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

learn more… | top users | synonyms

3
votes
1answer
81 views

how to compute this complex integral with high order polynomial?

Compute $$ \lim_{R \to \infty} \frac{1}{2\pi i} \int_{|z|=R} \frac{(2z^2+z-1)P'(z)}{P(z)+3}dz $$ where $P(z)=z^{10}+2z^9+z^5+1$. It seems like I may use residue but it contains too high ...
4
votes
1answer
47 views

How to show $\int_0^1\frac{e^{e^{2\pi it}}}{e^{2\pi it}}dt=1$

I was trying to integrate the contour integral $$\int_\gamma \frac{\vert z \vert e^z}{z^2}$$ where $\gamma$ parametrizes the unit circle counterclockwise. I cannot use the Generalized Cauchy Integral ...
0
votes
0answers
26 views

Expressing principal value of integral as real/imaginary

How is it that we can express $$ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{\cos 3x}{x^2+4}=\Re \ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{e^{3xi}}{x^2+4} $$ while we cannot for $$ ...
0
votes
0answers
48 views

How to show that a given domain is simply connected?

I was studying simply and multiply connected domains in complex integration. I know that the domain $\{z: 1<|z|<2\}$ is multiply connected as it can't be squeezed to a point without going ...
0
votes
1answer
31 views

inequality in absolute value with exponential

Can you help me with this problem please ?? If $r>0$ . Show that $\left |{\displaystyle\int_{\gamma}e^{iz^{2}}dz}\right |\leq{\displaystyle\frac{\pi(1-e^{-r^{2}})}{4r}}$ where ...
1
vote
2answers
26 views

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented.

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented. The numerator is not analytic in $\Gamma$ so we can't use Cauchy ...
-1
votes
0answers
24 views

Integral of $\int_{-\infty}^{\infty}{exp(-a\lambda^6+ib\lambda)d\lambda}$; a,b>0 [closed]

How can integrate the following: $\int_{-\infty}^{\infty}{exp(-a\lambda^6+ib\lambda)d\lambda}$ where a,b>0
1
vote
1answer
17 views

Contour integration over a circle

$$\int_C \frac{\cos(\ z)}{(z)^2} dz$$ where C is any circle enclosing the origin and oriented counter-clockwise. z0 = o of order 2 , f(z) = cos z $$\int_C \frac{\cos(\ z)}{z^2} dz$$ = $2 \pi i ...
2
votes
1answer
35 views

evaluating a contour integral where c is $4x^2+y^2=2$

Consider the integral $$\oint_C \frac{\cot(\pi z)}{(z-i)^2} dz,$$ where $C$ is the contour of $4x^2+y^2=2$. The answer seems to be $$2 \pi i\left(\frac{\pi}{\sinh^2 \pi} - \frac{1}{\pi}\right)$$ but ...
1
vote
2answers
39 views

Evaluate the complex integral [closed]

Evaluate the below integral: $$ \int_{0}^{\infty}{x^{\alpha - 1} \over 1 + x}\,{\rm d}x $$ How to start ?.
0
votes
1answer
20 views

complex integral over a line

The value of line integral over C of dz/(z^2+4) along the line x+y=1 in the direction of increasing x is ___ The answer is pi/2 I am not sure how to arrive at this answer .i had a guess that maybe i ...
0
votes
0answers
23 views

Could anyone help me to solve this integral question?

Could anyone help me to solve this integral question ? $$ \ \int_a^b t^{k-z-1}(1+mt^{-z})^{-(n+1)}dt $$
21
votes
2answers
455 views

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
3
votes
3answers
79 views

Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$

How to find the Cauchy principal value of the following integral $$\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$$ How to start this problem?
0
votes
1answer
33 views

Find the contour integral around unit circle.

Evaluate the below integral by turning it into a contour integral around a unit circle: $$\int_{0}^{\pi}\frac{\cos2\phi}{1-2a \cos\phi + a^2} d\phi$$ $where\;a\neq \pm1$
2
votes
1answer
26 views

evaluate the complex integration

How to evaluate the below integral $$\oint_{c} \frac{dz}{e^{z}-1}$$ where $C$ is the circle $|z|=1$ Can this be done by Cauchy's formula? If yes how? Or do I need to do something else in order to ...
2
votes
2answers
28 views

Evaluate the contour integration

Evaluate the below integral: $$\oint_{c}\frac{e^{2z}}{(z+1)^4}dz$$ where $C$ is the circle, $|z|=3$
0
votes
1answer
37 views

Evaluate the contour integral

How to evaluate this, $$\oint_{c} \frac{\sin\pi z^2+\cos\pi z^2}{(z-1)(z-2)}dz$$ where $C$ is the circle, $|z|=3$ I tried below things I believe 1 and 2 are simple poles here and the equation can be ...
1
vote
2answers
15 views

finding poles for a complex rational function

So in working out the details of a trig integration with complex integrals problem, I have ended up with an integrand of $$\frac{z}{z^4+6z^2+1}$$ I need to find the roots of $z^4+6z^2+1$ to use the ...
0
votes
2answers
25 views

Question about how to get the residue for infinite amount of poles

So I am asked to find the residue for each pole such as $$ f(z) = \frac{z}{1-\cos(2z)} $$ I understand pole of order 2 with $z= 2\pi k$ excluding zero. I also understand that residue equals to ...
0
votes
1answer
53 views

Why does this function residue equal 0?

$$ f(z) = \frac{e^{2z}}{(z-1/2)^{2013}} $$ Why does this residue equal 0? If I expand Laurent series, the right side will have $\dfrac{a_{2013}}{(z-1/2)^{2013}}$ $$ + \frac{a_{-2012}}{(z- ...
2
votes
1answer
25 views

solve a complex integral

I stumbled on this integral, the problem says to solve it with contour integration. Any insights on how to solve this in function of $n$? \begin{equation} \int_{0}^{2\pi}\cos^{2n}(\theta)d\theta ...
0
votes
0answers
56 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
1
vote
1answer
32 views

How can I calculate this complex integral?

The integral is the following: $$\int_{|z|=r} \frac{z+1}{z(z^2+4)} dz , r>0, r \neq 2 $$ I'm a little bit lost, I know that its partial fraction expansion is $$ \frac{z+1}{z(z^2+4)} = ...
0
votes
0answers
23 views

Integration of a complex integration

Let $C$ be the contour $|Z|=2$ oriented in the anti-clockwise direction.What is the value of the integral $\oint_{C}{ze^{\cfrac{3}{z}}}$$dz$ ? I don't know how to start. Please tell me which formula ...
2
votes
1answer
26 views

Conformal mappings to polygons: why is my integral conformal?

I'm learning about conformal mappings into polygons in a class,(undergrad complex analysis) and am having trouble understanding one of the examples given in my book. (Stein & Shakarchi) Here it ...
1
vote
3answers
15 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
86 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
48 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 ...
2
votes
0answers
26 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
29 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
3answers
103 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
41 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
44 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
34 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
154 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
27 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
51 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
63 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
27 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
46 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
46 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
28 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
55 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
20 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
11 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
65 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
37 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
47 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 ...