# Tagged Questions

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### I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
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### compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
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### computing integral using residue theory [duplicate]

I want to compute the integral $\int_{-\infty}^\infty {x^4\over {1+x^8}}dx$ by using residue theory. I find the zero of $Q(x)$ is $i^{1/4}$. Do I have to factor the denominator into 8 different ...
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### Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
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### prove $\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}}$

prove $$\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}}$$ I tried with $2\cos x =z+\frac1z$ then use residue theorem but I faced some troubles my try is : $2\cos x =z+\frac1z$ ...
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### Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
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### Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
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### Using residue to find a complex integral

Given the following: $$\int_{\varGamma_R} {z\,dz\over e^{2\pi iz^2}-1}, \ \ \ \varGamma_R=\{z\in \Bbb C:|z|=R\},\quad n<R^2<n+1,\,n\in\Bbb N.$$ I want to use the residue for this, but I can't ...
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### Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
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### About the integral $\int_{0}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}$

I need to prove the following identity: $$\forall a>0,\qquad\int \limits_{-\infty}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}=\pi(1-e^{-a}).$$ I think it can be proven using Laurent series. I tried ...
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### Integrating $t^{2r-1} / t^{2k} (1+t^2)^{r+1}$

Let $k$ and $r$ be natural numbers such that $1 \leq k \leq r$. I want to calculate $$\int_0^\infty \frac{t^{2r-1}}{t^{2k}(1+t^2)^{r+1}} dt.$$ Since the integrand is an odd function the standard ...
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### Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
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### Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
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### $\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
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### What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?

Why is it that when one in considering contour integration of a real function, such as $$\int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following: ...
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### Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx.$$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
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### Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so ...
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### Evaluating $\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$

I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when ...
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### Can the direction of Contour Integral be affect to the result of integration?

Now i doing the home work about Residue Integration and i doubt that "Can the direction of Contour Integral be affect to the result of integration?" I mean with the same shape of contour but direction ...
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### Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
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### Evaluation of the integral.

$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$ The integration is taken along the imaginary axis, an integer ...
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### I can't find out where did I go wrong in solving: $\int_{|z|=1}\frac{4+z}{(2-z)z}dz$

I can't find out where did I go wrong in solving: ...
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### Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,dx$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ...
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### Finding a definite integral by residue integration?

I have a probability distribution of the form $$\frac{k_1}{(k_2 x^2+k_3 x+k_4)^n}$$ and I want to find the mean and variance—but am running into problems with that. The mean would be ...
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### Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
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### How to calculate $\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
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### Hint to compute the following integral

Can someone give a hint on how to solve the following integral? $$\int_0^{2N\pi}\frac{(-R\cos t)(\xi t-r)+\xi R\sin t}{(R^2+(\xi t-r)^2)^{3/2}}dt$$ I've tried some substitutions. First I've splitted ...
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### Help in calculating the following integral $\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x.$

I was asked to calculate this: $$\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x.$$ My idea was to change the integration limits to $|z|=1$ in the complex plane and to ...
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### Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?

How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
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### Calculating integral with branch cut.

I'm learning how to calculate integrals with branch points using branch cut. For example: $$I=a\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{(1+\xi^{2})\sqrt{\frac{2}{m}\left(E-U_{0}\xi^{2}\right)}}$$ where ...
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### Finding a generalization for $\int_{0}^{\infty}e^{- 3\pi x^{2} }\frac{\sinh(\pi x)}{\sinh(3\pi x)}dx$

$\;\;\;\;$I was reading the introduction of Paul J. Nain's book "Dr. Euler's fabulous formula" where he talks about the sense of beauty in mathematics and quotes the G.N.Watson as saying that a ...
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### Determine the integral $\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$ using residues.

Determine the integral $$\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$ using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications. In order to do this. We ...
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### $\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ 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 ...
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### 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 ...
### 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 ...
### Integral $\int_0^\infty \exp(ia/x^2+ibx^2)dx$
Compute the integral: $$\int_0^\infty \exp\left(\frac{ia}{x^2}+ibx^2\right)\,dx$$ for $a$, $b$ real and positive. I tried complex variables, but don't really know how to ...