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

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Conditions for changing the order of integration for contour integral.

I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$ where the function $f(x)$ can be represented as a contour integral in complex plane: $$f(x)=\oint_\Delta ...
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64 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 & = ...
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1answer
54 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 ...
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0answers
29 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 ...
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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 ...
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76 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 ...
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2answers
41 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 $-r$
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2answers
94 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
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2answers
85 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 ...
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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. ...
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1answer
32 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
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3answers
56 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
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0answers
55 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})] ...
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1answer
56 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 ...
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0answers
47 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 ...
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0answers
54 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
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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 $$\int_\gamma\frac{1}{z\pm i}dz$$
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2answers
52 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 ...
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1answer
86 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 ...
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1answer
21 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
63 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
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1answer
54 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
64 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
54 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}
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1answer
53 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 ...
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1answer
22 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} ...
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1answer
62 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
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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
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1answer
49 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
50 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 ...
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1answer
26 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
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1answer
42 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.
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0answers
43 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
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1answer
101 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
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1answer
172 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
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 ...
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1answer
56 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
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2answers
96 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
30 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 ...
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1answer
34 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
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 ...
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1answer
18 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
66 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 ...
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0answers
35 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 ...
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3answers
62 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
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1answer
108 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
87 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
72 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
168 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 ...