3
votes
1answer
162 views

How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over ...
0
votes
1answer
29 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 ...
0
votes
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} ...
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 $-1$.
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} ...
2
votes
1answer
109 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 ...
4
votes
1answer
116 views

The substitution $y = ix$

Given $\int_0^\infty f(x) dx$, under what conditions would the substitution $y = ix$ not change the limits of integration, so to speak? Since setting $y = ix$ changes the range of integration to a ...
4
votes
1answer
138 views

How do you integrate $\int_0^\infty \exp(it^k)\,\mathrm dt$ for $k \in \Bbb N$?

My problem is with the integral $$\int^\infty_0 e^{it^k}\,\mathrm dt$$ with $k\in\mathbb{N}$. Somehow it can be evaluated by use of Cauchy's theorem. But I don't see how. The best thing I can ...
0
votes
1answer
67 views

Is this OK: $\int_a^b \!\mathrm{d}x \,\,f(x) =^? \int_{\mathrm{i}\,a}^{{\mathrm{i}\,b}} \!\mathrm{d} (\mathrm{-i}y)\,\,f(\mathrm{-i}y).$ Any proof?

This is related to Wick rotation in QFT but it is not exactly it. I'll take a 2-dimensional spacetime to be brief but usually there are more. I've checked with a few functions and with finite ...
8
votes
5answers
456 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 $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?