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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.

The concept of definite integral of real functions does not directly extend to the case of complex functions, since real functions are usually integrated over intervals but complex functions are integrated over curves.

Surprisingly complex integration is not so complex to evaluate, often simpler than the evaluation of real integrations. Some real integrations which are otherwise difficult to evaluate can be evaluated easily by complex integration, and moreover, some basic properties of analytic functions are established by complex integration only.

References:

https://en.wikipedia.org/wiki/Contour_integration

"Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable" by Lars Ahlfors

"Complex Variables and Applications " by James Ward Brown and Ruel V. Churchill

3259 questions
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How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity and then integrating along the entire real line. I…
Argon
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25
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2 answers

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and $\displaystyle\int_0^{\infty}\displaystyle\frac{\sin{ax}}{x^3+1}dx$ while the constant $a$ can be…
Layman33
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3 answers

I think I don't truly understand Cauchy's Integral theorem

Cauchy's theorem states that closed line integral of some holomorphic functions yields zero, in some good regions (i.e. simply connected domain). More explicitly, $$ \oint_\gamma f(z) d z=0. $$ Many textbooks use Goursat's theorem or Green's theorem…
SunnyMath
  • 349
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Show $|\int f(z)\, dz|\leq4$

$$\bbox[30px,border:2px solid black] {{\textbf{With great sadness the MSE community has taken} \\ \textbf{notice of the death of Christian Blatter.} \\ \textbf{This is Christian's last recorded question,} \\\textbf{as he passed…
Christian Blatter
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23
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2 answers

Summation using residues

In reference to this question about showing that the following interesting series takes on the value $$\sum_{n=0}^\infty \frac{1}{(2n+1)\operatorname{sinh}((2n+1)\pi)}=\frac{\log(2)}{8}$$ I tried the approach of using the residue theorem, from which…
Ron Gordon
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What is the geometric intuition for the $\bar \partial$-Poincare lemma, or for $\bar \partial$ more generally?

The one variable $\bar \partial$-Poincare lemma is proven in Huybrechts and Forster and so on in essentially the same way: one shows that for a local form $f d \bar z$, with $$g(z) := \frac{1}{2\pi i} \int_{B_\varepsilon} \frac{f(w)}{w-z} dw \wedge…
forget this
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19
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Integration of $\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$ by means of complex analysis

Dear all: this time I have the integral $$\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$$and we must try to solve it using complex integration, residues, Cauchy's Theorem and the whole lot. (BTW, does anyone have any idea whether this integral can be…
DonAntonio
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Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty e^{-x^2} \text{d}x$. Problem is I feel insecure…
Anguepa
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Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually?

Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually? (Like $\Gamma(\frac 12)$) If there is/are, could you show me how to calculate it? I found that $\Gamma(i)$ cannot be calculated by hand, but only can be…
Hyeonseo Yang
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Discussing the Integral of $\exp(-x^n)$

I am aware that $$\int_{0}^{\infty}e^{-x^2}\ dx = \frac{\sqrt{\pi}}{2}$$ but I was wondering if there was a general case for other exponents. Particularly: $$\int_{0}^{\infty}e^{-x^n} \ dx$$ where $n$ is a real number greater than $1$ (although I…
Gizmo
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Evaluate $\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt$

$$\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt=-\frac{{\pi}}{4}\ln2+\frac{1}{2}K$$ I ran across this integral while investigating the Catalan constant. I am wondering how it is evaluated. I know of this famous integral when the limits of integration are…
Cody
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4 answers

Numerical Integration for integrable singularity

Till this time i have learned three numerical technique to find the definite integration. They are Simpson, Trapezoidal and Gauss-legendre formula. The sad thing is that I can't apply these theorem directly of my integration has any integrable…
Numerical Person
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17
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3 answers

If $f$ is entire and $\lim_\limits{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant

I'm learning Complex Analysis and need to verify my work to this problem since my textbook does not provide any solution: If $f$ is entire and $\lim_\limits{z\to\infty} \dfrac{f(z)}{z} = 0$ show that $f$ is constant. My work and thoughts: From the…
glpsx
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16
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4 answers

Integral of $e^{ix^2}$

How does one evaluate $$\int_{-\infty}^{\infty} e^{ix^2} dx$$ I know the trick how to evaluate $\int_{-\infty}^{\infty} e^{-x^2}dx$ but trying to apply it here I get a limit which does not converge: $I = \int_{-\infty}^{\infty} e^{ix^2}dx =…
Christian
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4 answers

Cauchy-Riemann equation analogue but for the quaternions

given a function over the quaternions $$ U(x,y,z,t)+iV(x,y,z,t)+jW(x,y,z,t)+kR(x,y,z,t)=f(x,y,z,t) $$ what are the analogues of the Cauchy Riemann equation for the quaternionic plane so the function defined above is analytic ?? what happens with the…
Jose Garcia
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