied the Cauchy Riemann conditions to it: and found that this condition is true. Is that sufficient and does it make sense?
0
votes
1answer
80 views

Find the residue(s) of this function at each pole that lies in the contour?

Going through past papers and found this residue question I can't do. The question asks you to find the residue at each pole that lies in the contour shown. I've got as my answer for the poles ...
0
votes
1answer
45 views

Complex integrals over an ellipse instead of circle?

I was looking through past papers and found this integral: Which should be evaluated over an ellipse with I've done these plenty of times over a circle with |z| = 2 etc, but where do I start in ...
1
vote
0answers
97 views

evaluate the integral $I =\int_0^{+\infty} e^{ix^2}dx$

"Evaluate the integral $I= \int_{0}^{\infty} e^{ix^{2}}\, dx$. Let R > 0 and consider the closed contour $C_R = C(1)_R + C(2)_R + C(3)_R$ where $C(1)_R$ is the segment of the positive real axis from ...
3
votes
1answer
131 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
1
vote
1answer
63 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
3
votes
1answer
53 views

Find the analytic continuation of the $ f(z) = \int_{0}^{\infty} \frac{exp(-zt)}{1+t^2} dt$

Find the analytic continuation of the function $f(z)$ defined by $ f(z) = \int_{0}^{\infty} \frac{\exp(-zt)}{1+t^2} dt$ , $ |\arg(z)| < \pi/2$ to the domain $ -\pi/2 < \arg(z) < \pi$ I ...
1
vote
1answer
286 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
4
votes
1answer
96 views

Using Cauchy integral formula to calculate $\int_\gamma \frac{\cos{z}}{z^n}$

Let $\gamma(\vartheta)=\mathrm{e}^{i\vartheta},\,\vartheta\in[0,2\pi]$, and consider the integral $$I(n)=\int_\gamma \frac{\cos{z}}{z^n},$$ where $n\in \{0,2,4,6,...\}$. Is there any way to prove ...
3
votes
0answers
55 views

Branch-point order and Cauchy representation

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. Suppose we have the following representation: ...
10
votes
1answer
248 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$$ It appears to be ...
0
votes
1answer
37 views

Finding the complex integral along an arc

How can we evaluate complex expressions like these$\int_C(Z-Z^2)dZ$ where $C$ is the upper half of the circle $|Z-2|=3$
0
votes
1answer
89 views

Very difficult contour integral

I have to compute this integral and I don't have any idea how to get further on: $$\frac{1}{2 \pi i} \int_{\mid z \mid = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7}dz$$ I tried Rouché to maybe ...
6
votes
1answer
90 views

Line contour integral of complex Gaussian

Say I have the entire function $$f(z)=e^{-\frac{1}{2}z^2}.$$ I would like to consider the integral $$I=\int_\Gamma f(x)dz,$$ where $\Gamma$ is a line with negative slope $<1$ in $\mathbb{C}$ (so if ...
6
votes
1answer
103 views

Contour integration of a meromorphic function

Given a meromorphic function $f$ which is uniformly bounded on the upper half plane. Assume that $\int_{-\infty}^{+\infty} f(x)dx$ is absolutely integrable. Then Cauchy's integral theorem suggests ...
3
votes
1answer
270 views

Integrating $\int \frac{e^{ipx}}{(\cos x)^{a}} \frac{dx}{x- \xi}$

Let me quote the passage from the book, and then I'll explain the notation. Let us integrate $$ (i) \ \int \frac{e^{ipx}}{(\cos x)^{a}} \frac{dx}{x- \xi}$$ $$ (ii) \ \int ...
11
votes
1answer
153 views

Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration

EDIT: Instead of expressing the integral as the imaginary part of another integral, I instead expanded $\sin^{3}(x)$ in terms of complex exponentials and I don't run into problems anymore. $$ ...
4
votes
1answer
143 views

Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
0
votes
0answers
53 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with τ, λ, and k all real and ...