# Tagged Questions

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### fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
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### Calculating this integral?

I'm trying to calculate $$\int\limits_{-\pi}^0e^{-x}\cos(nx)\,\mathrm{d}x$$ as part of a Fourier series calculation. My problem is the calculations seem to loop endlessly - I'm integrating by parts ...
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### Use Fourier transform to calculate double integral of harmonic function

Let $$P_y(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty e^{-y|t|}e^{ixt}dt=\dfrac{1}{\pi}\dfrac{y}{x^2+y^2}.$$ Then $P_y(x)$ is harmonic in the upper half-plane $y>0$ and for $f\in L^1(\mathbb{R})$, ...
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### Show $\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0$

Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that ...
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### Closed form for integral $\int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?
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### Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$