0
votes
1answer
39 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}} ...
0
votes
1answer
27 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
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
32 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} ...
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 $-1$.
1
vote
2answers
107 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
105 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
112 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
52 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
121 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
61 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 ...
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
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
1answer
69 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 ...
4
votes
2answers
203 views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
6
votes
3answers
132 views

Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
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
1answer
84 views

Integral of $te^{-i\omega t}$ and Fubini's Theorem

I have a question about the value of the integral $\int_0^\infty t e^{-i \omega t} dt$. I stumbled upon it while trying to compute the Fourier transform of $|t|$. On one hand, we have that $|t e^{-i ...
3
votes
1answer
349 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 ...
1
vote
1answer
132 views

Need help integrating $\tan x$ and $\tan^n x$ using reduction

I have tried to use integration by parts taking $u$ as $\tan x$ and $v$ as $1$: $$\int \tan x \,dx = \int \tan x \cdot 1\; dx = \tan x \cdot x - \int \sec^2 x \cdot x\; dx$$ then by taking $u$ as ...
1
vote
1answer
148 views

Finding the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis

Trying to find the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis, I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $ d\theta=\dfrac{dz}{iz}$. I am trying ...
0
votes
2answers
194 views

a question about Cauchy integral formula

I'm new in the complex analysis and I'm stuck with this integral : $I=\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{(z^2+9)(z+9)} $ the exercise is about Cauchy integral, I don't want the whole ...
1
vote
4answers
281 views

Integrate $\int_0^\infty \frac{\sqrt{x}}{x^{2}+1}\, \mbox{d} x$

I've been trying to integrate the following $$\int_{0}^{\infty} \frac{\sqrt{x}}{x^{2}+1} \mbox{d} x$$ on half an annulus in the upper half plane. I keep getting $\frac{\pi}{\sqrt{2}}\ i$, which ...
8
votes
5answers
443 views

Showing that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left (\sqrt{a^2+1}-1\right)$.

How can I show that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
1
vote
3answers
288 views

Equality of absolute values of complex integrals

It was pretty hard finding a short and precise title. Here is my problem: The equation $$\bigg|\int_\gamma f(z)\text{d}z\bigg|\le\int_\gamma\big|f(z)||\text{d}z|$$ holds if $f$ is integrable (where ...
5
votes
3answers
254 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$/ Any help will be much appreciated.
3
votes
2answers
145 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
-1
votes
2answers
91 views

$\int_{-1}^{1}\exp(-at^2)/t^2 dt$ using residue theorem

How to calculate $$\int_{-1}^{1}\frac{e^{-at^2}}{t^2}dt$$ where $a>0$, using residue theorem?
4
votes
2answers
280 views

Show the existence of a complex differentiable function defined outside $|z|=4$ with derivative $\frac{z}{(z-1)(z-2)(z-3)}$

My attempt I wrote the given function as a sum of rational functions (via partial fraction decomposition), namely $$ \frac{z}{(z-1)(z-2)(z-3)} = \frac{1/2}{z-1} + \frac{-2}{z-2} + \frac{3/2}{z-3}. ...