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1answer
32 views

Complex Integral - exponential divided by a monomial

How does one solve integrals like this- $$I=\int^\beta_0 dx \frac{\exp(i\omega_nx)}{x-a}$$ where $\omega_n=\frac{\pi n}{\beta} $. EDIT: $\beta$ is a finite, real ...
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0answers
23 views

Second Derivative of Holomorphic function According to Cauchy's

I am asked to prove that, supposing f is holomorphic on the region G, w $\in$ G and $\gamma$ is a positively oriented, simple, closed, smooth, G-contractiblecurve such that w is inside $\gamma$ , we ...
1
vote
1answer
42 views

Integrating the function Im(z) on a variety of contours.

I've been asked to evaluate $\int_C Im(z) dz$ for a variety of contours, which I've had no issue in doing. For the sake of clarity, these contours included the upper and lower halves of the circle ...
0
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2answers
43 views

Complex Analysis, Integral over a Square

Given that $C$ is the boundary of the square with corners at $\pm4 \pm4i$ (sorry my formatting always seems to be stubborn, but that is plus or minus 4 plus or minus 4i, I am asked to compute $$\int_C ...
1
vote
1answer
32 views

A problem with Cauchy Theorem

I want to resolve the folowing contour integral, using the Cauchy theorem: $$ \oint_C \cot(\pi z)\,dz $$ where $C$ is rectangle defined by $x=\frac{1}{2},x=\pi, y=-1, y=1 $ I do understand that ...
0
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0answers
28 views

Contour integral going to zero on a limit

I've been asked to prove the following; $$\lim_{R \rightarrow \infty}\int_{C_R} \frac{z^2 + 8z + 42}{(z^2+4)(z^2-4z+5)}dz=0$$ Given that $C_R$ is a circle of radius $R$ centered at $0$. I thought ...
0
votes
1answer
31 views

Complex Integration-Computing winding number of a curve

I need to compute the winding number of $\alpha$ with respect to the point p=(1/2,0) Where $\alpha: [0,2\pi] \to \mathbb R^2$ $\alpha(t)=((2 Cos[t] - 1)*Cos[t], (2 Cos[t] - 1)*Sin[t])$ The winding ...
0
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1answer
51 views

Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
5
votes
2answers
115 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
5
votes
1answer
77 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
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3answers
57 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 ...
4
votes
1answer
140 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
74 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
44 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 ...
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votes
0answers
45 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
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2answers
113 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
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1answer
33 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
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1answer
43 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 ...
2
votes
2answers
83 views

Liouville's Criterion and Integration of $\sin(z)/z$ using elementary functions

Liouville's 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
39 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
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0answers
33 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
31 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
32 views

Integration by Parts help?

Show $\frac{1}{\sqrt{4\pi kt}}\int\limits_{\mathbb{R}}e^{-\sigma^2/(4kt)}(\sigma^2 +2\sigma x+x^2)d\sigma = 2kt+x^2$
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0answers
34 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
86 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 $|Z-i|=1$
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2answers
372 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
54 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 ...
0
votes
2answers
163 views

Prove a function has a removable singularity at $z=0$.

Let $f$ be a holomorphic function on $\mathbb{C}\smallsetminus \{0\}$. Suppose $\int_{|z|=1}z^nf(z)\,dz=0$ for any $n=0,1,2,\ldots$. Prove that $f$ has a removable singularity at $z=0$. How to prove? ...
2
votes
1answer
57 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
66 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
144 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
138 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 ...
1
vote
3answers
227 views

Possible values of $\int \frac{dz}{\sqrt{1-z^2}}$ over a closed curve in a region?

This is related to Ahlfors' problem #5 following section 4.4.7. Let $\sigma$ be a path in $\mathbb{C}$ starting at $-1$ and ending at $+1$. Let $\gamma$ be a closed curve in $\mathbb{C}$ which does ...
0
votes
2answers
123 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
57 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
55 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} ...
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votes
1answer
134 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 ...
3
votes
2answers
145 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
78 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
104 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
214 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
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1answer
93 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
137 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
85 views

problem about complex integration

The question is to find $\displaystyle\int \frac{z^2-z+1}{z-1}dz$ over $|z|=1$. My solution is : Using cauchy's integral formula we have $\displaystyle f(1) = \frac{1}{2\pi i}\int ...
2
votes
2answers
41 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
51 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
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0answers
124 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
50 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
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0answers
32 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
46 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 ...