# Tagged Questions

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

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### Double Integral with Residues

I'm trying to solve the integral $$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$ where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ ...
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### Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
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### Contour integral along a parabola

The question reads: Evaluate $$\int_\gamma f(z)dz$$ where $$f(z)=x^2: x,y \in \mathbb{R}$$ and $\gamma$ is the parabola $y=2x^2$ from $x=0$ to $x=2$. This is the first question I've encountered ...
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### How would you plot this equation (in the complex plane)?

So I am currently looking at how to calculate the integral of a complex function $f(z)$ within a contour $\gamma$. That is, an integral of the form $$\int_{\gamma} f(z) \; dz$$ Where the contour is ...
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### Mistake while evaluating the gaussian integral with imaginary term in exponent

I am trying to evaluate the integral $I=\int_0^\infty e^{-ix^2}\,dx$ as one component of evaluating a contour integral but I am dropping a factor of $1/2$ and after checking my work many times, I ...
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### Integrating $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$.

I'm supposed to find the value of $\int_{-\infty}^{\infty}\frac{e^{ax}}{1+e^x}dx$ for $0<a<1$. I wanted to integrate over the upper semicircle of radius $R$, and take the limit as $R\to\infty$. ...
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### Deformation of Gamma function integral contour

Terence Tao has described the gamma function as the inner product of a multiplicative and an additive character with respect to the Haar measure on $\Bbb R^+$. The gamma function is defined as follows:...
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### Contour integral of continuous but not holomorphic functions

This question was transferred here following Mathoverflow suggestions. Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). ...
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### Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
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### Contour Integral of $\int\limits_0^{2\pi}\frac{d\theta}{1+a\cos\theta}$ for $a^2<1$ (textbook wrong?)

My book is telling me that the answer is $\frac{2\pi}{\sqrt{1-a^2}}$. I'm getting an extra a on the numerator. Could somebody verify if I'm wrong, or if it's my book (it has been wrong numerous times)....
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### Prove the integral is always imaginary

Show that if f is analytic on D and γ is a closed curve in the region then the integral $$\int \overline{f(z)}f'(z)$$ is purely imaginary. I think this problem would use some extension of cauchy ...
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### $\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}\frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta \sin\omega t)^5} dt$

I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution. A charge $e$ moving along a straight line ...
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### Cauchy integral formula with singularities

I am stuck on this question. $$\int \frac{e^{sin z^{2}}}{(z^{2}+1)(z-2i)^{3}}dz$$ along the path γ where γ is a circle centered at the origin of radius different from 1 or 2. I initially thought of ...
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### Big-O Notation?

The problem is to to evaluate the following contour integral along a path $C$ defined/parameterized as $z(t)=εe^{it}$: $\int_C \frac{e^{iz}}{z} dz$ The solution for the problem proceeds to say ...
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### Evaluation of complex integral?

I'd like to verify the result of this integral, or find if I've made a mistake. In the following, $\mathbf x, \mathbf a, \mathbf b$ are all real vectors in $\mathrm R^3$. I do the following: group ...
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### $x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$

I need to find a value and "surface" of a body which is contained in the following contours: $x^2+y^2\le 1$; $z=\sqrt{x^2+y^2}$; and $x^2+y^2=4-z$. Some hints and directions will be helpful. Sorry for ...
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### Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
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### Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
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### Basic contour integration

I am trying to evaluate $$\int_{\gamma(0;2)}\frac {e^{i\pi z/2}}{z^2-1}\, \mathrm d z$$ using the Cauchy integral formula The problem is it is not holomorphic at $1$ and $-1$. My textbook suggests ...
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### Contour Integral over a Closed Circle (Complex Analysis)

I'm having trouble understanding the difference, other than notation, between a contour integral over an open curve and a contour integral over a closed curve. So far, it seems to me that the ...