Transcription of THE GAUSSIAN INTEGRAL
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THE GAUSSIAN INTEGRALKEITH CONRADLetI= e 12x2dx, J= 0e x2dx,andK= e numbers are positive, andJ=I/(2 2) andK=I/ 2 . notation as above,I= 2 , or equivalentlyJ= /2, or equivalentlyK= will give multiple proofs of this result. (Other lists of proofs are in [4] and [9].) The theoremis subtle because there is no simple antiderivative fore 12x2(ore x2ore x2). For comparison, 0xe 12x2dxcan be computed using the antiderivative e 12x2: this INTEGRAL is Proof: Polar coordinatesThe most widely known proof, due to Poisson [9, p. 3], expressesJ2as a double INTEGRAL andthen uses polar coordinates.
where the interchange of integrals is justi ed by Fubini’s theorem for improper Riemann integrals. (The appendix gives an approach using Fubini’s theorem for Riemann integrals on rectangles.) Since Z 1 0 ye ay2 dy= 1 2a for a>0, we have J2 …
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