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 Incoming links: Full-line Gaussian moment recurrence

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 13 / b / i / Solution 2026-10-18

Retain the original PDF's full-line integration limits. For

n \geq 2

, write

x^{n} e^{- x^{2}} d x = - x^{n - 1} d (e^{- x^{2}}) /2

. Integration by parts gives

I_{n} = [- \frac{1}{2} x^{n - 1} e^{- x^{2}}]_{- \infty}^{\infty} + \frac{n - 1}{2} \int_{- \infty}^{\infty} x^{n - 2} e^{- x^{2}} d x .

Gaussian decay dominates every power, so the endpoint term vanishes, so

I_{n} = \frac{n - 1}{2} I_{n - 2} .

This is the full-line Gaussian moment recurrence; its initial values differ from the half-line version.

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