5
votes
1answer
51 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
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
68 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
67 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
34 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
42 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
1answer
33 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
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
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
108 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 ...
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
103 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 ...
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 ...
3
votes
1answer
74 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
111 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
71 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
3answers
90 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 ...
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 $w=\lambda z$ ...
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 ...
0
votes
1answer
81 views

Find all functions $f(z)$ that are analytic everywhere in the entire complex plane that satisfy $f(2-i)=4i$

This question is apparently related to Cauchy's Integral Formula and related theorems, but I honestly don't know how to start, other than potentially saying that $f(z)$ is every function such that ...
0
votes
4answers
127 views

Evaluate $\oint_C\frac{dz}{z-2}$ around the circle $|z-2| = 4$

I don't completely understand how to approach these questions. I suppose the notation $\oint_C$ is something I'm not used to. So far, I have $\oint_C\frac{dz}{z-2} = \log(z-2)$. From here, I suppose ...
0
votes
1answer
95 views

Evaluate $\oint_C |z|^2 dz$ around the square with vertices at $(0,0), (1,0), (1,1), (0,1)$

I don't think I quite understand how to go about this. My solution so far: $\oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy$. ...
1
vote
0answers
91 views

Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
0
votes
1answer
97 views

Complex double integral

I'm having trouble calculating following (complex) integral. $$\int_D z^n dA$$ where $D=\{ z \in \mathbb{C} \mid \lvert z \rvert \leq 1 \}$. I know how to calculate complex (line) integrals and real ...
1
vote
0answers
39 views

Integral of the determinant of the Jacobian in complex variables?

Let $f$ be a function that maps a disc $D$ onto a unit square $A$ such that $f(D) = A$. $f$ is holomorphic and one-to-one. Why does $\int_D|f'(x+iy)|^2 \, dx \, dy = 1$?
2
votes
2answers
54 views

Does $\int_a^b f'(\gamma(t))\gamma'(t)\,dt$ depend on the path for meromorphic functions?

Given a meromorphic function $f:\mathbb C\to\mathbb C$ and a smooth curve $\gamma:[a,b]\to\Gamma\subset\mathbb C$ with $\gamma(a)\neq\gamma(b)$, I am tempted to think the fundamental theorem of ...
0
votes
1answer
101 views

Complex integral using residue theorem

I have $$\int_{|z|=1} z^m \sin\left(\frac{1}{z}\right)~dz,$$ for $m = 0,1,2,\dots$ I know that there is a singularity at $z=0$, and this singularity is within the curve, thus the residue theorem ...
5
votes
1answer
111 views

How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
0
votes
2answers
64 views

Change of variable x=iy in improper integral

I'm trying to solve the following question: Let $I=\int_0^{\infty}\exp(-x^4)dx$. Take $x=i y$ to get $i\int_0^{\infty}\exp(-y^4)dy=i I$. Explain. This change of variable implies $I=iI$, with $I$ ...
4
votes
4answers
134 views

Intuitive reason for why many complex integrals vanish when the path is “blown-up”?

It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along ...
1
vote
1answer
310 views

Complex Analysis: Fundamental Theorem of Calculus

I was given the following question on a complex analysis exam but didn't answer it correctly: Evaluate $$\int \sqrt{z-1} dz$$ about the the unit disk $|z|=1$ using only the Fundamental Theorem of ...
0
votes
0answers
35 views

Complex Integration with two variables

Let $ z_1, z_2 \in \mathbb{C}^+ \equiv \{ \forall z \in \mathbb{C}^+, \Im(z)>0 \} $. Also define $ m_1, m_2 $ such that for $i=1,2$ the following is true $$ z_i=-\dfrac{1}{m_i} + ...
0
votes
1answer
103 views

Evaluate this integral without using the residue theorem

I want to evaluate this integral $$ I = \int_c \tan z + \frac{\csc z}{z} dz $$ $$ c :|z| = 1 $$ apparently tanz is analytic in this region so its integral equals to zero now $$ I = ...
2
votes
0answers
32 views

$ g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm {d}b$

Do you have any idea how to compute this function? $$ \displaystyle g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm ...
3
votes
1answer
343 views

Change of variables in a complex integral

I want to evaluate this integral using Residue Theorem $$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$ $$ C : |z| = 1 $$ so I substitute letting $$\ W = z ^ {2 } $$ $$ dw = 2z dz $$ and the ...