Gamma density from repeated exponential convolution

For

F (x) = e^{- x} 1_{x \geq 0}

, induction in the convolution integral gives

F^{* n} (x) = \frac{x ^{n - 1} e ^{- x}}{( n - 1 )!} 1_{x \geq 0} .

Its Fourier transform in the angular-frequency convention is

(1 + ik)^{- n}

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