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

learn more… | top users | synonyms

0
votes
3answers
10 views

Not sure how this inequality is formed - $\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$

I have the following inequality in my notes - $$\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$$ We can start as follows ...
3
votes
1answer
78 views

Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$ We see that the only poles are at $x=\pm bi$. Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ ...
1
vote
2answers
44 views

Integral of $((x^2+1)((x-1)^2+1))^{-1}$

Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+1)(2-2x+x^2)}dx$$ So I am going to integrate this using a semicircular contour. Is it safe to say that on the curved part, the integral vanishes? because ...
1
vote
0answers
19 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
1
vote
0answers
24 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 ...
0
votes
0answers
25 views

circular contour integral with complex numbers [closed]

Let gamma(w,R) denote the circular contour t maps to w + Re^it where 0 < t < 2Pi. Evaluate the integral of 1/1+z^2 when gamma is gamma(i; 1)
1
vote
3answers
75 views

Geometrical Interpretation of the Cauchy-Goursat Theorem?

The Cauchy-Goursat theorem is really non-intuitive and is very astounding. Can someone geometrically explain to me why its true? I'm specifically talking about this version of the theorem: For ...
2
votes
1answer
39 views

Limit of an integral that arose in Fourier Analysis

$$\lim_{\alpha\to\infty} \int_{0}^{\delta}\frac{\sin(\alpha x) \sin(\lambda\alpha x)}{x^2}dx$$ where $\lambda,\delta \in \mathbb{R}, \lambda,\delta > 0$. Appreciate your help in finding this limit. ...
2
votes
2answers
42 views

How could I evaluate the integral of type $\int_{-\infty}^{\infty} e^{iax}\cdot e^{-(bx^2 + cx)}$?

I have an integral of type: $$\int_{-\infty}^{\infty} e^{iax}\cdot e^{-(bx^2 + cx)}~ dx$$ And I have no clue on how to integrate that properly. What I've tried so far is writing everything above ...
2
votes
2answers
31 views

Find $\int_\Gamma\frac{3z-2}{z^2-z}dz$, where $\Gamma$ encloses point $0$ and $1$.

Find $\int_\Gamma\frac{3z-2}{z^2-z}dz$, where $\Gamma$ is a simple, closed, positively oriented contour enclosing point $0$ and $1$. This question is on Page 197 of Fundamental of Complex Analysis by ...
5
votes
2answers
117 views

Why do we need a branch cut for $\int_0^{\infty} \frac{x^{\frac{1}{2}}}{{(1 + x)^2}}dx$?

What is the significance of the $x^{\frac{1}{2}}$ in the numerator of this integral. I have read this kind of integral requires taking a branch cut. Why do we need a branch cut, what does it enable us ...
0
votes
0answers
22 views

Plemelj like relation

How to prove the identity: $\frac{1}{(x+i\epsilon)^{n+1}} = P \frac{1}{x^{n+1}} - i \pi \frac{(-1)^n}{n!} \delta^{(n)}(x)$ that holds when integrating ($\epsilon$ is infinitesimal, x, $\epsilon$ are ...
5
votes
1answer
49 views

Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$

I tried to evaluate $$\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$$ when $r>0$. Clearly $\log z$ is not continuous at $z=-r$. So is this integral meaningful then? Since the function is continuous and ...
1
vote
4answers
58 views

Poisson Integral is equal to 1

Show $$ \int_{-\pi}^{\pi}P(r, \theta)d\theta = 1 $$ Let $\alpha(r) = \frac{r^2 - 1}{2r}$ and $\gamma(r) = -\big(\frac{r^2 + 1}{2r}\big)$. Then $$ \frac{1}{2\pi} ...
1
vote
0answers
26 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
1answer
40 views

Homework: Complex integral equation

Suppose $f(z)$ is analytic in the open region $D$, and $C$ is a simple closed curve in $D$. For any $z_0\in D\setminus C$, prove: ...
1
vote
2answers
41 views

$\int_{|z|=2}^{}\frac{1}{z^2+1}dz$

I tried finding the integral of $\int_{|z|=2}^{}\frac{1}{z^2+1}dz$ but not sure whether it is correct. $\gamma(t)=2e^{it},t\in[0,2\pi]$ ...
1
vote
2answers
27 views

$\int_{\gamma}^{}\frac{1}{z}dz$, $\gamma$ is the ellipse $x^2+4y^2=1$ traversed once with the positive orientation

I am unable to find the integral $\int_{\gamma}^{}\frac{1}{z}dz$, $\gamma$ is the ellipse $x^2+4y^2=1$ traversed once with the positive orientation. This maybe possible to be done using Cauchy-Goursat ...
1
vote
1answer
51 views

How to integrate $\int_{-1}^1\frac{1}{a + bx }dx$, where $a,b\in \mathbb{C}$ without using branch cuts.

Is there a way to integrate $$\int_{-1}^1\frac{1}{a + bx }dx,\,\,\,\,(*) $$ where $a,b\in \mathbb{C}$ without using branch-cuts? I was approached with such an integral relatively early in my text, and ...
0
votes
0answers
19 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 $S_n=\sum_{i=1}^{n}X_i$. ...
0
votes
0answers
10 views

Contour Integral squared function

$\gamma$ is a contour that goes from $-i$ to $-1$ and is contained in the third quadrant. Calculate $$\int_{\gamma}{z^{\frac{1}{2}}}dz$$ It is obvious that the primitive of the function is $\dfrac{2 ...
4
votes
2answers
46 views

Show that $\int_{|z|=1} \frac{e^z}{z^k} dz = \frac{2\pi i}{(k-1)!}$

I'm supposed to show that $$\int_{|z|=1} \frac{e^z}{z^k} dz = \frac{2\pi i}{(k-1)!}$$ where $|z|=1$ is traversed counterclockwise and $k>0$. We can parametrize this path as $\gamma(t)=e^{it}$ for ...
7
votes
0answers
63 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
34 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}} = ...
1
vote
1answer
32 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
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
votes
1answer
32 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
votes
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} ...
0
votes
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: ...
0
votes
0answers
44 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
37 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
36 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
65 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
31 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
16 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
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
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) ...
-1
votes
1answer
105 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
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 ...
1
vote
2answers
82 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
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 ...
1
vote
0answers
28 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
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
vote
1answer
88 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 ...
7
votes
2answers
145 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 ...
1
vote
1answer
28 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
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
votes
2answers
67 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
40 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
57 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 ...