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

learn more… | top users | synonyms

5
votes
0answers
31 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
1
vote
1answer
32 views

Complex Integration and deduce that function is constant

Let $f$ be an entire function, $z_{1}$, $z_{2}$ $\in$ $C$, with $z_{1} \neq z_{2}$ and $R>\max{(|z_{1}|,|z_{2}|)}$. Prove that $$2\pi i\dfrac{f(z_{1})-f(z_{2})}{z_{1}-z_{2}} = ...
0
votes
0answers
8 views

Parametrizing shapes, curves, lines in $\mathbb{C}$ plane

I've been struggling with parametrizing things in the complex plane. For example, the circle $|z-1| = 1$ can be parametrized as $z = 1 + e^{i\theta}$. I'm not sure how this was done. I understand how ...
1
vote
1answer
25 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
2
votes
1answer
30 views

Half-Fourier transform, relation to Delta function

so the Fourier transform of the Kronecker Delta function is (up to sign conventions / normalisation) $$\int_{-\infty}^\infty dt\; e^{i t \omega} = \delta(\omega).$$ Can one say anything about the ...
0
votes
0answers
32 views

How to prove that the integration contour of one integral is the subset of the other integral? [closed]

I have the two integrals which have the same non negative integrand. For example the following two integrals, $\int_{-l}^{l}f(x)dx$ ....... (1) and $\int_{a}^{b}f(x)dx$ ....... (2) What we ...
3
votes
1answer
55 views

Compute $\int_{\gamma} \frac{z}{z^3-1} dz$ where $\gamma$ is circle centered at origin of radius 2

Compute $$\int_{\gamma} \frac{z}{z^3-1} dz$$ where $\gamma(t)=2e^{it}$, $t\in[0,2\pi]$. The first part of the problem had me compute the same integral over the path $\gamma(t)=\frac{1}{2} ...
0
votes
1answer
33 views

Show that $\frac{1}{2\pi} \int_0^{2\pi} e^{ik\theta}d\theta = 1$ or $0$ depending on $k$.

I'm asked the problem (restating from the question title), $$\textrm{Show that }\frac{1}{2\pi} \int_0^{2\pi} e^{ik\theta}d\theta = \begin{cases} 1,\, k=0\\ 0, \, k\neq0 \end{cases}$$ My Attempt: ...
0
votes
0answers
36 views

Complex Integration Over an Ellipse

How do we evaluate $\int1/\sqrt{1-z^2}dz$ over the ellipse with the standard form with $a^2$$-$$b^2$=$1$$?$ I was trying to use the Cauchy's Integral Formula and the fact that a circle is homotopic to ...
0
votes
2answers
34 views

Show that $\int_{|z|=3} \frac{1}{z^2-1} dz = 0$

Here's a homework problem I'm having some trouble with: Show that $$ \int_{|z|=3} \frac{1}{z^2-1} dz = 0$$ So far, I've shown using Cauchy's Integral Formula that $$ \int_{|z-1|=1} ...
0
votes
1answer
34 views

Complex integral with imaginary exponent: $\int_0^\pi i \exp((i\theta)^{1+i}) d\theta$

How to approach the integral $$ \int_0^\pi i e^{(i\theta)^{1+i}} d\theta $$ I know I can't multiply the exponents, but what can I do? Am I at least right that the above is equivalent to $\int_0^\pi ...
1
vote
3answers
61 views

Using Residue Theorem to calculate the integral

for $$I=\int_{|z|=1}{z^m \cos\left(\frac{1}{z}\right)}\,dz$$ where $m=0,1,2...$ Is the singularity $z=0$ or there are some other singularities? if it is $z=0$, what's order of pole?
0
votes
1answer
29 views

Compute $\int_\gamma\overline{\zeta} \, d\zeta$ using Cauchy’s Integral Formula

Let $\gamma$ be the circle of radius $1$ and centre $0$, equipped with the counterclockwise orientation. Compute $$\int_\gamma\overline{\zeta} \, d\zeta$$ using Cauchy’s integral formula. Any hints ...
1
vote
1answer
11 views

Complex integral, absolute value of integrand

I want to integrate $f(z)=\frac{1-\mathrm{e}^{\mathrm{i}z}}{z^2}$ over the indented semicircle in the upper half-plane positioned on the $x$-axis as pictured below. The book (Complex Analysis by ...
1
vote
0answers
17 views

Complex Analysis- Integral Gamma

Calculate $\int_\gamma (1-e^z)^{-1} dz$ if $\gamma (t)=2i+e^{it}$ Need help getting started.. hints?
2
votes
1answer
30 views

Show $\int_0^{\infty} \frac{e^{-x}-e^{-xw}}{x} dx = \ln{w}$ for $\operatorname{Re}({w})>0$

I want to show that for $\operatorname{Re}({w})>0$, $$\int_0^{\infty} \frac{e^{-x}-e^{-xw}}{x} dx = \ln{w}.$$ I've tried setting the problem up as: $$\int_\gamma \frac{e^{-z}}{z} dz = 0,$$ where ...
0
votes
0answers
22 views

Change of variable in complex integral

When I want to evaluate the following integral $$\int_\Gamma e^{-z^2}dz~~~~~~~~~~~\Gamma:|z|=R,0\leq\arg{z}<\frac{\pi}{4}$$ so I subtitude, letting(choose a single-valued analytic branch) ...
-1
votes
1answer
74 views

I wanna know another method of Integration int 1/(a+bsinx) dx

$$\int_{0}^{2\pi} \frac{1}{a+b\sin(x)} dx = \frac{2\pi}{\sqrt{a^2-b^2}} ~ \text{if} ~ a^2 > b^2 $$ I know the trick substituting $y=\tan(x/2)$ But I'd like to know another method. For example ...
3
votes
2answers
45 views

How to integrate $\int_{0}^{\infty} \frac{dx}{\sqrt{x}(x^{2}+1)}$ using the residue theorem.

He was doing this integral using the formula $$\int_{0}^{\infty} \frac{dx}{\sqrt{x}(x^{2}+1)}= \frac{2\pi i}{1-e^{-2\pi i\alpha}}(\sum(Res(\frac{F(z)}{z^{\alpha}};z_{k})))$$ where ...
1
vote
2answers
78 views

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$

Evaluate $\begin{align} \int^{\infty}_{0} \dfrac{x^\alpha}{(1+x^2)^2}\end{align}dx, \ -1 < \alpha<3.$ May I verify if my solution is correct? Thank you. Consider ...
1
vote
1answer
34 views

General Cauchy theorem aplication

Let $a \neq b\in \mathbb{C} $ and $U := \mathbb{C} -[a,b] $ Let $\Gamma$ be a cycle in $U$. The following equality is true? $$\int_{\Gamma} \frac{1}{(z-a)(z-b)}dz=0$$ I saw it some notes of a ...
1
vote
0answers
26 views

How to integrate $\int\limits_C{\frac{sin\pi z}{(z^2-1)^2}}dz$, where $C: |z-1|=1$ using Cauchy's formula?

How can evaluate $$\int\limits_C{\frac{sin\pi z}{(z^2-1)^2}}dz$$, where $$C: |z-1|=1$$ by using Cauchy's formula. I have to use Cauchy's formula. Cauchy's formula $$f(z_0)=\frac{1}{2\pi ...
0
votes
0answers
33 views

Boundary line integral

I was trying to do this integral $$ \oint_{\left\vert\,z - 2\,\right\vert\ =\ 2}z^{4}\sin\left(\, z\,\right)\,{\rm d}z $$ by the definition of line integral $\displaystyle\int_{a}^{b}{\rm ...
1
vote
1answer
68 views

Calculating the complex line integral along a square

Calculate the complex line integral of the holomorphic function g(z)=1/z along the counterclockwise-oriented square of side 2, with sides parallel to the axes, centred at the origin. I parametrized ...
2
votes
0answers
57 views
+50

Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
1
vote
1answer
25 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
1
vote
0answers
20 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 $k\in\left(0,1\right]$. ...
5
votes
2answers
64 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
0
votes
0answers
36 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
1
vote
2answers
54 views

How to integrate $x\times \frac{\sin(x)}{x^2+a^2}$ from zero to infinity

I am trying to evaluate $\int_0^\infty\frac{x \sin(x)}{x^2+a^2} dx$. I get $\frac{\pi}{4} \sin(ia)$ using residue theorem. I integrated over the path that goes from -R to R along the real axis and ...
1
vote
1answer
35 views

complex integration-how to solve the given problem

how do we calculate the value $\frac{1}{2\pi i}\int\frac{\sum_{n=0}^{15}z^n}{(z-i)^3}dz$ in $C$:|z-i|=2 ? the answer for this is 1+15i.. how to get it? can someone please explain?
1
vote
1answer
35 views

Estimate of complex integral

Prove that $$ \left|\int_c (2-\frac{e^z}{z-\log 2}) dz \right| <\frac{2}{3} $$ when C is the part of circle $\left| \frac{z}{\pi} -1 \right|^2 =2$ where $Re(z)\geq 0$. ($\log$ means natural ...
2
votes
1answer
42 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
1
vote
2answers
45 views

Contour expression explanation

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$ contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and ...
3
votes
1answer
40 views

Is this Integral calculation correct?

Can someone confirm if my solution is right or if I have done something that is not permitted $$ \begin{align} & \int_\gamma e^{\pi z}=\int_\gamma \left( \frac{ e^{\pi z}}{\pi}\right)' \, dz ...
0
votes
1answer
46 views

Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
0
votes
1answer
46 views

How to find the area where $\frac{1}{z^2-4}$, $z \in \mathbb{C}$ is holomorphic?

Suppose that you are given a problem of finding the following complex integral: $$\int_\tau \frac{1}{z^2-4} dz$$ where $\tau = \{z \in \mathbb{C}: |z|=4 \}$. My question is (in the context of this ...
1
vote
2answers
51 views

Contour integral of complex logarithm

Evaluate $$\int_C Log(z) dz$$ where $Log(z)$ is the principle branch of the complex logarithm (Arg$(z)\in(-\pi,\pi)$) and $C$ is the contour given by the horizontal line connecting $z=i$ to $z=i+1$, ...
2
votes
1answer
164 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
2
votes
1answer
94 views

Complex integral involving logarithm

I've been working on this integral for quite a while and I think I've been able to progress but now I'm stuck. So I have to prove that $$\int_C f(z)\ dz =\int_C\frac{2z}{(1+z^2)\log(2+z^2)}dz =\pi ...
2
votes
1answer
39 views

can't follow the steps of a specific complex integration

Hi: I already asked this question on the complex analysis tag but nobody answered it so then I found this complex-integration tag and was hoping that someone might be able to answer it here. It is ...
0
votes
0answers
84 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
1
vote
4answers
86 views

Help with Complex integration

I have to calculate the following integral $$\int_{-\infty}^{\infty} \frac{\cos(x)}{e^x+e^{-x}} dx$$ Anyone can give me an idea about what complex function or what path I should choose to calculate ...
1
vote
1answer
44 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
0
votes
2answers
31 views

Cauchy integral formula or something else?

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$ with $C_1(0):\left|z\right|<1$ positive. Additionally ...
3
votes
2answers
76 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
3
votes
1answer
70 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
1
vote
2answers
40 views

Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$

Let $a_1, \dots ,a_n$ be arbitrary complex numbers. Define: $f \left( x \right)=\sum_{k=1}^{n}a_ke^{2 \pi ikx}$. I wish to show that there exists an $x\in\left[ 0,1 \right]$ s.t. $f \left( x ...
6
votes
2answers
97 views

The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$

How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$. I can easily take the residue in ...
0
votes
0answers
40 views

Solving an ordinary differential equation with two functions

Please I am trying to solve this differential equation but I do not seem to get the correct answer. Please help thank you. My equation is; $\frac{dc(x)}{dx} = 1+ b\frac{y(x)-y(x-1)}{y(x-1)}$ where b ...