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

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1answer
27 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
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1answer
36 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 ...
3
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1answer
60 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} ...
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1answer
34 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: ...
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0answers
46 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 ...
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2answers
42 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} ...
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1answer
37 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 ...
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3answers
68 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?
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1answer
34 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 ...
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1answer
18 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 ...
2
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1answer
31 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
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0answers
23 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) ...
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1answer
132 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 ...
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2answers
46 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 ...
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2answers
87 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
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1answer
38 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 ...
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0answers
30 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 ...
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0answers
35 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
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1answer
105 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 ...
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2answers
159 views

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 ...
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1answer
29 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} ...
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0answers
21 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
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2answers
68 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 ...
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0answers
43 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 ...
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2answers
59 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
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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
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1answer
39 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
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1answer
46 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? ...
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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
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1answer
44 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
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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
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1answer
48 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 ...
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2answers
56 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
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1answer
225 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
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1answer
103 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
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1answer
40 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 ...
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0answers
96 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 ...
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4answers
88 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
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1answer
47 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
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2answers
32 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
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2answers
81 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
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1answer
74 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 $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
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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
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2answers
100 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 ...
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0answers
41 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 ...
1
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2answers
84 views

Finding $\int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt$

I'd like to ask something about the following integral: $$ \int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt $$ I rewrote and took another variable. $$ ...
4
votes
1answer
90 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
0
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1answer
37 views

Complex exponent integral - prove $\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $

How to prove the exponent integration rule: $$\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $$ In the complex version of it - that is, when $\lambda \neq ...
4
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2answers
152 views

Complex integration, any ideas?

I need to solve the following integral of a function analytic on an open environmemt of the unit circle, but i dont know how to get that answers $$\frac{1}{2\pi ...
2
votes
3answers
70 views

Help with equality of complex integrals

I need to prove this equality of integrals...but i dont know how to begin, so if anyone can give an idea... Let f a continuous function on $\overline{D}=\{z : |z|\leq 1\}$. Then: ...