Introduction to Quantum Mechanics (2nd ed.). integration by parts, show that E ( X2 ) 1, and hence that var ( X ) 1. Weisstein, Eric W.'Gaussian Integral'. Show also that E ( X2 ) 2/22 and hence find an expression for var ( X ).'Introduction to integral discriminants'. ^ 'Reference for Multidimensional Gaussian Integral'.^ a b 'The Probability Integral' (PDF).'Integration in Finite Terms with Special Functions: the Error Function'. 'The History of the Normal Distribution' (PDF). These integrals turn up in subjects such as quantum field theory. The n + p = 0 mod 2 requirement is because the integral from ââ to 0 contributes a factor of (â∡) n+ p/2 to each term, while the integral from 0 to +â contributes a factor of 1/2 to each term. Since the exponential function is greater than 0 for all real numbers, it then follows that the integral taken over the square's incircle must be less than I(a)2
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