Riley hobson bence · Problem 2.45

Preliminary calculus

Mathematics — Methods · integration-reduction-formulas

Problem

If JrJ_r is the integral ∫0∞xrexp⁡(−x2) dx\int_0^\infty x^r \exp(-x^2)\, dx show that ((a)) J2r+1=(r!)/2J_{2r+1} = (r!)/2, ((b)) J2r=2−r(2r−1)(2r−3)⋯(5)(3)(1) J0J_{2r} = 2^{-r}(2r-1)(2r-3)\cdots(5)(3)(1)\, J_0.

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