# Tagged Questions

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### Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
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### If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...
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### How to calculate this basic Fourier Transform?

I am trying to calculate the Fourier Transform of $g(t)=e^{-\alpha|t|}$, where $\alpha > 0$. Because there's an absolute value around $t$, that makes $g(t)$ an even function, correct? If that's ...
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### fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
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### Solving the integral $\int_{0}^{\pi/2}(\sin[2n+1]t) \cdot \frac{\sin(t)-t}{sin(t) \cdot t}$

How should I solve this integral ?: $$\int_{0}^{\pi/2}\sin\left(\vphantom{\Large A}\left[2n + 1\right]t\right)\, {\sin\left(t\right) - t \over t\sin\left(t\right)}\,{\rm d}t$$ How can I use Fourier ...
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### Integral $\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$\int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k))$$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
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### Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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### Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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### Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
### Evaluate the integral $I=\int\limits_{-\infty}^{\infty} \frac{\sin^{2}u}{u^{2}}du$
To evaluate this integral I have got to use Parseval's Theorem and the fourier transform of $$s(x)=\begin{cases} 0 & x\leq -a \\ 1 & -a<x<a \\ 0 & x \geq a.\end{cases}$$ This ...