Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

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43 views

How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (residue theorem)

How to solve the following integral $\int_0^1 \dfrac{x^4}{\sqrt{x(1-x)}}dx$ (although we may use normal real integral to solve, I wonder if contour analysis can also help?) The question offers a hint ...
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1answer
52 views

How do I calculate the definite integral $\int_{-1}^1\frac{\sqrt{1-x^2}}{1+x^2}dx$ using complex variables?

I have tried solving the integral $\oint_{C}\frac{\sqrt{1-z^2}}{1+z^2}dz$ using the upper semi-circle contour; I am getting the poles $z=\pm i$. Only $z=i$ exists within the contour and I have ...
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19 views

Integral Value through Complex Integration (residue theorem)

I'd like to know how to evaluate the integral $$I=\int_0^\infty\frac{e^{-s^2}\sin(s)}{s}\,ds=\frac{\pi}{2}\text{erf}(1/2)$$ through the residue theorem. My first steps were to expand $\sin$ as ...
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63 views

Evaluating $\int_{(c)}\frac{x^{s}}{s^{k+1}}ds$

Here, $(c)$ is the path $c+it$, $c\in\mathbb{R}_{>0}$, $-\infty < t < \infty $ oriented 'upwards', $x$ is a positive real number and $k$ is a positive integer. The answer is given as $0$ for $...
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22 views

Find Stationary Points of Function in 4 complex variables

I have four contour integrals to evaluate around the origin for a function in the variables $z=(z_1,...,z_4)$. I want to evaluate such integrals by saddle point approximation, i.e. I need to find the ...
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1answer
69 views

Calculate $ \oint_\gamma \frac{\ln(1 - \overline z)}{z - w} dz$

I'm trying to calculate $$ \oint_\gamma \frac{\ln(1 - \overline z)}{z - w} dz$$ where I'm taking the principal branch of the logarithm, $\gamma$ is a smooth curve in the complex plane and $w \in \...
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1answer
45 views

How to calculate this complex integral (with poles on the contour)

I've come up with calculating this complex integral: Compute $\displaystyle\oint_C \dfrac{z}{z^2+4z+3} \mathrm{d}z$, where $C$ is the circle with center -1 and radius 2. The function has a pole, ...
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1answer
20 views

Complex Variable Integration over the quarter-circle C from $z=4$ to $z=4i$

I have to integrate three functions over the quarter-circle from $z=4$ to $z=4i$. The functions are: $z^2$, $|z|^2$, and $\bar{z}$. I'm trying to parameterize them because the circle $C(t) = 4\cos t + ...
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1answer
32 views

A non vanishing function has an nth root

If $V$ is a simply connected domain, then will any continuous function $f:V\to\mathbb{C}$ with no zeros in $V$ have a continuous $n^{th}$ root? Now, if the function were holomorphic, then we could ...
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1answer
34 views

Integral yielding auxiliary function for sine/cosine integrals

The Abramowitz & Stegun section on exponential integrals and related functions includes the following for $\Re(z) \ge 0$: $$ \int_0^\infty \frac{\sin t}{t+z} \mathrm{d}t = \int_0^\infty \frac{e^{-...
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Unable to deduce an upper bound for an complex integral to get the order of growth

I am getting stuck on some detail part of a practice problem. I am trying to deduce an upper bound for $|\int_{-\infty}^{\infty} e^{-|t|^\alpha} e^{2\pi izt}dt|$ to get the order of growth in the form ...
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1answer
26 views

Contour integral with path being the bottom half of circle followed by a line segment

Calculate $$\int_\gamma \sin^2(z)\cos(z) dz,$$ where $\gamma$ is the path from $3\pi$ to $i$ consisting of the following two pieces: The bottom half of the circle $|z-2\pi| = \pi$ followed by the line ...
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35 views

Complex integration path notation

I'm struggling to interpret the notation for this integral and can't find a direct definition for it. $\int_{(1/\mathcal{L})}f(s)ds$. Here, $\mathcal{L}:=log(x)$ for a large $x\in\mathbb{N}$. What ...
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1answer
49 views

Solving the Integral: $f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt$ when $L$ is odd

I want to solve the following integral when $L$ is odd: $$ f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt $$ which can be simplified to: $$ f(...
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32 views

Limit of Complex Integral is 0

How can I show that the $ \lim_{R \rightarrow \infty}$ $\int_{|z|=R} \frac{dz}{(z)(z-3)^2}$ = 0? I have already proved that by ML - inequality, $\Big| \int_{|z|=R} \frac{dz}{(z)(z-3)^2} \Big|$ $\leq \...
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Establish an upper bound for $\zeta'(s)\over\zeta(s)$?

Using Perron's formula, I am able to show that $$ \psi(x)={1\over2\pi i}\int_{a-i\infty}^{a+i\infty}\left[-{\zeta'(s)\over\zeta(s)}\right]{x^s\over s}\mathrm ds $$ where $\psi(x)$ is the Chebyshev's ...
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65 views

Computing $\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz$ using Cauchy integral formula

Let $\alpha(t) = re^{it}$ where $|a|<r<|b|$ and $t \in [0,2\pi]$. I'd like to compute $$\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz \ \ \ \ n, m \in \mathbb{N}.$$ It appears that the answer is $$2\...
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47 views

Calculate $\oint_\gamma \frac{\ln(\overline{z} - b)}{z - a} dz$ using the Cauchy-Pompeiu formula

I'm trying to learn how to calculate $$ \oint_\gamma \frac{\ln(\overline{z} - b)}{z - a} dz$$ using the Cauchy-Pompeiu formula: $$ f(a) = \frac{1}{2\pi i}\int_{\gamma} \frac{f(z) \,dz}{z-a} - \frac{1}{...
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23 views

Abel-Plana formula applied to $\frac{1}{(x^2+a^2)^2}$

This is a follow-up on a previous post, where from the comments I realized the remainder term in the Euler-Maclaurin formula applied to $f(x)=\frac{1}{(x^2+a^2)^2}$ does not vanish, as also a ...
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41 views

How can I calculate $\int_{-\infty}^{\infty}e^{-x^2}\cos(2bx)\,\mathrm{d}x$ using residue theorem?

I have the following integral $$\int_{-\infty}^{\infty}e^{-x^2}\cos(2bx)\,\mathrm{d}x$$ How can I prove this integral equals $\sqrt{\pi}e^{-b^2}$ by using the residue theorem? For $b>0$ and $\...
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7 views

Integral of the product of derivatives of a function and its complex conjugate

Let z(x) be a complex function of x and z*(x) its complex conjugate. I wish to evaluate the following: ∫[(dz*/dx)×(dz/dx)]dx The limit of integration is minus infinity to plus infinity. How do I go ...
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16 views

Complex integral of $\int_{\alpha_r}f = 0$ when $f$ is even

Let $D \subset \mathbb{C}$ be a domain with the property $z \in D \Rightarrow -z \in D$ and $f: D \to \mathbb{C}$ is continuous an even $(f(z) = f(-z))$. Let $r > 0$ and let $\overline{U}_r(0)$ be ...
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35 views

An application of the Plemelj's formula

I don't quite understand the following example from this paper (Page 488 after theorem 4.1) https://epubs.siam.org/doi/abs/10.1137/1007105?journalCode=siread: Find $$ P\int_0^1 \left(\frac{1-t}{t}\...
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42 views

When is a complex function orthogonal to its derivative

Consider the complex valued function $f(t)$ where $t \in [0, 2\pi)$. Under what conditions are $f(t)$ and $f'(t)$ orthogonal to each other? I'm defining orthogonal here to be $$ \int_0^{2\pi} \...
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1answer
37 views

Why I can change the order of summation and integral here?

Before asking, here I bring Theorem 10.7, Rudin's RCA. (Theorem begins.) Let $X$ be a measure space with a complex measure $\mu$. Let $\varphi$ be a complex measurable function on $X$. Let $\Omega$ ...
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1answer
61 views

How to integrate this function? $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}(1+x^2)}\,dx$

How to find definite integral $$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}(1+x^2)}\,dx$$ using complex intergral? And if $$ f(z) = \frac{1}{\sqrt{1-z^2}(1+z^2)}\,$$ There are simple poles at $$ z = i , z = -i ...
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52 views

Integrating $\log(z − 1)/z^{n+1}$ anticlockwise around the unit circle.

Integrate $\log(z − 1)/z^{n+1}$ where $n$ a positive integer, anticlockwise around the unit circle. If we consider the principal branch of $\log(z-1)$ the integral will be very difficult to evaluate, ...
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22 views

Compute $\int_{x=-\infty} ^\infty \frac{\tanh (x)}{\prod_{k=1}^N (x-(x_k+i))}dx$

I am trying to compute the following complex integral $$ \int_{-\infty}^{\infty} \frac{\tanh\left(x\right)} {\prod_{k = 1}^{N}\, \left[\,{x - \left(\,{x_{k} + \mathrm{i}}\,\right)}\right]}\,\mathrm{d}...
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1answer
25 views

Finding Laurent series and residue at the origin [closed]

Can someone help me with two problems about Laurent series? I`m trying to find the Laurent series of the following functions: $$f(z)=\frac{3 e^{5z}}{z^3}+1 \text{ and } g(z)=\frac{z}{3z^2+7}.$$ I ...
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28 views

Magnitude of difference of two complex exponentials

I am in a Fourier Transforms course, and I ran into some trouble understanding how my professor went from this step in his solution: $$\int_{-\pi}^\pi \left|{\frac{1}{\sqrt{2\pi}} e^{imx} - \frac{1}{\...
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1answer
28 views

Complex integral $\int_C{e^{z^3}z^2}dz$ over a parameterized curve $z(t)=t+t^{10}i$

Compute $\int_C{e^{z^3}z^2}dz$ where C is given by $z(t)=t+t^{10}i$, where $t\in [0,1]$. I think this is easier than I am making it, but can we just do u-substitution on this, and then integrate the $...
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46 views

Asymptotic expansion for $\int e^{i\lambda x^2}e^{-x^2}x^j\text{d}x$

I want to prove Theorem 2.5 on page 9 from these lecture notes by following the guidelines provided in Exercise 13 on page 18. Using complex integration one shows that \begin{equation}\int_{\mathbb{R}}...
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31 views

How to show an integral of a complex function is analytic?

The question is, with an interval $I=[a,b]\subset\mathbb{R}$, as well as a function $f:I\to\mathbb{C}$ how to show the following function: $$F(z)=\int_a^b\frac{f(t)}{t-z}\,dt$$ is analytic on the ...
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1answer
36 views

Complex integration over a surface

I am trying to define complex integration over a surface. I have found some notes and books here and there but nothing that defines it rigorously. So suppose we have the complex integral of a function ...
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Why does the argument principle of an 'unknown' complex-valued function always give non-zero values?

This question is motivated by a physics problem, and so uses non-standard notation. I have a 2D parameter space $\mathcal{M}=(x,y)$. Over it, I choose a smooth, closed, non-intersecting curve $\...
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1answer
59 views

Evaluate : $\displaystyle\int\limits_{\gamma }\frac{\log (1+z)}{1+z}dz$

Computer $$\displaystyle\int\limits_{\gamma }\frac{\log (1+z)}{1+z}dz$$ Where : $$\gamma =\{ |z|=1~ ; ~\Re z≥0,\Im z≥0 \}$$ I try : $z=e^{it}$ then $dz=ie^{it}dt$ And $t\in [0,\frac{π}{2}$ then ...
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33 views

Complex line integral in terms of x and y

Compute the complex line integral $\int (x^2 + iy^2)dz $ over the line segment from 0 to 2+i. I am struggling to understand what to do with the $x^2 +iy^2 $ term. I have been able to do these more ...
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1answer
36 views

Estimating $|\int_{\beta}\exp(iz^2)\ dz|$

Let $R > 0$ and consider a curve $$\beta(t) = R\exp(it), \ \ \ \ \ \ \ \ \ 0 \leq t \leq \pi/4.$$ I need to show that $$\left|\int_{\beta}\exp(iz^2)\ dz \right| \leq \frac{\pi(1-\exp(-R^2))}{4R}.$$ ...
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31 views

Cauchy-Goursat Theorem with singularities in $\partial R_0$

Consider $\overline{R}:[-a,a]\times [-b,b]$ and $\varphi: [-a,a] \rightarrow [-b,b]$ a continous map. Let $K$ be the graphic of $\varphi$. if $f: int(\overline{R}) \rightarrow \mathbb{C}$ is continous ...
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44 views

Complex Integration (Residue Theorem)

How do I integrate $ \oint_{C:\left | z \right |= R}^{}\frac{e^{^{\frac{1}{z}}}}{z^{2}+1}dz $ , with $ 0< R< 1 $ ? I am supposed to use the residue theorem but there's no Laurent series around z=...
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2answers
39 views

Evaluate $\int_{|z| = r} \frac{1}{a-\overline{z}}dz$

Evaluate $\int_{|z| = r} \frac{1}{a-\overline{z}} dz$ where $|a|\neq r$. I was trying to find a way to connect this integral to $\int_{|z|=r} \frac{1}{a-z}dz$. However this method does not work, and ...
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25 views

Cauchy's integral theorem proof from Green's theorem

I am studying Cauchy's integral theorem from shaum's outline,the theorem states that Let $f(z)$ be analytic in a region $R$ and on its boundary $C$. Then $$\oint_{C}f(z)dz=0$$ After the statement of ...
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1answer
168 views

how to calculate the integral $\int_{0}^{+\infty}\displaystyle\frac{x^\beta\cos(ax)}{x^2-b^2}dx$

How to calculate the following definite integrals $?$: $$ \int_{0}^{\infty}{x^{\large\beta}\cos\left(ax\right) \over x^{2} - b^{2}}\,\mathrm{d}x\,\,\,\mbox{and}\,\,\, \int_{0}^{\infty}{x^{\large\beta}\...
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1answer
65 views

Evaluating a definite integral using complex integration

How would I solve (evaluate) an integral using methods of complex integration, in particular $\int_{0}^{\pi}\frac{\sin x}{2+\cos x}dx$ ? If the boundaries went from $0$ to $2\pi$, I could use the ...
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46 views

Find the Laurent expansion of $f(z)=\frac{1}{(z^2-1)^2}$

Find the Laurent expansion on $0<|z-i|<2$ of: $$f(z)=\frac{1}{(z^2-1)^2}$$ I would like to try this exercise by imitating another that is similar: $$f(z)=\frac{1}{(z^2+i)^2}$$ where the Laurent ...
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2answers
35 views

Find the region of convergence

Find the region of convergence $$i) \sum_{n=1}^{\infty} \frac{(-1)^nz^{2n-1}}{(2n-1)!} $$ $$ii) \sum_{n=1}^{\infty} n!z^n $$ $$iii)\sum_{n=1}^{\infty} \frac{(-1)^nz^{n(n+1)}}{n} $$ I got the following:...
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1answer
55 views

Is there any way to tackle this integral (generating functions)?

I came across this integral when trying to get the coefficient of a generating function: $$\int_0^{2\pi} {(1-p + 2p e^{it})^n\over 1-e^{-3it}/8}\;dt$$ Here $p\in [0,1]$ and $n$ is a positive integer. ...
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2answers
42 views

Calculate the following integrals of complex variable

I am beginning to compute integrals of complex variable functions. As soon as I have them I will share my solutions, so that you can give me your opinion. $$ i) \int_{|z|=1} \frac{e^z}{z^n} dz$$ $$ ii)...
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1answer
42 views

Finding $M_0$ and $k>1$ such that $|\frac{\ln(z+i\sqrt{5})}{z^{2} +7}|\leq \frac{M_{0}}{R^{k}}$

I need to find $M_0$ and $k>1$ constants such that $|\frac{\ln(z+i\sqrt{5})}{z^{2} +7}|\leq \frac{M_{0}}{R^{k}}$ with $z=Re^{i\theta}$. This is what I got so far: Since $|z|^{2} -7 \leq |z^{2}+7|$, ...

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