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

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2
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0answers
69 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
6
votes
1answer
55 views

Definite integrals and möbius transformations

In examples I have seen for solving an infinite integral from $-\infty$ to $\infty$ using contour integration, the real axis becomes part of the contour of integration in the complex plane, and the ...
0
votes
0answers
32 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
0
votes
0answers
41 views

Method for evaluating $\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$

I have a problem where I must evaluate $$\int_{|z| = 1} \dfrac{z^2}{\sqrt[4]{P(z)}} dz$$ Where $P(z)$ is a polynomial with degree at least four and has exactly four roots in the unit circle. I know ...
3
votes
0answers
86 views

An interesting identity involving powers of $\pi$ and alternating zeta series

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
1
vote
2answers
44 views

Definite integrals with complex analysis

Can anyone explain me why: $$\int_{C_r}\frac{1}{z} \mathrm{d}z+\int_{C_r}\frac{e^{iz}-1}{z}\mathrm{d}z\stackrel{r \to 0^+}{=}\pi i$$ $C_r$ is half circle from $r$ to $-r$
2
votes
2answers
107 views

Inverse Laplace with $\ln$

How can I compute the inverse Laplace of 1) $\ln\left(\dfrac{s+1}{s-1}\right)$ 2) $\ln\left(\dfrac{s-1}{s}\right)$. Can someone please hep me to do these two problems
1
vote
2answers
101 views

Contour integral $\int^\pi_{-\pi}(a-\cos\theta)^b\exp(c\cos\theta)d\theta$ assuming $a>1$, $b>0$, $c>0$

Under the condition $a>1$, $b>0$, $c>0$, is there any good function to express the following integral? $$ \int^\pi_{-\pi}\left(a-\cos\theta\right)^b\exp\left(c\cos\theta\right)d\theta $$ I ...
2
votes
1answer
46 views

Showing integral on contour tends to zero

I'm trying to prove: $$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$ Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients. ...
0
votes
1answer
33 views

What is the intuition behind contours and their geometric properties

What is the the intuition behind contours? Can someone explain whar are contours, their geometric properties in simple manner
0
votes
3answers
66 views

Integrating $1/\sqrt{z^{2}-1}$ on some contour

If I wanted to integrate $$\oint \frac{1}{\sqrt{z^{2}-1}}$$ Say around a circular contour radius $2$ centre $0$, how would I do that? Does the function have poles at $\pm 1$ or are they just "branch ...
2
votes
0answers
67 views

inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
1
vote
1answer
62 views

Contour Integral of $\int_0^{\infty} \frac{1}{x^4+1} dx $ - Missing a factor of 2

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{1}{x^4+1} dx $$ Consider $$ \oint \frac{1}{z^4+1} dz = \oint \frac{1}{(z - \frac{1-i}{\sqrt 2})(z + \frac{1-i}{\sqrt 2})(z - \frac{1+i}{\sqrt ...
0
votes
0answers
51 views

Complex Integration: $\int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $

I'm supposed to evaluate: $$ \int_0^{\infty} \frac{\sin x}{x(k^2x^2 +1)} dx $$ Attempt Consider $ \int_0^{\infty} \frac{e^{iz}}{x(k^2x^2 +1)} dz $ Simple poles at $z = \pm \frac{i}{k} $, simple ...
1
vote
0answers
59 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
2
votes
1answer
53 views

If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
0
votes
2answers
58 views

Contour integral of $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$

I'm supposed to evaluate $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$ Using a contour of a unit circle, $z=e^{i\theta}$. This is the same as: $$2i \oint \frac{1}{z^2 - Az + 1 } dz $$ The ...
1
vote
1answer
93 views

Calculate $\int_0^\infty\frac{\sin x}xdx$ by integration of a suitable function along given paths [duplicate]

How can I calculate $$\int_0^\infty\frac{\sin x}xdx$$ by integration of a suitable function along the following paths: where $R$ and $\varepsilon$ are the radius of the shown outer and inner ...
0
votes
1answer
23 views

Show that $f(a)=\frac{1} {2\pi}{\int_C {\frac{(R^2-a \overline a)f(z)}{(z-a)(R^2-z \overline a)}dz}}$

The function $f(z)$ is regular when $|z|<R'$ Show that if $|a|<R<R'$ then $$f(a)=\frac{1} {2\pi}{\int_C {\frac{(R^2-a \overline a)f(z)}{(z-a)(R^2-z \overline a)}dz}}$$ Where $C$ is the ...
3
votes
1answer
68 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
1answer
58 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 $$ ...
2
votes
1answer
70 views

Complex Contour Integration - Complex Analysis

I'm just practising for my upcoming exam, and I've come across a question I'm having a bit of difficulty with. I've been asked to show the following; $$\int_{0}^{\infty} \frac{dz}{\cosh(z)} = ...
2
votes
2answers
55 views

A Contour Integral I

What is the value of the integral \begin{align} \int_{-a}^{c} \sqrt{ \frac{a+x}{c-x} } \ \frac{dx}{(d-x)(x-b)} \end{align}
1
vote
1answer
67 views

Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx $$ Attempt Considering $$ \oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz $$ So first I find the branch points of the function. This ...
0
votes
1answer
30 views

Contour Integration of this path circling branch point

If we let the semi-cricle blow up to infinity and the radius of the tiny circle encircling the branch point at origin go to zero, by residue theorem we have: $$\int_\gamma + \int_{AB} + \int_{BC} ...
0
votes
1answer
64 views

Contour Integration: What is the function?

I know that the integral around a closed path = 0 since there are no poles. Why is the integral along the slanted path $\int_0^R e^{-x^2w^2} w dx$? If $w$ is defined to be along the slope, ...
1
vote
1answer
38 views

Help in understanding contour integration

I would like help in understanding the process of contour integration. As an (hopefully straightforward) example, I have chosen the calculation of Bernoulli number $B_2$. I should be very grateful ...
1
vote
1answer
60 views

Finding definite integral using contour integration

Wanting to find the value of the integral $\int_{0}^{\infty} \dfrac{1}{\cosh (x)} dx$ and know I have to find the residue at $\dfrac{\pi i}{2}$ which I find to be $-i$. So far so good. So then, I know ...
2
votes
1answer
55 views

Finding a definite integral using complex analysis.

Now, I want to integrate $\int_{0}^{\infty} \dfrac{\cos (2x) -1}{x^2} \mathrm{d}x$, now I attempted to set $f(z)=\dfrac{e^{i2z}-1}{z^2}$ and then integrate around a similar contour to the classical ...
0
votes
1answer
27 views

Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
0
votes
1answer
43 views

Compute $\int_{\gamma} x dz$

Let the perimeter of the square formed by the points $0$, $1$, $1 + i$, $i$ and $z = x + iy$. How can i compute $$\int_{\gamma} x dz$$. Some help to compute this complex integral please.
1
vote
0answers
75 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
2
votes
1answer
103 views

Evaluation of tricky integral

I want to evaluate the integral $$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$, where $b$ and $s$ are positive real numbers. I thought of ...
5
votes
1answer
173 views

Evaluate: $\int_{W(-1/\gamma)}^{W(1/\gamma)}\frac{e^{-u} \,\text{d}u}{\sqrt{1-(\gamma u e^{u})^2}}$

Evaluate the integral $$ P(\gamma)=\int_{W(-1/\gamma)}^{W(1/\gamma)}\frac{e^{-u} \,\text{d}u}{\sqrt{1-(\gamma u e^{u})^2}} $$ where $\gamma$ is a real number not equal to $0$ and has whatever ...
3
votes
2answers
59 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 ...
1
vote
1answer
57 views

How would Cauchy calculate $\int_{\partial B_1(2i)}\frac{e^{z^2}}{2i-z}dz$?

Please consider the following curve integral: $$I:=\int_{\partial B_1(2i)}\frac{e^{z^2}}{2i-z}dz$$ where $$B_r(z_0):=\left\{z\in\mathbb{C}:|z-z_0|<r\right\}$$ Let $\gamma :[a,b]\to\Omega$ denote ...
5
votes
2answers
108 views

Riemann Zeta function Analytic continuation integral

Following Riemann paper about analytic continuation of Zeta Function: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf I can't understand the contour integral step: "If one now ...
0
votes
1answer
31 views

Evaluate $ \int_{C(0,5)} \frac{1}{i-\cos z}dz $.

How do I evaluate $$ \int_{C(0,5)} \frac{1}{i-\cos z}dz? $$ Do I have to find the poles first, and then use residue theorem, or find where the function is holomorphic and then integrate using ...
-1
votes
1answer
39 views

Quick question on Jordan's Lemma

The key equation in Jordan's Lemma is: $$I_\Gamma = \int_\Gamma e^{imz}f(z) dz \rightarrow 0$$ as $R \rightarrow \infty$. Why is $|\exp(imz)| = |\exp(-mR\sin\theta)|$?
3
votes
1answer
34 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 ...
0
votes
1answer
21 views

How do we find a path $\gamma$ with winding number $1$ and $2$ relative to points $1$ and $2$, respectively?

Let $\gamma :[a, b]\to\Omega\subseteq\mathbb{C}$ denote a parametric piecewise continuously differentiable path in $\Omega$ and $$\text{ind}_{\gamma}(z):=\frac{1}{2\pi ...
3
votes
1answer
75 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 ...
0
votes
0answers
42 views

Choice of Branch Cut in Contour Integral

Suppose we have to evaluate an integral: \begin{align*} \int_C f(z)\,dz \equiv\int_{C} \frac{e^{-iz}}{z^2-(a-bi)}dz\ ,\ a,b \in \mathbb{R}, a,b >0 \end{align*} where the contour $C$ is closed in ...
0
votes
3answers
63 views

Contour integrals using residues

The question I'm working on is the following: Let $C_R$ be a contour in the shape of a wedge starting at the origin, running along the real axis to $x=R$, then along the arc $0 \leq \theta \leq ...
1
vote
1answer
122 views

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
3
votes
1answer
90 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 ...
0
votes
1answer
27 views

Residue of $\frac{\cot{ax}}{x^2-b^2}$?

I am interested to find the residue of $$\frac{\cos{ax}}{(x^2-b^2)\sin{ax}}$$ at $x=b$. How would I go about doing this? I can see that the pole is second order, and so the formula $$\text{res} = ...
2
votes
0answers
85 views

Numerical integration of function with singularities

I am currently trying to solve a semi-infinite integral containing a set of singularities lying on the real axis numerically. The process I am using is breaking the integral into small steps $\Delta ...
8
votes
4answers
173 views

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$ using complex analysis.

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$. This is the last question in our review for complex analysis. Hints were available upon request, but being the student I am, I waited until the ...
1
vote
1answer
35 views

Complex Integral - exponential divided by a monomial

How does one solve integrals like this- $$I=\int^\beta_0 dx \frac{\exp(i\omega_nx)}{x-a}$$ where $\omega_n=\frac{\pi n}{\beta} $. EDIT: $\beta$ is a finite, real ...