Tagged Questions
1
vote
1answer
51 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
10
votes
1answer
114 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
4
votes
1answer
84 views
Integrate: $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
Can this integral be solved with contour integral or by some application of Residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catlan constant}$$
It has two ...
3
votes
2answers
26 views
Problem involving the computation of the following integral
I was solving the past exam papers and stuck on the following problem:
Compute the integral $\displaystyle \oint_{C_1(0)} {e^{1/z}\over z} dz$,where $C_1(0)$ is the circle of radius $1$ around ...
2
votes
1answer
28 views
Evaluate the following contour integral
I was solving old exam papers and I am stuck on the following question:
Evaluate the contour integral $\displaystyle \oint_{C} \frac{dz}{(\bar z-1)^2}$ where $C$ is the semi-circle $|z-1|=1, \Im ...
2
votes
1answer
33 views
Evaluating Complex Line Integrals
Calculate $\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$ and $\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$ when $\gamma$: $|z|=1$ is positively oriented.
This is what I have tried to do, starting ...
0
votes
1answer
45 views
$ \int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx} $
I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct.
$$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$
For this one, ...
3
votes
4answers
76 views
Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?
How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
1
vote
1answer
50 views
Fourier transform of $\frac{1}{1+x^2}$
we know that the Fourier transform of $\frac{1}{1+x^2}$ is $f(y) = \int_{-\infty}^{\infty} \frac{1}{1+x^2} e^{-2\pi i x y} dx $ . Here is the idea used in my textbook, for y<0 :
We calculate the ...
1
vote
0answers
19 views
Is there an extension to this circular contour?
If we are given a finite, multivariate series:
$$f(x) = \sum_{j=a}^b{\sum_{k=c}^d{h_{j,k}x^{j+k i}}}$$
where the $h_{j,k}$'s are constants, and $j+k i$ is a complex number, $a$, $b$, $c$, and $d$ ...
2
votes
1answer
59 views
Cauchy integral theorem, of $e^z /( z-1)$?
So I'm doing some problems on Cauchy Integral theorem, and one of the questions is to find this integral about the circle $|z| = 2$
of $\exp(z)/(z-1)$
I don't think it's possible because it's not ...
1
vote
2answers
139 views
$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ using residues
How do I find the following integral by converting it into a complex integral and then using residue theorem?
$$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$$
My approach is as ...
1
vote
0answers
51 views
Complex Integral using Residues
This is the question:
Find the integral using residue theorem.
$$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta} $$
I solved it like this :
$$\int_0^{2\pi}{d\theta \over1+8\cos^2\theta}=\int_0^{2 \pi} ...
5
votes
1answer
73 views
Problem with calculating a winding number
I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$.
...
1
vote
0answers
30 views
Please help with evaluating an integral using the Residue Theorem [duplicate]
Use the Residue Theorem to evaluate $\displaystyle\int_{0}^{∞} \frac{\sin^2(x)}{x^2} \, dx$.
Using the trig identity, this is how far I've gotten:
let $F(z)=\dfrac{1}{2}\dfrac{1-(e^{2iz})}{z^2}$, and ...
1
vote
2answers
77 views
Complex analysis: contour integration
Evaluate by contour integral:
$$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$
Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
2
votes
0answers
28 views
Understanding estimate
I'm reading some lecture notes and trying to understand the step where they go from the estimate
$$
\log |P(z)| \le C|z|^q \Big (\int \limits_0^{|z|} \frac{n(t)}{t^q}\,dt + |z|\int \limits ...
4
votes
3answers
136 views
Use the Residue Theorem to evaluate the integral:
$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$
I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
2
votes
1answer
51 views
Evaluate $\int_{-\infty}^\infty x\cdot\exp(-x^2+ix)\,dx$ using complex analysis.
It is easy to evaluate $\int_{-\infty}^\infty x\cdot\exp(-x^2+ix)\,dx$ without using complex analysis, i.e.,
$\int_{-\infty}^\infty x\cdot\exp(-x^2+ix)\,dx=\exp(-\frac{1}{4})\int_{-\infty}^\infty ...
2
votes
1answer
63 views
Use the Residue Theorem to evaluate the following integral:
$$\int_{-∞}^{∞} \frac{x^4}{1+x^8} dx$$
I've found the zeros in the upper half plane to be
$$e^{i \pi/8}, e^{i 3 \pi/8}, e^{i 5 \pi/8}, e^{i 7 \pi/8}$$ (right?)
But then the calculation got really ...
2
votes
3answers
73 views
Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.
Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$.
The contour is ...
1
vote
2answers
84 views
Evaluation of the contour integral $\int_\beta \frac{e^z}{e^z-\pi} dz$
Suppose $\beta$ is a loop in the annulus $\{z:10<\left|z\right|<12\}$ that winds $N$ times about the origin in the counterclockwise direction, where $N$ is an integer. Determine the value of ...
0
votes
0answers
34 views
Is it possible to convert this circular contour into a different one?
It's fairly well known that we can eliminate all traces of a variable $x$ by contour integration. For instance, suppose we have a function:
$$f(x) = c_0 x^0 + c_1 x^1 + c_2 x^2 + \dots +c_n x^n$$
...
0
votes
0answers
20 views
Is this Wick rotation correct? (self-intersecting closed contour)
I wonder if what is shown in figure 9.1 here is correct?
Doesn't the contour self-intersect, i.e. it's not a simple closed curve hence the Residue theorem shouldn't apply to this closed contour, ...
0
votes
0answers
32 views
theorems on analytic extension
I am a probability student and I have forgotten much of complex analysis I have learnt when I was an undergraduate.
I have recently seen integrals evaluated using analytic extensions techniques when ...
5
votes
4answers
325 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
4
votes
1answer
118 views
Integral of $\ln |\sin(x)|$
Does anyone have a real formula for the integral $$\int\ln |\sin(x)|\,dx ?$$
Neither Maple nor Mathematica give a real answer.
Using integration by parts and the series for $x\cot x$, I get $$x\ln ...
6
votes
1answer
89 views
Where is the mistake in my integration of $\sin^{2n}(\theta)$?
Note: it's most likely in steps $3-6$.
$1.$ Deriving a useful formula:
$$-i e^{i2\theta}+i=-i \cos(2 \theta)+\sin (2 \theta)+i=2 \sin(\theta) \cos (\theta)+i(1-\cos (2 \theta))$$
$$=2 \sin ...
14
votes
5answers
501 views
Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
3
votes
1answer
51 views
Evaluating the (complex) integral $\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$ using residues.
I am trying to evaluate the following integral.
$$\int_\gamma \frac{e^{z+z^{-1}}}{z}dz$$
where $\gamma$ is the path $\cos(t)+2i\sin(t)$ for $0\leq t <4\pi$.
So, $\gamma$ is an ellipse ...
1
vote
2answers
56 views
Complex analysis, evaluating a path integral
Evaluate the integral
$\int_{\gamma}e^{z^2}+ \overline{z} \ \ dz, $
where $\gamma$ is the positively oriented unit circle.
1
vote
0answers
37 views
Is always $\mathbb{D} \subset U$ in Poisson's integral formula?
Is always $\mathbb{D} \subset U$ in Poisson's integral formula? I mean for example for following example: Let $U=\{z|\operatorname{Im}(z)>0\}$ represent corresponding Poisson integral formula ...
1
vote
2answers
111 views
Evaluate $\int\limits_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx$
I would like to show that
$$\text{PV}\int_0^\infty \frac{\cos(ax)}{\cos(bx)}\frac{1}{1+x^2}dx = \frac{\pi}{2}\mathrm{sech}(b)$$
using complex analysis. $a$ and $b$ are real numbers and $a \neq b$.
...
1
vote
2answers
55 views
Calculating $\int_{C^+(2,2)} \frac {e^\sqrt z} {(z-2)^2}dz$ and $\int_{0}^{\infty} \frac 1 {1+x \sqrt x}dx$
I want to calculate $$\int_{C^+(2,2)} \frac {e^\sqrt z} {(z-2)^2}dz\quad\mbox{and}\quad\int_{0}^{\infty} \frac 1 {1+x \sqrt x}dx$$ using complex integration. In the first part $\sqrt z$ denotes the ...
3
votes
1answer
62 views
Evaluation a this integral
If $f$ ia a continuously differentiable function on the unit circle and
$$
g(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{f(x+t)-f(x-t)}{2\tan\frac{1}{2}t}dt
$$
evaluate
$$
...
1
vote
2answers
55 views
Integral between $-\pi$ and $\pi$ [duplicate]
How can I show that $\int_{-\pi}^{\pi}\sin (mt) \sin (nt) {dt}=\begin{cases}\ 0 \mbox{ if } m \neq n\\ \pi \mbox{ if } m=n \end{cases}$.
I want to prove the above property by expressing sinAsinB as a ...
0
votes
0answers
61 views
Contour Integrals and counterclockwise
$\int_C (z-z_0)^{(n-1)}\ dz$ for any integer $n$, where $C$ is the contour once around the circle $|z-z_0|=1$ counterclockwise and $z_0$ is any point in the plane. Also give the values of the ...
2
votes
1answer
70 views
Evaluate a complex integral using power series expansions
Using power series expansions, evaluate the integral
$$\int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz.$$
where $\gamma_r:[0,2\pi]\rightarrow \mathbb C$ is given by $\gamma_r(t)=r(\cos t + i\sin ...
1
vote
1answer
42 views
Analytic continuation of function given as integral
I have a function $I(D)$ defined by the following integral representation
$$
I(D)=\int_0^\infty\mathrm{d}\alpha\,(1+2\alpha)^{-D/2}
$$
which is clearly only sensible for $D>2$. The result of the ...
0
votes
4answers
51 views
Help me to prove this integration
Where the method used should be using complex analysis.
$$\int_{c}\frac{d\theta}{(p+\cos\theta)^2}=\frac{2\pi p}{(p^2-1)\sqrt{p^2-1}};c:\left|z\right|=1$$
thanks in advance
6
votes
4answers
182 views
Need help proving this integration
If $a>b>0$, prove that :
$$\int_0^{2\pi} \frac{\sin^2\theta}{a+b\cos\theta}\ d\theta = \frac{2\pi}{b^2} \left(a-\sqrt{a^2-b^2} \right) $$
7
votes
2answers
272 views
Definite Integral $\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$
I want to prove that
$$\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$$
4
votes
1answer
109 views
Integrals of functions related to the sinc function
For postive integers $m$ and $n$, is it possible to find formulas for $\displaystyle \int_{0}^{\infty} \frac{\sin^{2m} x}{x^{2n+1}} \ dx \ (m > n)$ and $\displaystyle\int_{0}^{\infty} ...
10
votes
2answers
239 views
Fourier series of function $f(x) = \begin{cases}0 & \text{if }-\pi<x<0 \\ \sin(x) & \text{if }0<x<\pi \end{cases}$
$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\
\sin(x) & \text{if }0<x<\pi.
\end{cases}$$
My attempt:
I went the route of expanding this function with a complex Fourier series.
...
1
vote
1answer
50 views
another Fresnel-like integral
$\displaystyle \int_{0}^{\infty} \sin \left(ax^{2}-\frac{b}{x^{2}} \right) \ dx $
Maple returns numerical results for different values of the parameters that don't agree at all with my answer, and ...
2
votes
1answer
146 views
Line integral on unit circle over non-holomorphic function
In my lecture notes on complex analysis, there are a few "challenges" whether or not you can find a continuous function $f$ on the disc $D(0,1)$ for which
$ \int_{\partial D(0,r)}f(\zeta)d\zeta = 0$
...
0
votes
1answer
44 views
Prove that complex modulus has no primitive
After having done a (small) course on complex analysis from a "physics point of view", I'm now doing a larger course on it from a mathematical perspective. However, early on in my lecture notes, it ...
8
votes
4answers
250 views
Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
I need to show that
$$
\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}
$$
I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result ...
2
votes
2answers
131 views
How do I solve this integral using complex analysis?
I'm having a hard time working on this practice problem. It says:
Compute the integral:
$$\int\limits_{-\infty}^{\infty}\dfrac{1}{y^4+1}\,\mathrm{d}y$$
-1
votes
1answer
51 views
Explain how this reveals the number of roots in $f(z)$
If this takes place in a simple and smooth closed curve $\gamma$, that doesn't cross itself explain why for the polynomial, f(x),
$\frac{1}{2\pi i}\int_\gamma\frac{f'(x)}{f(x)} dx$ tells you the ...
