Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSDeduce the value of∫ ∞(u +2) 2J =du.0 (u 2 +4)5/218.15 The complex function z! is defined byz! =∫ ∞0u z e −u dufor Re z>−1.For Re z ≤−1 it is defined by(z + n)!z! =(z + n)(z + n − 1) ···(z +1) ,where n is any (positive) integer > −Re z. Being the ratio of two polynomials, z!is analytic everywhere in the finite complex plane except at the poles that occurwhen z is a negative integer.(a) Show that the definition of z! forRez ≤−1 is independent of the value ofn chosen.(b) Prove that the residue of z! at the pole z = −m, wherem is an integer > 0,is (−1) m−1 /(m − 1)!.18.16 For −1 < Re z0, occurs in some statistical mechanics problems. By first consideringthe integral∫ ∞J = e iu(k+ia) du,and a suitable variation of it, show that I =(π/a) exp(a 2 )erfc(a), where erfc(x)is the complementary error function.18.18 Consider two series expansions of the error function as follows.(a) Obtain a series expansion of the error function erf(x) in ascending powersof x. How many terms are needed to give a value correct to four significantfigures for erf(1)?(b) Obtain an asymptotic expansion that can be used to estimate erfc(x) forlarge x (> 0) in the form of a seriesaerfc(x) =R(x) =e −x2 nx . n n=0Consider what bounds can be put on the estimate and at what point theinfinite series should be terminated in a practical estimate. In particular,estimate erfc(1) and test the answer for compatibility with that in part (a).18.19 For the functions M(a, c; z) that are the solutions of the confluent hypergeometricequation,0644∞∑