Formulas and Theorems for the Special Functions of by Dr. Wilhelm Magnus, Dr. Fritz Oberhettinger, Dr. Raj Pal

By Dr. Wilhelm Magnus, Dr. Fritz Oberhettinger, Dr. Raj Pal Soni (auth.)

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11·12 + + ... J ' C(7) 7·11 ·12 ·13 ·14 +... J. 36 I. The gamma function Power series expansions of some trigonometric and hyperbolic functions -f= sm z zcotz= . 2 i ,,=0 (-It (1 - 22n - 1) B2"1 z2n, (2n) . B L (-It 22n~z2" ,,=0 (2n). 00 = - 2~ C~2~) z2n, Izl < n, tan z = ~ (_1),,+1 22n (22n _ 1) B2"1 ,,~ = 2 i' (22:~ (2n). z2n n(2n)! JzJ < ' 1/;2' tanh z = 2 coth (2z) - coth z 1: 2 (22n - 1) B 2n z2n-1 2 1: (_1)n+1 (22"1/;~ C(2n) z2n-1, = 2" (2n)1 n=1 = 1) ,,=1 z coth z = z [2 Jzl < ~ , + e'2z-1 sinh ( z -2--- ~)] 1 + log (1 _ log - - - - = -z 2 e-Z) ~~ z 00 _ "" B 2" z2n , z I J < 2n.

Rez - (2m The following is a contour integral representation of C(z, IX). C(z, IX) = - r in-: z) f (0+) ( [1 - 00 e- r 1 1 (-W- 1 e- txl dt, Re IX> 0, Iarg (-t) I < n, z =F 1, 2, 3, ... Some integrals associated with C(z, IX) 1) 2 8 1+l¥) fOO[l +e -1]-1 'b.... 1"(z'21X- 1"( .. ) - C(2)J . J Vlo J ~ dt ~ [n2log 2 + 4 (log 2)3 + C(3, ~) o 1 (log t)2 t2 2 8' 2 1 = - J 1 (log t)1I+l dt = - 10 2 t2 g .

1 Linear transformations 2Fl(a, b; c; z) = (1 - zy-a-b ~l(C - a. c - b; c; z), 2F1 (a, b; c; z) = (1 - z) -a ~1 (a. - 1) , 2Fl (a, b; c; z) = (1 - Z)-b ~l (b. 1-)z a. c - b'• c - a - b + 1'1 . - z) , b =1= ± m, m = 0, 1, 2. • . ) _ r(c) r(c - a - b ) . 2F d a• b. c, z - r(c _ b) r(c _ a) ~da. b. a + (1 - z )C-a-b r(c) r(a) r(a + b - c) F ( r(b) 2 1 C- larg (1 - z) I <~, c - a - 48 II. The hypergeometric function 2F( 1 a, 1) ·) ( )-a 2F 1 ( a, a - e +1',a - b+1·'z b· ,e, z = T(c)T(b-a) T(b) T(c _ a) -Z T(c) T(a - b) + 1,.

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