# Questions tagged [parsevals-identity]

This tag is for questions regarding Parseval's Identity, an important result in the study of Fourier Series.

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### Parseval-Plancherel type identity for probability generating function

Assume that $f,g \in L^2(\mathbb R)$ and define the Fourier transform of $f$ by $$\hat{f}(\xi) = \int_{\mathbb R} \mathrm{e}^{-i\,x\,\xi}\, f(x)\,\mathrm{d}\xi, \quad \xi \in \mathbb R.$$ The well-...
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### Using Parseval's Identity to Calculate an Integral

The exercise is to use Parseval's identity to solve the following integral: $$\int_{-\pi}^{\pi}|\sum_{n=1}^{\infty}\frac{1}{2^n}e^{inx}|^2 dx$$ Now, I know that the ...
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### Calculate $\sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^2}$ using Parseval

The exercise asks me to calculate $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^2}}$ using the Fourier series of f(x) = \begin{cases} 0 & -\pi < x < 0 \\ ...
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### Use parseval's identity to prove $\int_0^\infty {\sin^4t \over t^2} dt = {\pi \over 4}$

I know how to calculate $$\int_0^\infty {\sin^4t \over t^4} dt$$ by taking the function $f(x)=1-|x|$ and it will be $\pi \over 3$. But here the denominator is not raised to power $4$ but $2$. How ...
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