CAS PROJECT. Error Function (21) 2 rX erfx = -- J e- w2 dw

Chapter 12, Problem 12.6

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CAS PROJECT. Error Function (21) 2 rX erfx = -- J e- w2 dw ~ 0 This function is imp0l1ant in applied mathematics and physics (probability theory and statistics. thermodynamics. etc.) and fits our present discussion. Regarding it as a typical case of a special function defined by an integral that cannot be evaluated as in elementary calculus. do the following. (a) Sketch or gmph the bell-shaped curve [the curve of the integrand in (21 )J. Show that erf x is odd. Show that I b ~ e-w2 dw = 2 (erfb - erfa), a b I e- w2 dw = .y:;;: erf b. -b (b) Obtain the Maclaurin series of erf x from that of the integrand. Use that series to compute a table of erfx for x = OCO.OI)3 (meaning x = O. O.OL 0.02, .... 3). (c) Obtain the values required in (b) by an integration command of your CAS. Compare accuracy. (d) [t can be shown that erf (x) = 1. Confirm this experimentally by computing erf x for large x.(e) Let J(x) = 1 when x> 0 and 0 when x < O. Usingerf(X) = 1, show that (12) then gives1 x 2u(x, t) = I I e- Z dz v 7f -x/(2cVtJ(t> 0).1 IX 2 (g) Show that (t) = =-- e-s /2 ds"27f -ex:569(1) Express the temperature (13) in tenns of the errorfunction.Here. the integral is the definition of the "distributionfunction of the normal distribution" to be discussed inSec. 24.8.