# Questions tagged [contour-integration]

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

2,534 questions
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### Does there exist an entire function $F$ such that $F'(z)=f(z)$ for all $z \in \Bbb C \setminus \{0 \}$?

Let $f : \Bbb C \setminus \{0 \} \longrightarrow \Bbb C$ be a holomorphic function such that $\int_{\gamma} f(z)\ dz = 0$ for any closed curve $\gamma$ in $\Bbb C \setminus \{0 \}.$ Does there exist ...
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### Integral problems for practice

Hello can someone give me a few problems with double and triple integral that include polar and not polar form approach and also contour integrals problems, because i am end of examples from textbook ...
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### Calculate $\int_{-\infty}^\infty{x^2\,dx\over (1+x^2)^2}$

The question:\, Calculate $$\int_{-\infty}^\infty{x^2\,dx\over (1+x^2)^2}.$$ Book's final solution: $\dfrac\pi 2$. My mistaken solution: I don't see where is my mistake because my final solution ...
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### contour integral of $1/\sqrt{z(z-1)(z-2)}$ [on hold]

How to find contour integral of $1/\sqrt{(z(z-1)(z-2)}$ around 0 and 1?
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### Contour integration problem with sin and cos

so I'm revising contour integration for an upcoming complex analysis exam. I have been asked to integrate $$\int_0^{2\pi}\frac{\sin^2x}{a+b \cos x}dx$$ I thought the sensible thing to do here would ...
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### Explanation of coefficient when evaluating contour around a branch for fractional version of Cauchy's Integral Formula

I am working on fractional derivatives which are defined by taking the Cauchy Integral formula and letting the order of the derivative be non-integer. Specifically, \begin{equation} f^{(\alpha)}(z)=\...
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### Inv. Laplace $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$

What would be the inverse Laplace of the following function? $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$ ...
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### Show that $\int_{\mathbb \Gamma}\lambda^kd\lambda = 0 , \forall k \ge 0$

Let $\mathbb \Gamma$ denote the boundary of a convex polygon with vertices $w_1, ..., w_n$ in $\mathbb C$. Show that $$\int_{\mathbb \Gamma}\lambda^kd\lambda = 0 , \forall k \ge 0$$ I've found some ...
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### Integration by transforming to complex

Evaluate the following by transforming it into a complex integral: $$\int_{-\infty}^{\infty} \frac{\cos 4x}{x^4+5x^2+4}dx.$$ Could someone show me where to start? This is not homework, it's a study ...
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### Inverse Laplace transform of $f(s)={\frac{1}{s^{3/2}}}$ using complex integration

I want to find the inverse Laplace transform of $$f(s)={\frac{1}{s^{3/2}}}$$ Refer to the Laplace transform table, and I found that the result is $$F(t)=2\sqrt{\frac{t}{\pi}}$$ But I do not know how ...
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### Calculating $\oint_{\gamma}\bar z^ndz$ for a known path

I wish to calculate the following integral: $$\oint_{\gamma}\bar z^ndz$$ $\gamma$ is a triangle with vertices at $0,1,i$, in the positive direction and $n\in \mathbb Z$. Since $\bar z^n$ is not ...
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### Help with complex valued integral

I'm given that $f(z)=z \bar{z}=x^{2}+y^2$ with $z=x+iy$ and $r(t)= <cost,sint>,\quad 0 \leq t \leq 2\pi$. I was able to evaluate the line integral by parametrization of f using r(t). I am ...
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### Formulations of Cauchy's theorem that don't seem consistent

So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ...
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### Evaluation of the integral $\int \frac{x^{\frac{1}{3}}}{1+x^3 } dx$ [closed]

I'm looking to solve this integral right here: $\int \frac{x^{\frac{1}{3}}}{1+x^3 } dx$ I would like to know what approaches I could take to solve this using complex analysis.
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### Analytic continuation and complex integration of one variable of a multivariate function

Consider a function $f:(x_0,x_1,...,x_n)\in\mathbb{R}^n\rightarrow f(x_0,...,x_n)\in\mathbb{R}$. $f$ is a ratio of polynomials in $x_0,...,x_n$ which only has simple poles in the variable $x_0$, whose ...
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### Finding the infinite sum involving $\coth$ function using contour integration

I am looking to show: $\sum_{n=1}^∞ \frac{\coth(nπ)}{n^3} = \frac{7π^3}{180}$ There is a hint earlier that you are supposed to be using the function $f(z)=\frac{\cot z\coth z}{z^3}$. I have ...
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### Computing $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$

Compute the integral using residues: $\int_{|z-i|=\frac{3}{2}}\frac{e^{\frac{1}{z^2}}}{z^2+1}$ Inside the circumference there are the following singular points $-i$ which is a pole of order 1 and $0$ ...
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### $\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$ difficult integral with two branch cuts

$$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$$ I defined two branch cuts along the real axis: $[-\infty ,-\frac{1}{a}]$ & $[0,\infty]$ with the following contour: I defined the $arg{(z)} =0$ ...
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### Integrating $\int_0^{\infty} \frac{(\log x)}{(x + a)^2 + b^2} \operatorname d\!x$

I'm trying to show that $\int_0^{\infty} \frac{(\log x)}{(x + a)^2 + b^2} \operatorname d\!x = \frac{1}{b}\arctan \sqrt{a^2 + b^2}.$ However, I am a bit confused applying the key hole "method." I ...
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### Construct $\varphi (z)$ such that $\int_{|z|=1} \frac{\varphi (z)}{z-w} dz =0$

I have this problem to complex analisis. Construct $\varphi (z)$ a continous function nonzero in $S^{1}$ such that $$\int_{|z|=1} \frac{\varphi (z)}{z-w} dz =0$$ for $|w|<1$. I have the idea to ...
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### $\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$ Cauchy principal value

$$\int_{-\infty}^\infty \frac{e^{pz}}{e^z-1}dz$$ I started by defining the following contour: rectangular contour It is easy to show that the integrals along the 2 vertical sides of the rectangle ...
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### how $\int_a ^ b |f'(x)|$ gives the length of the arc of the contour $f$ : $(f(x) : x \in [a , b])$

I got to know that $\int_a ^ b |f'(x)|$ gives the length of any contour. Where $f(x)$ is a piece-wise differentiable function from $[a,b]$ to $\mathbb R^2$. I was reading complex integral . Can anyone ...
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### Cauchy Principal Value for Complex Oscillating Function

I'm not particularly well-versed in how to compute Cauchy principal values, so any help here would be appreciated. I am trying to evaluate the PV for the following integral: \begin{align} \int_{-\...