Supplementary Problems GAMMA FUNCTION F(7) (a) 2 r(4) r(3) r(3) T(3/2) (6) T(9/2) 9.28. Evaluate (с) г(1/2) г(8/2) г (5/2). te- dx, e- dz, 9.29. Evaluate (a) (b) (e) S* dz, (b) 9.80. Find (a) (c) 9.31. Show that > 0. 9.32. Prove that l(n) = da, n> 0, L' (a In zj* dz, S' Vin (1/2) dz. 9.33. Evaluate (a) (In x)* dz, (b) (0) 9.34. Evaluate (a) г(-7/2), (ь) г(-1/3). 9.35. Prove that lim r(*) = * where m = 0,1,2, 3, ... Prove that if m is a ponitive integer, r(-m + 1) = (-1)m 2" V 1-3-5... (2m – 1) 9.36. 222 GAMMA, BETA AND OTHER SPECIAL FUNCTIONS 9.37. Prove that r'(1) = e-- In z de is a negative number (it is equal to -y, where y is called Euler's constant). Obtain the miscellaneous result 4 on page 211 from the result (4) of Problem 9.23, [Hint: Expand e/(sVn) + ... in a power series and replace the lower limit of the -«.) 9.38. BETA FUNCTION 9.39. Evaluate (a) в(3, 5), (6) в(3/2, 2), (c) B(1/3, 2/3). 9.40. Find (а) x(1- 2)" da, (6) (4- *)3/2 da. 9.41. Evaluate (a) {r(1/4)} dy J. Tai-y 9,42. Prove that %3D 9.43. Evaluate (a) Bin e con e de, (b) cor e de. 9.44. Evaluate (a) sins e de, (b) cos e sin? e de. 9.45. Prove that Vtan e de = z/y2. z dz y dy S, , N 9.46. Prove that (a) (b)