5. (a) The gamma function is defined as r(x) = /ex-ldt First show by integrating by parts that T(x + 1) = al(x) Using this, show that for positive integer n: T(n + 1) = n! Show that I (2 ) = VT Evaluate I(3/2) using the recurrence formula (r + 1) = x(x). Lastly show that r(n+;) =1.3... (20-DrCH (2n)! 2n = 22nn! V (b) Consider the integral First show that I = 7 /2. Now one can formally transform the volume element dri dx2 . . . day to N-dimensional spherical (hyper-spherical) coordinates to get where S is the factor that arises upon integration over the angles. Show that S2 = 27 and S3 = 47. SN can be determined for any N by writing I in hyper- spherical coordinates: 1 = e-12 , N -'SN dr Show that I= SNI ( 2 ) Equate these two values for I to get 27N/2 SN = T(N/2) Show that this reduces correctly for N = 2 and 3. Lastly, convince yourself that the volume of an N-dimensional sphere of radius R is given by and show that this reduces correctly for N = 2 and 3