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

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6
votes
0answers
158 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
4
votes
0answers
46 views

Integrate complex function over $\mathbb{C}^2$

I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and ...
3
votes
0answers
2k views

Integrate complex conjugate

I'm doing some exercises on complex integration, and stumbled over this: $$\int_\gamma \bar{z} dz,$$ Where $\gamma$ is any closed $C^1$ curve which is the boundary of a bounded, connected, open $U ...
2
votes
0answers
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 ...
2
votes
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 ...
2
votes
0answers
119 views

Complex line integral for closed homotopic curve.

I try to show that the following analogous result for $\textbf{closed curve}$ : Let $\gamma_0, \gamma_1$ be continuous paths in a domain $D \subseteq \mathbb{C}$ and $\omega$ be a closed form on $D$. ...
2
votes
0answers
33 views

How can I solve $\int \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $ for complex functions $f_i$?

The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$. How can solve this integral? $$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$ ...
2
votes
0answers
57 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
2
votes
0answers
35 views

$ g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm {d}b$

Do you have any idea how to compute this function? $$ \displaystyle g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm ...
2
votes
0answers
277 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
2
votes
0answers
133 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
1
vote
0answers
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 $$G(s,t) = f(s)g(t)H(s,t)$$ ...
1
vote
0answers
31 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} ...
1
vote
0answers
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: ...
1
vote
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 ...
1
vote
0answers
26 views

Complex integral difficulty

We need some sort of analytic expression for the integral: $$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$ ...
1
vote
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 ...
1
vote
0answers
63 views

Using residue theorem find $\int_{\theta-\pi}^{\theta-2\pi}1/(r^2+1-2r\cos u)$

Find the value of $I$, using residue theorem. $$ I= \int_{\theta-\pi}^{\theta-2\pi} \frac{du}{r^2+1-2r\cos(u)} - \int_{\theta}^{\theta-\pi} \frac{du}{r^2+1-2r\cos(u)} $$ with $r<1$ and ...
1
vote
0answers
30 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
1
vote
0answers
45 views

What is the meaning of this integral?

Does anyone know the meaning of this type of integral? $\displaystyle{\int f(z) \,\overline {dz}}$. I think this means $\displaystyle{\int u\,dx + v\,dy+i\int v\,dx -u\,dy}$ where $f=u + iv$
1
vote
0answers
35 views

Using Multiple Branch Cuts in a Contour Integral

I have the integral $$I=\int_0^{\infty}\!\frac{x^{1/2}}{\left(x^3+a^3\right)^{1/2}\left(2x^3+a^3\right)^{1/6}}\,\mathrm dx$$ which I am trying to integrate using complex integration. I know that ...
1
vote
0answers
48 views

Complex integration, limits, arctan

$\left.\frac12i\;\text{Log}\frac{1-(-i+e^{it})}{1+(-1+e^{it})}\right|_0^{2\pi}=\frac12i\left(\log\left|\frac{i}{i}\right|+i\arg 1-\log|1|-i\arg1+2\pi ik\right)$ could someone explain how this is ...
1
vote
0answers
42 views

If f is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ then $|f'(0)|\leq 4$

I need to prove that if $f$ is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ for z all $z\in D_1(0)$ then $|f'(0)|\leq 4$. This is my proof and I need to verify this. Let $n\in ...
1
vote
0answers
32 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
1
vote
0answers
26 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
1
vote
0answers
24 views

Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
1
vote
0answers
32 views

Find natural number to satisfy inequality given with integral

Find the number $N\in \mathbb{N} $ for which the following inequality is true, if $\sigma(t)=|z|=1 $, with $0\leq t\leq \pi/2$ and $\left | \int_{1}^{i} e^{z^{-1}} \right | \leq N \frac{\pi}{2}$. ...
1
vote
0answers
77 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
1
vote
0answers
41 views

Second Derivative of Holomorphic function According to Cauchy's

I am asked to prove that, supposing f is holomorphic on the region G, w $\in$ G and $\gamma$ is a positively oriented, simple, closed, smooth, G-contractiblecurve such that w is inside $\gamma$ , we ...
1
vote
0answers
47 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
1
vote
0answers
159 views

Complex integration from zero to infinity at different directions

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)\ge 0$. Is it correct to substitute $w=\lambda z$ ...
1
vote
0answers
163 views

Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
1
vote
0answers
65 views

Integral of the determinant of the Jacobian in complex variables?

Let $f$ be a function that maps a disc $D$ onto a unit square $A$ such that $f(D) = A$. $f$ is holomorphic and one-to-one. Why does $\int_D|f'(x+iy)|^2 \, dx \, dy = 1$?
1
vote
0answers
240 views

Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0

The ultimate aim is to solve the following integral: \begin{equation} \label{eq:Icos1} \begin{aligned} I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
0
votes
0answers
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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
21 views

$|\oint_{\mu_2^R}\frac{ze^{iz}}{z^2+a^2}dz|\rightarrow 0$ 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 ...
0
votes
0answers
44 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 ...
0
votes
0answers
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 $$ ...
0
votes
0answers
54 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
0
votes
0answers
33 views

About the support of an equivalence class of chains

Let $\Omega$ be a non empty open set of $\mathbb{C}$. Let $\mathscr{C}(\Omega)$ be the set of continuous curves in $\Omega$. Let $H(\Omega)$ be the set of holomorphic functions defined on $\Omega$. ...
0
votes
0answers
29 views

Help with $\int _{R_0<|z|<R_1}\frac{1}{z} dz$.

Consider the integral in $\mathbb{C}\simeq \mathbb{R}^2$ $$ \int_{R_0<|z|<R_1} \frac{1}{z}\; dx_1 dx_2 $$ where $0<R_0<R_1$ and $z=x_1+i x_2$ and $|z|=(x_1^2+x_2^2)^{\frac{1}{2}}$. So ...
0
votes
0answers
10 views

Finding the limits when integrating a complex number

Evaluate $\int_c f(z) dz$ from $z(0,0)$ to $z=2+4i$ where $f(z)=x^2 -iy^2$ I know how to work this out and I know the answer is $24+\frac{8}{5}i$ However I do not understand why the limits for x are ...
0
votes
0answers
22 views

when is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t) \, dt = 0$, with $x \in \mathbb{R}$?

For what value of $x \in \mathbb{R}$ is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t)\,dt = 0$, where $a$ is some constant?
0
votes
0answers
36 views

Lebesgue integration in $\mathbb{C}$

I'm confused as to how we are supposed to integrate $$\frac{1}{\pi}\int_U\left[\frac{d}{dz}\left( \frac{z-\alpha}{1-\bar\alpha z }\right)\right]^2 \, dm$$ where $U$ is the unit disc, ...
0
votes
0answers
54 views

Contour integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
0
votes
0answers
51 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
0
votes
0answers
36 views

Substitution in complex integration

Let $\gamma$ be the circle $|z| = r$, $0 < r < \frac{\pi}{2}$, taken positively. Find $$ \int _{\gamma} \frac{1}{\text{tan}^{17}(z)} dz ~\text{and} \int _{\gamma} \frac{1}{\text{sin}^{15}(z)} dz ...
0
votes
0answers
42 views

(Complex) integral over a half circle

I read a book and find one example which I do not understand : Let $f(z) = \frac{e^{iaz}-e^{ibz}}{z^2}$ on $B(0,1)\setminus\{0\}$. Let $\gamma$ be the half circle over x-axis : $\gamma (t) = ...
0
votes
0answers
24 views

Compute the integrals $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$.

Given two smooth contours, $C_1$ and $C_2$, that respectively lie on the upper and lower half plane compute $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$. Let $a$ be a fixed real positive number. ...