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5
votes
1answer
49 views

Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $ \deg Q - \deg P > 2$. Is it possible to have a general formula for ...
1
vote
2answers
32 views

How to handle the complex integration of this function around a branch point

I have this complex integral to which I don't know if it's possible to assign a value: The integral is on a small circle around the origin. The function is $\frac{1}{(z-1)\sqrt{z}}$. The fact is ...
-1
votes
0answers
34 views

proving cauchy integral formula using integration by parts

I have found several proofs of Cauchy's generalized Integral formula, but I am looking to prove the first derivative case by using the fact that the first derivitave is analytic to state it as a ...
4
votes
1answer
66 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
2
votes
2answers
66 views

Find $\int_\gamma \frac{dz}{z^2}$ wihtout explicit calculations

Evaluate the following integral without doing any explicit calculations: $\int_\gamma \frac{dz}{z^2}$ where $\gamma(t) = \cos(t) + 2i\sin(t)$ for $0 \le t \le 2\pi$. This exercise comes along with ...
2
votes
0answers
42 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
0
votes
0answers
33 views

Liouville's theorem for functions not bounded on an isolated set

On Liouville's theorem. Prove that if $f(z)$ is defined, analytic, bounded in the entire complex plane, except for an isolated set of points, then $f$ must be constant. What happens if said set has a ...
0
votes
2answers
41 views

Cauchy's argument principle, trouble working simple contour integral

I'm trying to teach myself Cauchy's argument principle by doing a simple example, but apparently I'm missing something, because every time I try to do the contour integral I get 0. Cauchy's argument ...
0
votes
1answer
25 views

Contour Integrantion of a exponential function

I am trying to evaluate an integral of type $$ I = \int_{-\infty}^{\infty} \frac{e^{ikx}P(x)}{Q(x)} dx $$ Where ...
0
votes
0answers
23 views

Evaluate $\int f(z)\,dz$ with $f(z) = x^2 - iy^2$, where $z = x + iy$, and the curve $C$ is given by $C(t) = t - it^2, 0<t<1$

I began by assigning the $\Re z$ to be $x = t$ and the $\Im z = y = -it^2$. Then I computed $z'(t) = 1 - 2ti$. Then $f(z(t)) = t^2 - it^4$. Then I took $$ f(z(t)) z'(t) = (t^2 - it^4)(1-2ti) = (t^2 ...
0
votes
1answer
32 views

Complex integral (Cauchy's Theorem?)

I have $$\int_{\gamma}\frac{1}{4z^2-1}dz$$, where $\gamma$ is the unit circle in the complex plane. I said this integral equals to $$\int_{0}^{2\pi}\frac{ie^{it}}{4(e^{it})^2-1}dt$$ Then I let ...
0
votes
1answer
28 views

Lioville's criteria and Integration of sin(z)/z .

Lioville Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$. Now ...
1
vote
1answer
37 views

Integral of $e^{\overline{z}}$

So I'm suppose to integrate $e^{\overline{z}}$ along the unit circle where $z(t)=e^{it}$ and from t goes from 0 to $2\pi$. This is the work I've done so far $\int e^{\overline{z}}dz = ...
0
votes
0answers
28 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...
0
votes
1answer
22 views

Complex Integration by Parts help

Solve $\frac{1}{\sqrt{4\pi t}}\int\limits_{\mathbb{R}}e^{-\sigma^2/(4t)}(\sigma^2 +2\sigma x+x^2-1)d\sigma$ I am told the integral of the heat kernel is 1
0
votes
1answer
30 views

Integration by Parts help?

Show $\frac{1}{\sqrt{4\pi kt}}\int\limits_{\mathbb{R}}e^{-\sigma^2/(4kt)}(\sigma^2 +2\sigma x+x^2)d\sigma = 2kt+x^2$
0
votes
0answers
31 views

Relation of an integral over the entire real line to the same but with the integrand shifted by an imaginary amount

I would like to relate the following two integrals: \begin{align} I_1 &= \int_{-\infty}^\infty f(x) dx .\\ I_2 &= \int_{-\infty}^\infty f(x - i X) dx \text{ with } X \text{ real.} \end{align} ...
0
votes
1answer
50 views

Evaluate using cauchy's integral formula

How can we evaluate this expression using cauchy's integral formula $\int_C \frac{e^{\pi Z}}{ ( {Z^2 + 1}) ^2} dZ$ where $C$ is $|Z-i|=1$
1
vote
2answers
103 views

What is ML Inequality property of complex integral

What is ML inequality property in complex integral which says $|\int_{c}f(z)dz| \leq ML$. I can't understand a thing from this expression. I want to understand it conceptually(if that helps). How can ...
1
vote
2answers
39 views

Line integral of complex expression

How can we integrate expressions like these $\int_C \operatorname{Re}(Z) \, dZ$ where $C$ is the shortest path joining the points $1+i$ and $3+2i$. The $\operatorname{Re}(Z)$ in the expression is ...
2
votes
1answer
54 views

Integration of complex function

how do we do start doing the following integral? $$\int_{-\infty}^{\infty}e ^{-\pi(x^2a^2+2iux)}\,dx$$ Any help will be appreciated NOTE: $i$ is the square root of $-1$.
3
votes
3answers
59 views

Integrating $I(\alpha)=\int^{\infty}_{0} \frac{x^{\alpha}}{x^4+1}dx$

Here is the question: Let $P(x)$ be a polynomial of degree $d>1$ with $P(x)>0$ for all $x>0$. For what values of $\alpha \in \mathbb{R}$ does the integral $I(\alpha)=\int^{\infty}_{0} ...
1
vote
2answers
100 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
3
votes
1answer
101 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
0
votes
2answers
111 views

A contour integral for Fourier transform

How does one show the following, preferably with contour integral on the complex plane? $$\frac{\Gamma(\alpha)}{2\pi}\int_{-\infty}^\infty (ik)^{-\alpha}e^{-ikx}dk = (-x)_+^{\alpha-1},$$ where $x$ is ...
2
votes
1answer
51 views

Can I switch the order of integration and “Real(z)” operation?

Let $f(z,\eta)$ be an entire function. I need to calculate (numerically) the integral: $$\int\limits _{0}^{\pi}\mbox{Re}\left(\int\limits _{0}^{\pi}f\left(z,\eta\right)d\eta\right)dz$$ Can I switch ...
0
votes
2answers
48 views

find general solution to the Differential equation

Find the general solution to the differential equation \begin{equation} \frac{dy}{dx}= 3x^2 y^2 - y^2 \end{equation} I get \begin{equation} y=6xy^2 + 6x^2 y\frac{dy}{dx} - 2y\frac{dy}{dx} ...
-2
votes
1answer
120 views

Rate of change optimisation

Polonium-210 is a radioactive element whose time rate of decay is proportional to the quantity present at any time. A nuclear accident, confined to a single room of a nuclear research laboratory ...
2
votes
2answers
97 views

Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
0
votes
1answer
58 views

Double Integration Problem for Buffon's Needle experiment

Numberphile has a video about the Buffon's needle experiment (Video). I am writing an essay on determining $\pi$ using probability and I need to show my understanding of the topic. I kind of already ...
3
votes
1answer
73 views

Using argument principle to compute an integral

Let $f(z)=z^4-2z^3+z^2-12z+20$. Then evaluate the integral by using the argument principle $$\oint_C \frac{zf'(z)}{f(z)} \,ds$$ Where $C$ is the circle $|z|=5$. What I've tried: I tried using the ...
7
votes
1answer
110 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
0
votes
1answer
80 views

Integral of holomorphic function which tends to $0$

Let $R > 0 $, $z \in \mathbb{C}, \ f : D(z,R) \rightarrow \mathbb{C} $. $Re(f) \ $ and $Im(f) \ $ are $C^{1} $ on $D(z,R) \ $. Then f is complex differentiable in $z$ if and only if $$ \lim_{r ...
1
vote
1answer
109 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
3
votes
1answer
69 views

problem about complex integration

The question is to find $\int \frac{z^2-z+1}{z-1}dz$ over |z|=1. My solution is : Using cauchy's integral formula we have $f(1) = \frac{1}{2\pi i}\int \frac{f(z)}{z-1}dz$ but f(1) = 1. therefore, ...
2
votes
2answers
35 views

question related with cauch'ys inequality

We have a statement that if f(z) is analytic in a domain D and $C = \{z : |z-a| = R\}$ is contained in D. Then, $|f^n(a)|\leq \frac{n!M_R}{R^n}$ where $M_R = \max|f(z)|$ on C. What if the given D ...
3
votes
2answers
50 views

doubt in complex integration

I was doing problems on complex integration and got stuck at one question. The question is $$ \int_{\gamma}{{\rm e}^{{\rm i}\pi z}\left(z + i\right)^{2}\cos\left(nz\right) ...
0
votes
0answers
119 views

complex sum-integration

Let $a_n \in \mathbb{R}$. Show that, $$0\le \sum_0^N\sum_0^N\frac{a_na_m}{n+m+\frac{1}{2}}\le \pi \sum_0^Na_n^2.$$ Hint: Integrate on the semicircle with radius R, the sum ...
2
votes
1answer
44 views

Complex Integration of DTFT

Question A discrete-time signal $u \in \mathcal{l}^2(\mathcal{Z})$ has DTFT \begin{equation} \hat{u}(\omega) = \frac{5+3\cos(\omega)}{17+8\cos(\omega)} \end{equation} Use complex integration to find ...
1
vote
0answers
31 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
0
votes
1answer
45 views

Conditions for complex integrability

I was wondering if the Lebesgue conditions for Riemann integrability also hold for a complex path integral, and what that would mean exactly. I am assuming that $f(z)$ would have to be bounded and ...
1
vote
3answers
88 views

Find the residue of $\frac{e^{iz}}{(z^2+1)^5}$ at $z = i$ and evaluate $\int_0^{\infty} \cos x/(x^2+1)^5 dx$

I know the evaluation of $\int_0^{\infty} \cos x/(x^2+1)^5 dx$ requires that I solve the first part, but for some reason I'm stumped. I get that I should use $\lim_{z \to ...
1
vote
0answers
72 views

Evaluate the following integrals/ Cauchy integral theorem

So I have two questions. 1) Evaluate $ \oint_{|z|=1} \dfrac{cos(\pi z^2)}{(z-2)(z-4)^3} dz$ and 2) Evaluate $ \oint_{|z|=6} \dfrac{cos(\pi z^2)}{(z-2)(z-4)^3} dz$. Now I know the integrand is ...
0
votes
0answers
32 views

Complex Integration using polar coordinates

Consider the complex variable $y=re^{j\phi}$ with $r\in(0,\infty)$, $\phi \in (-\pi,\pi)$, and the complex integral $$ I=\int\limits_\mathbb{C} {f(y)\log(f(y))dy} $$ Does the following ...
0
votes
1answer
38 views

Integrating Real Function in the Complex Plane

Question: Evaluate the integral $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$ ...
1
vote
1answer
83 views

Dirac Delta — Symmetry

I had a curiosity question rise up in the middle of the night regarding the behavior of the Dirac Delta. Because it's not a function per-se, I am not sure how a concept like "integration" symmetry ...
1
vote
3answers
104 views

Calculating the integral expression $\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$ for complex-valued z

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)> 0$ and t is a real variable. Is it correct to ...
1
vote
0answers
85 views

Complex integration from zero to infinity at different directions

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)\ge 0$. Is it correct to substitute $w=\lambda z$ ...
1
vote
1answer
68 views

How can I use Cauchy formula to this Integral?

$$∫ \frac{5z^2 - 3z + 2}{(z-1)^3} dz$$ and the contour is any closed simple curve involving z=1 (sorry, I forgot to write this information) Need to solve it using Cauchy Integral formula Can anyone ...
0
votes
1answer
51 views

Prove that $f'(a) = \frac{1}{2\pi}\int_0^{2\pi}e^{-i\theta}f(a+e^{i\theta})d\theta$

I know this is to be derived from Gauss' Mean Value Theorem, but I can't get the $e^{-i\theta}$. Where am I going wrong? $f'(a) = \lim_{h \to 0}\frac{f(a+h) - f(a)}{h} = \lim_{h\to ...