Questions on the evaluation of integrals along a locus in the complex plane.

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3
votes
4answers
159 views

Evaluating a complex contour

I need to show the following result: $$ \int_{-\infty}^\infty \frac{1}{(1+x^2)^{n+1}}dx\, = \frac{1\cdot 3\cdot\ldots\cdot(2n-1)}{2\cdot 4\cdot\ldots\cdot(2n)}\pi $$ With n=1,2,3,... This function ...
3
votes
2answers
148 views

Integral $\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$

Hi I am trying to calculate, $$ \int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx $$ where $a,b$ are positive real constants. I Know $\ln(xy)=\ln x +\ln y$, but I do not ...
3
votes
2answers
128 views

Does $\lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0$?

Let $C_{R}$ be the upper half of the circle $|z|= R$. Does $ \displaystyle \lim_{R \to \infty} \int_{C_{R}} e^{iz} \ dz \to 0 $? Jordan's lemma is not applicable here. And I'm not sure how to get a ...
3
votes
2answers
62 views

what is the proper contour for $\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz$

what is the proper contour for $$\int_{-\infty}^{\infty}\frac{e^z}{1+e^{nz}}dz:2\leq n$$ I tried with rectangle contour but the problem which I faced how to make the contour contain all branches point ...
3
votes
3answers
93 views

Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues

Find $I:=\lim\limits_{R\to \infty}\int\limits_{-R}^R \frac{x \sin(3x)}{x ^2+4}dx$ using residues. Let $f(z)= \frac{z \sin(3z)}{z ^2+4}$. First define two contours: $$\Gamma_1: z=t \text{ where } ...
3
votes
1answer
50 views

Complex integral of $\frac{\cos x}{x^2+4} $

I want to evaluate: $$ \int_{-\infty}^{\infty}\frac{\cos x}{x^2 +4} dx $$ Using wolfram alpha, it gave an answer of $\frac{\pi}{2e^2}$. Wolfram Alpha is never wrong. Attempt $$ ...
3
votes
1answer
90 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
3
votes
2answers
106 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?
3
votes
2answers
151 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
3
votes
1answer
193 views

Evaluating $\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4 \right)}dx$

How can we evaluate $$\int_0^\infty \frac{\cos(ax)-e^{-ax}}{x \left(x^4+b^4\right)}dx \quad a,b>0$$ using Complex Analysis? This problem was given in a Complex Analysis book which I was reading. ...
3
votes
4answers
140 views

Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$

I want to compute $$ \int_{-\infty}^\infty \frac 1{ (1+x^2)^{n+1}} dx $$ for $n \in \mathbb N_{\geq 1}$. If I let $$ f(z) := \frac 1 {(z+i)^{n+1}(z-i)^{n+1}} $$ then I see that $f$ has poles of ...
3
votes
2answers
117 views

Complex integration help

The integral given is $$\int_{-\infty}^{\infty} \frac{\cos(x)-1}{x^2}\,dx $$ Ok, so, I've used the upper semi circular contour with the function $$f(z) = \frac{e^{iz}-1}{z^2}$$ Now the residue I ...
3
votes
3answers
531 views

contour integral computations

Let $C$ be the boundary of the square whose vertices are $1+i$, $1-i$, $-1 + i$ and $-1 -i$. Suppose that $C$ is oriented counterclockwise. How to compute a) $$\int_C \frac{e^z}{z-1/2} \, dz$$ b) ...
3
votes
2answers
55 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
3
votes
1answer
109 views

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

Prove the following using complex analysis $$\tag{1}\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx=\frac{\pi}{2}\ln(2)$$ I found this problem in Schaum's outlines of complex variables. I thought that we ...
3
votes
2answers
134 views

Integral $\int_0^\infty e^{imx^2}dx$

In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral), $$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int ...
3
votes
3answers
82 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 ...
3
votes
1answer
127 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 ...
3
votes
3answers
217 views

Are the integration contours of this improper integral properly selected?

I have been recently trying to review some topics on improper integrals. The Integral I am trying to solve is: $$ \int_0^\infty {log(x) \over x^2 -1} dx $$ The branch cut of the $log(x)$ is ...
3
votes
3answers
127 views

Evaluating $\int_{0}^{\infty} \frac{1}{1+x^{3/2}}\,\textrm{d}x$

I'm trying to evaluate this integral using contour integration (over a Riemann surface), but I'm stuck at the step where I need to calculate the residues. The roots of $1+z^{3/2}$ are $1$ and ...
3
votes
1answer
432 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
2answers
385 views

Complex analysis: contour integration

Evaluate by contour integral: $$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$ Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
3
votes
1answer
102 views

contour integration of $f(z) = z^{a-1}(z+z^{-1})^{b-1}$

I want to evaluate $\displaystyle f(z)= z^{a-1} \left(z + z^{-1}\right)^{b} \ (a >b >-1)$ counterclockwise around the right half of the circle $|z|=1$. So I close the contour with the vertical ...
3
votes
1answer
58 views

Contour integration of $\frac{(\ln z)^2}{z^2+1} $

I'm supposed to take the principal branch of $\ln z$ and evaluate this integral: $$ \oint \frac{(\ln z)^2}{z^2+1} $$ Attempt I suppose the integral they are talking about is something like ...
3
votes
2answers
173 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
3
votes
1answer
130 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
128 views

How to find this integral using Cauchy integral formula

How to obtain that $$\int\limits_{|z|=r} (\bar{z})^{-m} z^{-n-1}\, dz = \begin{cases} 2\pi ir^{-2m} &\text{if}\,\,n=m, \\ 0 &\text{if}\,\,n \neq m, \end{cases}$$ for $r>0$. I suppose I ...
3
votes
1answer
206 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
3
votes
1answer
41 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
3
votes
3answers
214 views

Evaluating an improper integral that involves $\exp(-|x|)$

I am trying to prove that the function $f:\mathbb C\setminus\mathbb R\rightarrow\mathbb C$ defined by $$ f(z) := \frac{1}{2\pi i}\int_{-\infty}^\infty\frac{\exp(-|x|)}{x-z}dx $$ is holomorphic. I ...
3
votes
1answer
718 views

Inverse Laplace transform using contour integration

I want to show by contour integration that $\displaystyle\mathcal{L}^{-1} \{\text{arccot}(s) \}(t)= \frac{\sin t\ }{t}$. In other words, I want to evaluate $\displaystyle \frac{1}{2 \pi i} \int_{a - ...
3
votes
1answer
236 views

Cauchy principal value of $\int_{-\infty}^{\infty} \frac{1}{x^{3}} \ dx $

By definition $\displaystyle\text{PV} \int_{-\infty}^{\infty} \frac{1}{x^{3}} \ dx = \lim_{\epsilon \to 0 ,\ a \to \infty} \Big(\int_{-a}^{0 - \epsilon} \frac{1}{x^{3}} \ dx + \int_{o+ \epsilon}^{a} ...
3
votes
2answers
636 views

contour integration and branch points

An exercise in a textbook says to evaluate $\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos (ax) \cos^{b} (x) \ d x \ (a > b > -1)$ by letting $\displaystyle f(z) = z^{a-1} ...
3
votes
2answers
50 views

find $\int _\gamma \frac{1}{z+\frac {1}{2}}dz$

I'm asked to find $$\int _\gamma \frac{1}{z+\frac {1}{2}}$$ where $\gamma (t)=e^{it}, 0\leq t\leq 2\pi$. To do this I deal with two different logarithms, one without the negative imaginary axis ...
3
votes
1answer
241 views

contour integration of logarithm function

I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem: $$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$ where $\mathcal{C}$ is a closed square ...
3
votes
1answer
312 views

Contour Integration: $\int_0^\infty\frac{1}{x^a(1-x)}\,dx$ for $0<a<1$.

I've been trying to calculate $$\int_0^\infty\frac{1}{x^a(1-x)}\,dx\quad\text{with }0<a<1.$$I haven't had much luck. I tried taking the branch cut with of the positive reals and estimating that ...
3
votes
2answers
183 views

Contour question in complex analysis

Let $C_1$ be the line segment from $-1-i$ to $3-i$, and $C_2$ be the portion of the parabola $x=y^2+2y$ joining the above points $-1-i$ and $3-i$. Show that $$\int_{C_1}zdz=\int_{C_2}zdz=4+2i.$$ So ...
3
votes
3answers
267 views

Find the residues of singularities of the following function:

Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$ and $\gamma_R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and the straight line $\gamma_2$ from ...
3
votes
2answers
306 views

Evaluate the following contour integral…

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t\leq 2\pi$. Evaluate: $$\int_{\gamma(0,1)} \frac{\cos(z)}{z^2}dz$$
3
votes
3answers
141 views

Calculate $\int_{-\infty}^{+\infty}e^{-\frac{(x-it)^2}{2}}dx$ using contour integration

When one wants to calculate the characteristic function of a random variable which is of normal distribution, things boil down to calculate: $$\int_{-\infty}^{+\infty}e^{-\frac{(x-it)^2}{2}}dx$$ There ...
3
votes
2answers
347 views

Contour Integral Question

I'm working through a contour integral question, which is rounded off by finding the integral: $$\int^{\infty}_{0} \frac{x-\sin(x)}{x^3} dx$$ I have already shown that the residue at $0$ of the ...
3
votes
2answers
310 views

Finding $\frac{1}{2\pi}\int_{0}^{2\pi} \cos^{2n} x dx$

I have a question that asks me to find the value of $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi} \cos^{2n} x dx$ $\ $ by considering the integral $$ \displaystyle \oint_{\gamma} \frac{1}{z}\left ( ...
3
votes
2answers
606 views

Proper application of Cauchy integral formula

I'm being asked to integrate $f(z) = \frac{e^z}{z^2 + 2z + 1}$ around a $5 \times 5$ square, centered at $i$, in the counter clockwise direction. It seems to me that applying Cauchy's integral formula ...
3
votes
1answer
56 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 $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
3
votes
1answer
31 views

Quick question on poles

Consider this function for $0 < a < b$: $$f_{(z)} = \frac{z^4}{z^2(z-\frac{a}{b})(z-\frac{b}{a})}$$ This function has a pole of order $2$ at $z=0$, a pole of order 1 at $z=\frac{a}{b}$, but ...
3
votes
1answer
61 views

Complex contour integral and partial fractions

I'm doing complex integration and I'm trying to evaluate: $$\int_C \frac{\cos{z}}{z^2 + 1} dz$$ Where $C$ is the clockwise boundary of a parallelogram with vertices $3i$, $2$, $-3i$, $-2$ (i.e. a ...
3
votes
1answer
84 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
3
votes
1answer
66 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}}$ ...
3
votes
2answers
114 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 / ...
3
votes
1answer
62 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 ...