Parseval identity

For

f, g \in L^{2} (R^{n})

and the convention

f (ξ) = \int f (x) e^{- i x \cdot ξ} d x

\int_{R^{n}} f (x) \overline{g (x)} d x = \frac{1}{( 2 π ) ^{n}} \int_{R^{n}} f (ξ) \overline{g (ξ)} d ξ .

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- Even rational Parseval integral Parseval identity

Even rational Parseval integral

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Applying the Parseval identity to the repeated exponential convolution gives

\int_{- \infty}^{\infty} \frac{d k}{( 1 + k ^{2} ) ^{n + 1}} = \frac{π ( 2 n )!}{2 ^{2 n} ( n ! ) ^{2}} = \frac{π}{4 ^{n}} (n 2 n) .

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Parseval identity

Even rational Parseval integral

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