3
votes
1answer
31 views

inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
0
votes
1answer
46 views

Use contour integration to calculate real integrals

I tried to prove this equality: $$\int_0^{\pi/2} e^{-x\cos \theta}\ \cos (x\sin \theta)\ d \theta=\frac \pi 2 - \int_0^x \frac {\sin u} u du$$ I calculated $$\int_{\gamma^+} \frac {e^{iz}} z dz$$ ...
3
votes
2answers
53 views

Integral of $\int_0^{2\pi} \frac{e^{-it }dt}{e^{it}-z}$

Sorry if this question seems stupid, but I am confused here: Does it follow that $$ I(z)=\int_0^{2\pi} \frac{e^{-it}dt}{e^{it}-z} = 0 $$ For every $z$ with $|z|<1$? I think this is true. I ...
0
votes
1answer
16 views

Contour Integral About A Circle

I have $\int_\delta \frac{z}{z^3 -1} dz$, where $\delta(t) = (\frac{1}{2})e^{it }$ with $t \in [0, 2\pi]$. It's clear that the winding number is $Ind_\delta(z_0) = 1$, where $z_0 = 0$. I'm just ...
1
vote
0answers
29 views

How to prove that the integration contour of one integral is the subset of the other integral? [on hold]

I have the two integrals which have the same non negative integrand. For example the following two integrals, $\int_{-l}^{l}f(x)dx$ ....... (1) and $\int_{a}^{b}f(x)dx$ ....... (2) What we ...
1
vote
2answers
60 views

Evaluate $\int_{\partial C} \frac{dz}{(z-a)(z-b)}$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are not on $\partial C$)

In discussing the possible outcomes of the integral $$\int_{\partial C} \frac{dz}{(z-a)(z-b)}$$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are complex and not on $\partial C$), ...
2
votes
1answer
27 views

Complex integration with real integral

If $\gamma$ is unit circles $A(0,1)$ parameterization of one positive rotation and $a\in\mathbb{R}$, $0<a<1$. Show that $$ \int\limits_0^{2\pi} \frac{dt}{1+a^2-2a \cos t}=\oint\limits_{\gamma} ...
-1
votes
2answers
37 views

Complex integration of exponential function

I am asked to find the integral of $z e^{z^2}$. I have applied the formula of multiplication but the factor of exp cannot be eliminated ofcourse. So how can i solve it. Sorry for such a basic question ...
1
vote
1answer
21 views

Using Cauchy's Theorem on Contour Integral

I need to solve $\int_\gamma (1-e^z)^{-1}$ if $\gamma (t) = 2i + e^{it}$. I would assume Cauchy's Integral theorem applies here, where $\gamma$ is a closed path on a convex open set. I'm having ...
3
votes
1answer
37 views

Integrals over closed contour

Calculate integral $$\oint_{\gamma} \frac{z+1}{z^4+2iz^3} dz$$ where $\gamma$ is parameterization of circle $B(0,1)$ along one positive rotation. I did something like this with Cauchy. \begin{align} ...
0
votes
1answer
26 views

Real value of a complex contour integral equal to the contour integral of the real value of a complex function?

Exactly as in the title: is it generally true that $Re(\int_\gamma f(z)dz) = \int_\gamma Re(f(z))dz$. If not, what would be a case in which it is false? I was thinking a counterexample would follow ...
1
vote
1answer
56 views

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$?

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$? By letting $z = e^{it}$, we get $$\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz = \int_0^{2\pi} \mathrm{Log}(e^{it}) ie^{it} dt = ...
0
votes
2answers
54 views

What is the relation between two integrals?

Let us suppose that we have two integrals, $I_1$ and $I_2$ with the same non negative integrand. The integration contour of $I_1$ is a subset of the the integration contour for $I_2$. What can we say ...
0
votes
0answers
26 views

Integration contour as points of set.

If B is the subset of A, I wounder do we have two integration contour or one for this? What will happen if we take AUB i.e. A union B, than do we have one integration contour? and what if we take A ...
1
vote
0answers
35 views

Confused about pochhammer contour?

I know some theorems about complex analysis such as the argument principle. But I do not get the pochhammer contour. I read about it on the wiki page of the beta function , but I do not understand a ...
0
votes
0answers
22 views

How to select the integration contour

In the following two figures which describe sets, How many possible integration contour we have for the figure 1 and how many integration contour we have for figure 2.
1
vote
0answers
32 views

Contour integrals for $f(z)= e^{3z}$

Integrate $f(z)=e^{3z}$ along line segment from $(0,0)\to(1,1)$ parabola $y=x^2$ from $(0,0)\to(1,1)$ circle $|z|=3$ once around its arc (positive $360^o$) First I parametrized with $z(t)=t+it$ ...
1
vote
2answers
71 views

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$ May I verify if my solution is correct? Thank you. Consider ...
0
votes
0answers
27 views

Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$ M(s) = \frac{2a^s}{s-2(1-a^s)} $$ We ...
1
vote
1answer
63 views

asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
0
votes
1answer
39 views

Value of the integral $\int_0^{2\pi}\int_0^{2\pi}\delta(k_2\cdot e^{i\theta}+k_3\cdot e^{j\phi} +z )d\theta d\phi$

I would like to compute the following integral, which arises from some physics problems, where $k_2$, $k_3$ are real, $z$ is in general complex, $$ \int_0^{2\pi}\int_0^{2\pi}\delta(k_2\cdot ...
1
vote
2answers
54 views

How to integrate $x\times \frac{\sin(x)}{x^2+a^2}$ from zero to infinity

I am trying to evaluate $\int_0^\infty\frac{x \sin(x)}{x^2+a^2} dx$. I get $\frac{\pi}{4} \sin(ia)$ using residue theorem. I integrated over the path that goes from -R to R along the real axis and ...
0
votes
0answers
30 views

Complex Integrals using a contour [duplicate]

Can anyone help me how to prove this integral
0
votes
0answers
52 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
4
votes
2answers
98 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
10
votes
1answer
229 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
3
votes
2answers
77 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
2
votes
0answers
118 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
4
votes
2answers
123 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
0
votes
0answers
75 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
2
votes
1answer
26 views

Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
0
votes
0answers
45 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
2
votes
1answer
33 views

$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ where $f(z) = z^{1/2}$ with branch such that $\Re f(z) \geq 0$

As the title states, the definite integral in question is $$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz,$$ where $f(z) = z^{1/2}$ with branch cut such that $\Re f(z) \geq 0$, i.e., the cut is the ...
0
votes
2answers
57 views

How should I calculate $\displaystyle\int_{-\infty}^\infty\exp\left\{-\frac{1}{2}(x-it)^2\right\}dx$?

I've read that the residue theorem would help to calculate $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\exp\left\{-\frac{1}{2}(x-it)^2\right\}}_{=:f(x)}dx$$ Since $f$ is an entire function ...
1
vote
3answers
89 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
2
votes
2answers
58 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
0
votes
0answers
33 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] ...
0
votes
1answer
29 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
3
votes
1answer
28 views

Contour Integrals evaluation verification

$$ \begin{align} \gamma(t)=2cost + isint, t\in[0,2\pi] \end{align} $$ Could someone verify my thinking and my results for the following Integrals and if I have properly justified my thoughts: $$ ...
4
votes
3answers
66 views

Can $\frac1{z^2}$ be integrated on $|z+i|=\frac32$ using Cauchy's theorem?

$$ \begin{align} \int_{|z+i|=\frac{3}{2}}\frac{1}{z^2}dz=0 \end{align} $$ Is it safe to say the Integral is $0$ due to cauchy's Theorem? Does this apply for any $z_0$ that lies inside the circle ...
1
vote
1answer
47 views

Is this Integral Calculation correct? $\int_{|z|=1}(z^2+2z)^{-1}dz=\pi i$

$$ \begin{align} \int_{|z|=1}(z^2+2z)^{-1}dz=\int_{|z|=1}\frac{1}{z(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+\int_{|z|=1}\frac{1}{2(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+ 0= \\ ...
1
vote
1answer
24 views

Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be ...
3
votes
1answer
47 views

Calculation of a Contour Integral

$$ \begin{align} \int_{|z|=1}(4-z^2)^{-1/2}dz \end{align} $$ The exercise hints at the usage of $e^{2Logz}$ , although any solution or methodology for such integrals would be welcome.
0
votes
1answer
31 views

Geometric explanation of a contour's image

$$\gamma(t)= t^2 + i\, t^4 , \quad t\in [ -1, 1]$$ What is the geometric explanation of the image of the above contour? Intuitively , I think it's ellipsoid-like, but I don't know how to put it in a ...
0
votes
1answer
34 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
1
vote
1answer
31 views

Contour Integration Confusion

I am trying to find the value of $\displaystyle\int_0^\infty\frac{(\log x)^2}{1 + x^2}\,dx$ using contour integration. My approach: I have calculated the residue at z = $i$ and have shown that ...
1
vote
2answers
45 views

Contour expression explanation

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$ contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and ...
3
votes
1answer
98 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
5
votes
2answers
80 views

Complex Numbers - Finding Roots

Hi there I was wondering if someone could help me? I am struggling to find the roots of the polynomial $z^4+2z+3=0$ It is not a quadratic so can't use the quadratic formula so am not quite sure ...
2
votes
1answer
18 views

Estimate of line integral of O(x^n) function

Let $f$ be an analytic function in some sector in the complex plane behaving as $$f(z)=\mathcal O(z^n)$$ for some $n$ as $z\to\infty$. Can one prove in general that line integrals of $f$ (in this ...