Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [contour-integration]

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

This is the procedure of calculating the contour integral around a given path/contour. It allows us to evaluate integrals on the real line $\mathbb{R}$ that are not able to be evaluated using real-variable methods.

Links:

Contour Integration at Wolfram MathWorld

3880 questions
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Contour integration to compute $\int_0^\infty \frac{\sin ax}{e^{2\pi x}-1}\,\mathrm dx$

How to show: $$\int_{0}^{\infty}\frac{\sin ax}{e^{2\pi x}-1}dx=\frac{1}{4}\frac{e^{a}+1}{e^{a}-1}-\frac{1}{2a}$$ integrating $\dfrac{e^{aiz}}{e^{2\pi z}-1}$ round a contour formed by the rectangle whose corners are $0 ,R ,R+i,i$ (the rectangle being…
meye
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Integration of $\exp(-z^2)/(z-z_0)$

I am interested into the following integral $I = \int_{-\infty}^{+\infty} \frac{f(z)}{z-z_0} dz$ where $f$ is typically a Gaussian function $e^{-z^2}$. I am tempted to simply use Cauchy formula to get $I = 2\pi i f(z_0) = 2\pi i \ e^{-z_0^2}$ But I…
J. Rouxel
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Contour integration . Biot-Savart, maths

The Biot-Savart law for the magnetic field at a point, r , in space due to a constant-current carrying conductor is given as : $$\boldsymbol{B}(\boldsymbol{r}) = \frac{\mu_0}{4\pi}I\int_c…
user341469
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Why is this contour integral undefined?

Define the following contours: I have a question which requires me to, if possible, compute the contour integral $$ \int_{\gamma + \delta + \epsilon} \cos (z) dz $$ However, in the mark scheme it says that this integral is undefined (with no…
M Smith
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Contour integration

In Penrose's 'Road to Reality', he states that for any integer n, $n \ne 1$, $ \oint z^n dz=0$. Qualitatively, why is this so, given that for any negative n poles in the complex graph exist (namely at z=0): integrating around them would surely yield…
Meow
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Contour integration for inverse hyperbolic function

I am trying to solve the integral involving trigonometry and inverse hyperbolic functions, like the following: $I=\int_{-\infty}^{\infty} f(x)\frac{\sin(dx/2)^2}{(dx/2)}dx,\;\;…
FreeMyMind
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Upper bound for $|\int_\gamma f(z)e^{iz} dz |$

Let $\Omega = \{ z \in \mathbb{C} | Im(z) > 0 \}$ and $f : \Omega \to \mathbb{C}$ a continuous function. We suppose that $\forall z \in \Omega : |f(z)| \leq M|z|^n$ We define $\gamma : [0,\pi] \to \mathbb{C}$ by $\gamma (t) = Re^{it}$ with $R…
Jeyrome Sapin
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Contour integral of $f(z) = \frac{1}{z^2+iz+6} $

Need help evaluating a certain contour integral. $f(z) = \frac{1}{z^2+iz+6} $ Steps so far: Poles: $ z^2+iz+6 \rightarrow \frac{-i \pm \sqrt{-1-24}}{2}=0 \rightarrow z_0 = +2i, -3i $ Residues: $ a_{-1} = \lim{z \to z_{i}} […
Zainy
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Contour integral of $\frac{x^{p-1}}{1+x}$

I am trying to find the integral $$\int_0^\infty\frac{x^{p-1}}{1+x}\;\mathbb{d}x$$ I know that this is easily expressible in terms of beta function. But i need to prove that it's value is $\dfrac{\pi}{\sin{p\pi}}$ using a good contour I guess. I…
Someone
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An inequality from complex contour integral on page 81 of Stein and Shakarchi's Complex Analysis

On page 81 of Stein and Shakarchi's Complex Analysis, there is an inequality, \begin{equation} \left\lvert \int_{A_{R}} f \right\rvert \leq \int_{0}^{2\pi}\left\lvert\frac{e^{a(R+it)}}{1+e^{R+it}} \right\rvert…
khalatnikov
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Contour integral of $\int_0^\infty \log(x) e^{-x} dx$

Is it possible to resolve this integral using integral contour? What should be the contour? \begin{equation} \int_0^\infty \log(x) e^{-x} dx = -\gamma \approx -0.577216 \end{equation} where $\gamma$ is the Euler‐Mascheroni constant.
Mauro
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Contour integral with signum function

I need to solve the following integral $$\int\limits_{\left| {s - a} \right| = \delta } {\frac{{{\mathop{\rm sgn}} (is)}}{{{e^{2\pi irs}}}}ds}$$ Where the contour is the semicircle in the upper half plane oriented clockwise. I attempted the…
mathematicus96
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Is the integration $\int_{C_R}e^{ikz}dz=0\ (\text{if}\ k>0)$ correct?

When I read P.W. Milonni's book "Fast light, slow light, and left-handed light", I encounter this problem. In chapter 2 of the book, the author introduces the integral $$G(z,\tau)=\frac{1}{2\pi}\int_{-\infty}^\infty d\omega e^{-i\omega\tau}e^{i…
zrysky
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How is half-contour integration possible?

You integrate in a loop around a singularity $z$ and get $2\pi i \text{Res}(z)$. Is there a path of integration such that the result is $\gamma 2\pi i \text{Res}(z)$ with $\gamma\in (0,1)$? If it helps, I know that if the singularity is simple, then…
Meow
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Wallis' Formula

How can I show the following, for $n\geq 0$: $$ \frac{1}{2\pi} \oint_{\ \Gamma} \frac{1}{z} \left(z + \frac{1}{z}\right)^{2n} dz $$ using a contour $\Gamma$ defined as the unit circle centered at the origin and oriented counterclocwkise. Ref.…
user14685
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