For questions about integration methods that use results from complex analysis and their applications

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Real integral done by complex methods [duplicate]

$\int_{-\infty}^{\infty} \frac{cosx}{x^2+25} dx $ = $ \frac{\pi}{5e^5}$ Any ideas?
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1answer
21 views

Applying Cauchy theorem to the integral of $\overline{z}^2$ over two different curves.

I have solved the following exercise, in which I had to compute $$\int _\gamma \! \overline z ^2 \, \mathrm{d}z,$$ where $\gamma$ is The circumference $\left| z\right| =1$ The circumference ...
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27 views

How to calculate the real integral below by contour integral?

In a paper, I meet two integrals: $\displaystyle I_1=\int_M^{M-\omega}\frac{dx}{1-\sqrt{\frac{2x}{r}}}=i\pi r$ and $\displaystyle ...
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1answer
124 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me ...
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2answers
115 views

How to solve $ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $

I am trying to solve this integral, I think that it could be solve using the complex. $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
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18 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
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1answer
63 views

Integration of a analytic function

here is the problem I currently try to solve: $$\int\limits_{-\infty}^{+\infty}\left((1+ixa^2)^{-\frac{n_1}{2}}\cdot(1+ixb^2)^{-\frac{n_2}{2}}\right)e^{icx} \mathrm{d}x $$ with $a,b,c\geq0$ (real ...
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2answers
56 views

Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$? [closed]

Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$ for positive real $n$? I've come to this answer by inverse Fourier transform. But since there is n, I am quite confused that I ...
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1answer
23 views

Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$.

Let $f$ be analytic in an open set $U \subseteq \Bbb C$ and let $K \subseteq U$ be compact. Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$ ...
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$|\oint_{\mu_2^R}\frac{ze^{iz}}{z^2+a^2}dz|\rightarrow 0$ as $R\rightarrow\infty$

I have been trying to solve the integral $\int_0^\infty\frac{x\sin(x)}{x^2+a^2}dx$ for $a>0$ by using contour integration. To do this, I defined $f(z):=\frac{ze^{iz}}{z^2+a^2}$, and am trying to ...
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49 views

Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
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61 views

Why does this example of global residue theorem not work?

This question is related to and inspired by a previous question What is the residue obtained from this integral? , but note that the appearing functions are slightly different. Consider the following ...
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1answer
42 views

What is the residue obtained from this integral?

Consider the following integral in two complex variables $z_1$ and $z_2$: $$\frac{1}{(2\pi i)^2}\oint_{{|z_1|=\epsilon}\atop{|z_2|=\epsilon}}dz_1 dz_2\frac{1}{z_1 ...
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1answer
24 views

Consider a positively oriented circle with $| z | = 2$. Calculate $\int z/[ (4-z^2)(z+i)]$.

Consider a positively oriented circle with $| z | = 2$. Calculate $$\oint \frac{z\;dz}{(4-z^2)(z+i)}$$ I did the following. $$\oint\frac{z\;dz}{(4-z^2)(z+i)}=\oint\frac{z}{4-z^2}\cdot ...
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1answer
34 views

Solve the following contour integral (Complex Analysis)

Compute the following integral: $$\int_{\delta D_1(0)} \frac{e^{z}}z dz $$ So I rewrote the formula in terms of $x$ and $y$ since $z = x + iy$ I got $$f(z) = \frac{e^xysin(y) + ...
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32 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
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1answer
23 views

Upper Bound on Complex Line Integral

I'm working through the second edition of Complex Variables by Stephen Fisher, and reached a proof involving the upper bound of line integrals, namely $$ \left| \int_\gamma u(z)\;dz \right| \leq ...
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2answers
41 views

Complex analysis proof about $|f(z)|$

I have to prove the following and have absolutely no idea where to start: If $f$ is holomorphic in $|z|>R$ and its limit at $\infty$ is $0$, then $\exists \; m \in \mathbb{N}$ such that $|f(z)| ...
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1answer
55 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
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2answers
73 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
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2answers
26 views

Prove that: $\oint_c{\dfrac{1}{z^{2n+1}}(z^2+1)^{2n}}=\binom {2n} {n} 2\pi i$

In a mathematical methods problem, where the $c$ is a the unit circle around the origin and in counterclockwise, I need to use a step that I'm not so sure about (Because I don't know how to develop ...
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1answer
122 views

Global Residue Theorem in CP^2.

Consider the following meromorphic form defined on $\mathbb{C}P^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
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45 views

Proof of Sokhotski-Plemelj theorem

Sokhotski-Plemelj theorem states $$ \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\varphi(\zeta) \,d\zeta}{\zeta-z}+\frac{1}{2}\varphi(z), \, \\ \phi_e(z)=\frac{1}{2\pi ...
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2answers
43 views

how to solve this complex exponential integration ??

During exercising and example of Fourier Series , I encountered with an integration : $$ \frac{E\omega_o}{4\pi j}\int_{0}^{\frac{\pi}{\omega_o}}\Big[e^{-j\omega_o (n-1)t}-e^{-j\omega_o ...
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1answer
24 views

A question about complex integration

Let $z_0\in \Bbb C$ and $R>0$. Let $f:B(z_0,R)\to\Bbb C$ be a complex function such that $f=u+iv$ (where $B(z_0,R)$ is the open disc centered at $z_0$ with radius $R$). If $u$ and $v$ have ...
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3answers
63 views

Evaluatig: $\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$

Evaluatig: $$\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$$ Where $a, b\in \mathbb R^+$ What i have done: Because $\cos(bx)=\Re(e^{ibx})$, we can note that: ...
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1answer
22 views

Problem in computing complex integrals for fourier transform

This is from a problem set of open course 8.02 by MIT OCW. I am not able to understand how the integral was solved. I have basic knowledge of Fourier transformation, and the Dirac delta function ...
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56 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
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2answers
69 views

Evaluating: $\int_{-\infty}^{\infty}{\frac{1}{\cosh(kx)}dx}$

How can you integrate: $$\int_{-\infty}^{\infty}{\dfrac{1}{\cosh(kx)}dx}$$ I know that: ...
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2answers
77 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
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0answers
25 views

Contour integration of logarithm: $g(\omega) \log[1 - \chi(q,\omega)]$

I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced ...
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0answers
26 views

Complex integral difficulty

We need some sort of analytic expression for the integral: $$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$ ...
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1answer
79 views

Using The Riemann Zeta Functional Equation

Riemann was able to establish the following link between the Riemann zeta function and the weighted prime counting function $J(x)$. $$\ln(\zeta(s))=s\int_1^\infty J(x)x^{-s-1}dx$$ Using the Mellin ...
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0answers
39 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
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1answer
34 views

Integration along a keyhole

(H. Priestley complex Analysis Chapter 7 Exercise 9) Suppose $f$ is holomorphic inside and on $\gamma(0,1)$. By integration around the usual keyhole like this one : Integration of $\ln $ around a ...
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1answer
89 views

Verifying the complex integral: $\int_0^{\infty}\frac{\cos{ax}}{1+x^4}dx$

Verifying the integral: $$\int_0^{\infty}\dfrac{\cos{ax}}{1+x^4}dx$$ I started considering: $$\cos{x}=\dfrac{e^{ix}+e^{-ix}}{2}\implies \cos{ax}=\dfrac{e^{iax}+e^{-iax}}{2}?$$ So: ...
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1answer
53 views

Complex integral with Fourier

So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral $$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$ ...
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2answers
44 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
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39 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
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2answers
72 views

Translated complex gaussian-type integral: $\int_0^{\infty} \exp(i(t-\alpha)^2) dt$

It's fairly straight forward to show that $$ \int_0^{\infty} \exp(it^2) dt = \frac{\sqrt{\pi}}{2}\exp\left(i\frac{\pi}{4}\right) $$ via complex contour integration over a contour shaped like a piece ...
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1answer
54 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
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0answers
33 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
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1answer
35 views

Complex integral using Residue Theorem with a regularised pole

I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds \ e^{-i\Omega ...
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1answer
36 views

Complex integration: normally on a closed contour?

I have been studying complex integration for a few months now, and it seems my textbook mostly considers integration on closed contours. Is there no interest in integration on non-closed contours ?
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3answers
56 views

Complex integration with trigonometric and logarithm

Show that $\int_0^{2\pi}\log\sin^22\theta dx=4\int_0^\pi\log\sin \theta d\theta=-4\pi \log2$ I did $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ ...
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2answers
180 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did ...
2
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2answers
45 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow ...
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1answer
68 views

Poles on the curve

Say I have this integral: $$\oint_\gamma f(z)\,{\rm d}z,$$and $f$ has a pole on $\gamma$. I understand that we "cut around" the pole with an arc of radius $\epsilon$ and then make $\epsilon \to 0$. ...
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2answers
40 views

Bounding a complex integral over a square

I'm solving the following exercise: Use the estimate lemma to prove that $$\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \leq 4\sqrt{10},$$where $\gamma$ is the square with vertices $\pm 1 ...
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1answer
31 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...