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

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Deriving Cauchy integral formula

Within in the proof of Cauchy Integral Formula there is this line $$f^{(k)}(z)=\frac{k!}{2 \pi i} \int_\Im \frac{f(\zeta)}{(\zeta - z)^{k+1}} d\zeta \quad (k=1,2,3,...)$$ My goal is to derive this ...
2
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1answer
63 views

Contour integral on the function $\frac{\log z}{z-z_0}$ [duplicate]

Let the domain $O=\mathbb{C}-(-\infty,0)$, the point $z_0 \in O$ and the circle $\gamma=C(0,r<|z_0|)$ in the positive direction. Compute $\int_C \frac{\log z}{z-z_0} dz$. So far we didn't see the ...
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1answer
57 views

Modified Bessel Function Integral representation proof $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{2}/4t}dt $

How do I proof the following integral representation for the Modified Bessel function of the second kind (Macdonald Function). $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{...
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1answer
36 views

Definition of the domain - Cauchy theorem - Contour integration

In general, in that kind of question, there isn't a domain defined (I think). If I ask you to tell me the answer of the contour integral $\int_{\gamma}\frac{1}{z-2} dz$ where $\gamma$ is simply the ...
4
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0answers
85 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
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0answers
19 views

System of equations - Contour integral

A problem ask to find the constants $a$, $b$ and $c$ if $f(z)=az^2+bz+c$ and $\int_{\gamma}\frac{f(z)}{z}dz=2 \pi i$, $\int_{\gamma}\frac{f(z)}{z+1}dz=4 \pi i$ and $\int_{\gamma}\frac{f(z)}{z-1}dz=8 \...
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3answers
22 views

Inequality - Contour integral

I would like to solve the inequality $|\int_{\gamma} \frac{1}{z^2}dz|\leq 2$ where $\gamma$ is the line $[i,2+i]$. I thought about using the Cauchy theorem in closing the path between $i$ and $2+i$, ...
1
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1answer
27 views

Contour integral over the unit circle

I have to evaluate the contour integral $\int_{\gamma} \frac{\sum_{k=0}^m a_k z^k dz}{z^{n+1}} = \int_{\gamma} \frac{f(z)dz}{z^{n+1}}$ with $n \geq 0$ and $\gamma = C(0,1)$ counterclockwise oriented. ...
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1answer
40 views

Contour integral - Unit circle

I have to compute the contour integral $\int_{\gamma} \frac{(z-1)}{z(z+1)(z-2)}dz$ where $\gamma$ is (a) the circle $C(0;1)$, (b) the circle $C(0,\frac{3}{2})$ and (c) the rectangle of vertex $-2-i$, $...
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1answer
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$r(\theta) e^{i\theta}$ - Parametrization of the square $\gamma$

In the question Contour integral - Circle instead of a square achille hui explains that $r(\theta) e^{i\theta}$ with $r(\theta)\min\left(\frac{1}{|\cos\theta|}, \frac{1}{|\sin\theta|}\right)$ is a ...
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1answer
60 views

Contour integral - Circle instead of a square

I would like to solve the integral $\int_{\gamma} \frac{1}{z \bar{z}} dz$. Here $\gamma \subset \mathbb{C}$ is a square centered at the origin and where his vertices are parallel to the axes. Could I ...
3
votes
1answer
38 views

Cauchy's theorem for contour integration

I have to compute $\int_C(z+\frac{1}{z})^{2n}\frac{1}{z}dz$, where $n \in \mathbb{N}$, and $C$ is the unit circle with positive orientation. So let $z(t)=\cos (t) + i \sin (t)$, with $-\pi \leq t <...
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1answer
74 views

Certain type of integrals

I would like to do the following integral $$ \int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}}, $$ where the $i\epsilon$ has been added to avoid some possible divergencies. ...
3
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1answer
89 views

Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

I'm working on the problem $$I=\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$$ I found 4 singularities and i would like to use the singularities in the 1st and 2nd quadrants to solve this integral; i.e. $...
2
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1answer
25 views

Is this dual behavior allowed in integration in Complex Analysis?

The integral $$I_1 = \int_{C} \bar zdz=4\pi i ,$$ when $C$ is the right-hand half $z=2e^{iθ},\ (-\dfrac{π}{2}≤θ≤\dfrac{π}{2})$, of the circle $|z| =2$ from $z =-2i$ to $z = 2i$. And, one also can ...
2
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3answers
75 views

Improper integral $\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$

How do I solve this? $$\int _{0+0}^{1-0}\frac{dx}{\left(4-3x\right)\sqrt{x-x^2}}\:dx$$ I know it's a type 3 improper integral, and I'm having issues with these. I think that I need to write it as a ...
3
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1answer
52 views

How to evaluate the integral $\int_0^{2\pi}\mathrm{d}\theta e^{ia\cos(\theta-\theta_1)}\cos^2(\theta-\theta_2)$

I have an integral: $$\int_0^{2\pi}\mathrm{d}\theta e^{ia\cos(\theta-\theta_1)}\cos^2(\theta-\theta_2),$$ where $a, \theta_1$ and $\theta_2$ are reals. Any idea on how to evaluate this integral.
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2answers
138 views

Problem over a definite integral, which surely needs contour integration

During my Master Thesis work I came up with an integral which I am going to consider as a hard challenge. I have been trying for days to crack it, but still nothing. The integral is the following $$\...
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1answer
28 views

Complex: evaluating integral with residues

Having a bit of trouble here. Having this integral $$ \int_{0}^{\infty} \frac{dx}{(x^{2}+1)(x^{2}+4)^{2}} $$ I can tell it's even, so it has symmetry. Thus, $$ \frac{1}{2} \int_{-\infty}^{\infty} \...
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1answer
52 views

Prove that the following function has a branch cut

I am given with a function $$\zeta(z)=\int_{-\infty}^\infty\mathrm{d}x \frac{f(x)}{z-x}.$$ Any idea to prove that $\zeta(z)$ is discontinuous across real axis for $f(x)\neq 0$?
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2answers
86 views

Integrate $\int_{-\infty}^\infty\frac{e^{-ik\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}dx$

I'm trying to evaluate the integral below for my research related to sound radiation. Assume $a$ is a positive constant. $$\int_{-\infty}^\infty\frac{e^{-ik\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}dx$$ First,...
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0answers
48 views

How to solve a contour integral?

I am trying to solve the contour integral $$\frac{1}{2\pi i}\int_C\frac{e^{t}}{t(2at+x^2)}{\rm{exp}}\left(\frac{ax^2}{2(2at+x^2)}\right)\,{\rm d}t $$ where the path of integration $C$ starts at $-\...
5
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2answers
119 views

How to evaluate integral $\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$?

I came across the following integral: $$\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$$ while trying to calculate the inverse Laplace transform $$ L_p^{-1} \left[ \frac{\sinh(\alpha\sqrt{...
2
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1answer
90 views

evaluate $\int _0 ^\infty \frac{1-\cos(ax)}{x^2}dx$

Im trying to evaluate for a given $a\in \mathbb R$$$\int _0 ^\infty \frac{1-\cos(ax)}{x^2}dx$$ I have noticed that since $1-\cos(ax)$ is analytic in $\mathbb C$, the integral $$\int _{C} \frac{1-\cos(...
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0answers
61 views

Evaluation of an improper integral with complex exponential

Are there any convenient ways to calculate an integral of the form $$ \int_{-\infty}^\infty\frac{a_1 e^{j\omega\alpha}+a_2e^{j\omega\beta}}{1 + a_1a_2e^{j\omega\gamma}}d\omega$$ where $a_1,a_2,\alpha$,...
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1answer
59 views

Compute, for all integers $m,n$, $\int_{|z|=2}z^n(1-z^m)dz$

Compute, for all integers $m,n$, $\int_{|z|=2}z^n(1-z^m)dz$. Isn't $z^n-z^{mn}$ analytic, so the value is 0? And if it isn't analytic, letting $z(t)=2e^{it},t\in[0,2\pi]$ gives $$\int_{0}^{2\pi}(2^ne^...
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1answer
38 views

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$ [duplicate]

Compute $\int_{|r|=2}\frac{1}{z^2+1}dz$. Is there a special trick to solving this? I tried letting $z(t)=2e^{it},t\in[0,2\pi]$, which gave me $$\int_{|z|=2}\frac{1}{z^2+1}dz=\int_0^{2\pi}\frac{2ie^{...
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0answers
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Contour integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$

Let $m\in\mathbb{N},t>0$ how to compute the integral $\int_{\gamma}e^{-t\mu}(-\mu)^{-m}\log\sqrt{-\mu}d\mu$, where $\gamma$ is contour $\{|\arg(\mu+1)|=\pi/4\}$ transversed upward? Here are my ...
3
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1answer
76 views

Cauchy Principal Value of $\int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx$

The problem here is to evaluate $$ \int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx $$ for $a,\mu >0.$ Clearly this integral doesn't converge in the usual sense, but we can calculate its ...
1
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1answer
42 views

Higher order derivatives of Cauchy integral

An excerpt from my course notes in introductory complex analysis. Given a curve $\Gamma$ in the complex plane and a continuous function $\varphi : \Gamma \to \mathbb{C}$, define its Cauchy integral ...
8
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2answers
235 views

How to evaluate this integral $\int_{0}^{\infty }\frac{\ln\left ( 1+x^{3} \right )}{1+x^{2}}\mathrm{d}x$

How to evaluate this integral $$\mathcal{I}=\int_{0}^{\infty }\frac{\ln\left ( 1+x^{3} \right )}{1+x^{2}}\mathrm{d}x$$ Mathematica gave me the answer below $$\mathcal{I}=\frac{\pi }{4}\ln 2+\frac{2}{3}...
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1answer
77 views

Possible to integrate $\ln |x-y| dy$ on a circle? [closed]

I am wondering what is the solution to $$\int_{\partial D}\ln |x-y| dy$$ when $D$ is the unit disc in $\mathbb{R^2}$ and $x \in \partial D$. Is this even possible analytically?
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34 views

Contour integral of 1/z

Consider the integral $\int_{-\infty}^\infty dx \frac{1}{x+i \epsilon}$, where $\epsilon$ is a positive real number. We can evaluate the integral by closing the contour in the complex plane and then ...
2
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0answers
35 views

Solving differential equation by transform with contour integral

In my notes, I have an example for solving for the Airy function in the equation: $$\frac{d^2y}{dx^2}-xy=0$$ So he uses the contour integral to represent the solution (with $C$ as a yet unspecified ...
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Integrals that don't coincide with the Riemann integral?

This is probably a lame question, but I was wondering if there exist integrals that do not coincide with Riemann's integral for function that are integrable with respect to both these integrals?
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1answer
63 views

Cauchy Residue Theorem Integral

I have been given the integral $$\int_0^ {2\pi} \frac{sin^2\theta} {2 - cos\theta} d\theta $$ I have use the substitutions $z=e^{i\theta}$ |$d\theta = \frac{1}{iz}dz$ and a lot of algebra to transform ...
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0answers
44 views

Contour Integration of Definite Integral of Sine and Cosine, 4th order pole

There is this problem from Complex Variables (Brown and Churchill) regarding the definite integral of a sine and cosine function. It's supposed to be integrated using residues. I've been trying to ...
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1answer
44 views

Complex integral of the function $f(z)=\dfrac{1}{z^4+1}$

I must calculate this integral $$\int_C\dfrac{dz}{z^4+1}$$ , where $C$ is the circle $x^2+y^2=2x$. My result is $\int_C\dfrac{dz}{z^4+1}=-\dfrac{\pi}{\sqrt{2}}$ , but my book "A collection of problems ...
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1answer
301 views

Compute the series $\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$

I need to compute $$\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$$ This an exercice of "Amar and Matheron, complex analysis". I proved the convergence and now to compute the sum, I follow the ...
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3answers
197 views

Evaluation of Bose-Einstein and Fermi-Dirac Integrals

Can anyone help me with the following integral, which has arisen from my study of cosmology? $$\int_0^\infty\frac{p^2dp}{e^{(p\pm\mu)/T}+ 1}$$ I have tried substitution and using standard integral ...
3
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1answer
68 views

Proving a modified version of Jordan's lemma?

I want to prove the version of Jordan's lemma which say's that: Around a contour as shown below: The integral: $$\int_{\Gamma} e^{az}f(z)dz$$ will go to $0$ if $$|f(z)|\le \frac{M}{R^\alpha}$$ for ...
3
votes
2answers
94 views

How to find $\int_{-\infty}^\infty \exp(-x^2) \, dx$ with contour integration?

Contour integration is a very powerful tool. But what if a function has no poles or zero's ? For instance : How to find $\int_{-\infty}^\infty \exp(-x^2) \, dx$ with contour integration?
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1answer
59 views

The complex integral of the reciprocal of polynomial is constant on sufficiently large circles

Let $P$ be a polynomial of degree $n ≥ 2$. Suppose all the roots of $P$ lie in the disk $D_{r}(0)$. Let $R > r$ and $$I(R)=\int_{|Z|=R}{1\over P(z)}dz$$ Prove that the integral is constant. I ...
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1answer
31 views

Am i evaluating this contour integral correctly?

Let $\gamma$ be the circle of radius 2 centered at the origin. $$ \int_{\gamma} \frac{1}{z^2+i}dz $$ I tried factoring the denominator out to where $ \int_{\gamma} \frac{1}{z^2+i}dz $ = $ \int_{\...
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1answer
24 views

Algebraic confusion

How does $$\frac{1}{b^2i}\oint\frac{1}{z}\frac{dz}{\lbrack k + \frac{1}{2i}\left( z - \frac{1}{z}\right) \rbrack^2}$$ become $$\frac{1}{b^2i}\oint\frac{1}{z}\frac{dz}{\lbrack \frac{1}{2iz} \left( z^...
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0answers
27 views

Two different answers with applying the Cauchy integral formula and parametrizing.

I have the integral $\int_{|z|=1}\frac{cosz}{z}dz$ By applying the Cauchy Integral Formula, I get that this equals $2\pi i*cos(0) = 2\pi i$ Is this correct? If I parametrize the integral with $z= e^{...
5
votes
1answer
132 views

Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz $

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds $$ It looks like a complicated ...
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0answers
44 views

Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...
1
vote
1answer
54 views

Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the upper-...
6
votes
2answers
151 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...