Questions on the evaluation of integrals along a locus in the complex plane.

learn more… | top users | synonyms

1
vote
1answer
29 views

Limit as $r$ tends to zero of integral $\int_C \frac{e^{iz}-1}z \mathrm dz$

Let $\mathcal C$ be a semi-circle of center $O(0,0)$ and radius $R$, such that $y \ge 0$. Find the limit as $R$ tends to zero of: $$\int_{\mathcal C} \frac{e^{iz}-1}z \mathrm dz$$ How can I find ...
1
vote
2answers
61 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
1
vote
0answers
29 views

Integral of function has different values depending on contour?

What can I say if the integral of my function has different values depending on contour? If my function were analytic on a domain it would evaluate to $0$ right? My contours give values like $\pi ...
3
votes
3answers
238 views

How to know if an integral is well defined regardless of path taken.

I can calculate \begin{equation*} \int_0^i ze^{z^2} dz=\frac{1}{2e}-\frac12, \end{equation*} but why can I calculate this irrelevant to the path taken? Is this since it is analytic everywhere - if ...
3
votes
2answers
53 views

Using Cauchy Integral Formula $\int_C \frac2{z^2 -1}dz$

I want to understand why I can't use Cauchy Integral Formula for the following problem: $$\int_C \frac2{z^2 -1}dz\text{ on the contour } |z-1|=\frac12$$ Now it says that I need $f$ to be analytic ...
2
votes
3answers
55 views

Contour integration of: $\int_C \frac{2}{z^2-1}\,dz$

I want to calculate this (for a homework problem, so understanding is the goal) $$\int_C \frac{2}{z^2-1}\,dz$$ where $C$ is the circle of radius $\frac12$ centre $1$, positively oriented. My ...
1
vote
0answers
50 views

Inverse Gamma function for integers (Hankel)

So I want to prove that for all integers $n \in \mathbb{Z}$ it holds that $$F(n):= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-n}e^{s} ds = \frac{1}{\Gamma(n)},$$ with $\gamma$ the 'Hankel'-contour: ...
1
vote
1answer
41 views

Evaluating the integral $\int_C \text{Re }z\,dz$ from $-4$ to $4$ via lower half of the circle

I want to evaluate the integral $\int_C \text{Re }z\,dz$ from $-4$ to $4$ via the contour being the lower half of the circle of radius $4$ centered at the origin. So I can take: ...
2
votes
1answer
60 views

Evaluating an integral across contours: $\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$

This is for an assignment, describing the procedure is most beneficial for me, rather that solely computing the result. I want to evaluate the following integral: $$\int_C\text{Re}\;z\,dz\,\text{ ...
1
vote
1answer
56 views

Residue theorem, double pole, sinh.

how can I use the residue theorem to calculate $$\int_{-\infty}^\infty dx\, \frac{e^{-i x}}{(\sinh x)^2}$$ Im confused about how to tackle the double pole at $x=in\pi$. Thanks!
1
vote
0answers
31 views

Evaluating the inverse Laplace transform of $1/(s^2-\sum_{n=1}^\infty{n!s^{3-n}x^n})$

I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of $$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$ and find out the $x^n$ ...
0
votes
1answer
16 views

Piecewise continuity for contours - Definition for complex analysis

We want piecewise continuity for any contours in complex analysis. What does this refer to? I imagine it refers to the nature of referring to each line arc being continuous. E.g. We want continuity ...
1
vote
1answer
110 views

Is this derivation of the Dirichlet Integral using a derivative under the integral sign, incorrect?

To find the integral of the Sinc function: Start with, \begin{equation} I(a)=\int_{-\infty}^{\infty}\frac{\sin\ ax }{x}dx %\hspace{20.0} ; (a>0) \end{equation} \begin{equation} \Longrightarrow ...
0
votes
1answer
24 views

Choosing contour to evaluate integral

I am practicing for a preliminary exam in complex analysis. I have struggled with the following problem for a while. I was hoping someone would have suggestions for contours to integrate over. (Or ...
0
votes
1answer
54 views

Gaussian Integral using contour integration with a parallelogram contour

I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function, $f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi ...
2
votes
3answers
95 views

Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$

I wish to compute $$\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx, \quad a>0$$ but have no contour to work with. Does anyone have ideas on how to compute this integral?
0
votes
1answer
127 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
4
votes
2answers
69 views

Computing the complex integral?

I am dealing with the following: $$\int_{0}^{\infty}\frac{x\sin(x)}{(x^2+a^2)(x^2+b^2)}dx$$ Furthermore, I know $a,b>0$ and I know $a\neq b$. I believe this is using Jordan's Lemma? I see that the ...
0
votes
0answers
25 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
1
vote
1answer
70 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
9
votes
2answers
198 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
1
vote
1answer
35 views

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$ $C=\partial D_1(\mathbf i/2)$ is the boundary of the disc with center $\mathbf i/2$ and radius $1$, then $\mathbf i$ is ...
0
votes
2answers
22 views

Finding the value of $I=\int_C \overline{z} dz$ along $|z|=2$ from $z=-2i$ to $z=2i$

I want to find the value of the integral $$I=\int_C \overline{z} dz$$ When $C$ is the right hand half of the circle $|z|=2$ from $z=-2i$ to $z=2i$ Refer to beautifully made picture: Now I am new to ...
0
votes
1answer
37 views

Improper integral (using methods in complex variables) [closed]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
4
votes
1answer
100 views

Method of Steepest descents integral

I am looking to evaluate the following asymptotic integral: Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of steepest descents ...
2
votes
1answer
42 views

Solve an Integral one open and one complex limit

If we pose an integral $2ie^{-\zeta^2}\int\limits_{-\infty}^{i\zeta} e^{-x^2} dx$ where $\zeta$ is a complex number. For imaginary $\zeta$ this comes down to the error function. I would appreciate ...
1
vote
1answer
41 views

Why is $\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$?

Why is $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$$ and not $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$
0
votes
1answer
67 views

Inverse $z$ transform - contour integration

Here is my task: Find inverse $z$ transform of $$X(z)=\frac{1}{2-3z}$$if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)=\dfrac{1}{3}\left (\dfrac{2}{3}\right ...
4
votes
2answers
92 views

Evaluating contour integral without using Residue Theorem

Find the value of the integration without using Cauchy integral formula/Residue theorem: $\int_{C}\cfrac{dz}{z^2+1}$ where C is a simple closed contour oriented in counter clockwise ...
4
votes
0answers
44 views

Integral representation of Bessel function K

Does someone have an idea how to connect the following function (appearing in the quantization of a real scalar field in a uniformely accelerated frame) : $$ K(x,y) = \int_{0}^{\infty} \frac{dt}{t} ...
0
votes
1answer
19 views

Evaluating this contour integral

Let $R$ be the rectangle with vertices at $-1$, $1$, $1+2i$, $-1+2i$. Compute $$\int_{\partial R} \frac{(z^2 +i)\sin(z)}{z^2+1}dz$$where the boundary of $R$ is traversed counterclockwise. Here is ...
0
votes
0answers
52 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
0
votes
2answers
49 views

Compute the following integral, where $C$ is the circle $|z|=3$

Evaluate:$$\int_{C} (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) dz $$ where $ C$ is a circle $|z|=3$ and $z \ \epsilon \ \mathbb{C}$ The function that is being integrated has ...
1
vote
3answers
101 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
1
vote
1answer
39 views

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$. The following construction is in Walter Rudin's Real and Complex Analysis Chapter 16, Exercise 11. ...
0
votes
1answer
56 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
3
votes
2answers
34 views

Computing $\int_{\gamma}e^zdz$, where $\gamma$ is a particular semicircle

How can I compute $\int_{\gamma}e^zdz$, if $\gamma$ is the semicircular arc depicted below? So, $\gamma=3e^{i\theta(t)}$, with $0\le\theta(t)\le\pi$, and then ...
1
vote
2answers
89 views

$\int\limits_{\gamma} \frac{z}{(z-1)(z-2)}dz$, $\gamma(\theta) = re^{i\theta}$, $2 < r < \infty$

For $0 < r < 2$, we can use Cauchy's integral formula and choose our holomorphic function to be $f(z) = \frac{z}{z - 2}$ since $z = 1$ is the only pole, but if $r > 2$, then both poles $z = ...
3
votes
1answer
44 views

$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$ - different answers depending on value of $t$?

After taking an inverse Laplace transform I have the following - $$y = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{1}{s^2}e^{s(t - \frac{1}{2}x^2)}ds$$ In my notes it says if $t ...
3
votes
3answers
62 views

Evaluation of real trigonometric integrals using the Cauchy Residue Theorem

$I = \int^{2\pi}_0 \dfrac{d\theta}{2 - \cos \theta}$ This is straight from a book I'm reading, which suggests to convert $\cos\theta$ into $0.5(z+1/z)$ and then solve the integral on the unit circle. ...
7
votes
2answers
281 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
0
votes
0answers
30 views

Integrate function with 2 branch points

Every example I see in textbooks so far has not shown me cases like this, so please help with the following question. I wish to integrate a function $f(z)$ around the contour shown below. $f(z)$ has ...
2
votes
0answers
58 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
4
votes
3answers
145 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
0
votes
1answer
42 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
1
vote
1answer
42 views

Why is $\int\limits_{\gamma} \frac{1}{z-1} \neq 2\pi i$, $\gamma = \{z : \lvert z \rvert = 1\}$?

$\int\limits_{\gamma} \frac{1}{z-1}$ $\gamma = \{z : \lvert z \rvert = 1\}$ I use Cauchy's integral formula, which says $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = \frac{2\pi i}{n!} ...
0
votes
3answers
31 views

$\int\limits_{\gamma} \frac{1}{z-1}$, $\gamma(\theta) = 2e^{i\theta}$, $0 \leq \theta \leq \frac{\pi}{2}$

$\gamma(\theta) = 2e^{i\theta}$ is a circle centered at $(0,0)$ with radius $2$, so $z = 1$ is inside this path and thus we have to use Cauchy's integral formula for $\int\limits_{\gamma} ...
0
votes
0answers
48 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
18 views

Contour Integration example check

I have this question and have solved that the residue is zero? hence the integral is zero by the residue theorem? could someone confirm this please?? Also would the answer to this integral be ...
0
votes
2answers
41 views

Contour integration example question

I'm currently trying to solve this however I get to the point where I have, $$\int_{0}^{2\pi} \frac{ie^{\exp(it)}}{\exp(it)+3}.dt$$ am I on the right tracks? if yes could you help with the ...