Transcription of THE GAUSSIAN INTEGRAL - University of Connecticut
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THE GAUSSIAN INTEGRAL - University of Connecticut
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. To start, writeJ2as an iterated INTEGRAL using single-variable calculus:J2=J 0e y2dy= 0Je y2dy= 0( 0e x2dx)e y2dy= 0 0e (x2+y2) this as a double INTEGRAL over the first quadrant. To compute it with polar coordinates, thefirst quadrant is{(r, ) :r 0 and 0 /2}.
The integral we want to calculate is A(1) = J2 and then take a square root. Di erentiating A(t) with respect to tand using the Fundamental Theorem of Calculus, A0(t) = 2 Z t 0 e 2x dxe t2 = 2e t2 Z t 0 e x2 dx: Let x= ty, so A0(t) = 2e 2t2 Z 1 0 te 2t2y dy= Z 1 0 2te (1+y )t2 dy: The function under the integral sign is easily antidi erentiated ...
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