Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 11 Special functions Lecture 12 Gamma, beta, error, and elliptic 1 2. The factorial function (usually, n : integer) 0 e ax dx 1 e x 0 0 1 1 x 1 x xe e dx . 2 0 0 0 2 23 2 x 3 x Similarly, x e dx 3 , x e dx 4 xe ax dx 1 0 0 n x xe dx n! n 1 0 0 x n e x dx n! 1 2 3. Definition of the gamma function: recursion relation (p: noninteger) p 1 x - Gamma function p 0 x e dx, p 0. n x n 1e x dx n 1!, 0 n 1 x n e x dx n!. 0 p x - Recursion relation p 1 0 x e dx p!, p 1. p 1 p p - Example 9 / 4 (5 / 4)5 / 4 5 / 41/ 41/ 4 so, 1/ 4 9 / 4 16 / 5. 3 4. The Gamma function of negative numbers p 1 p 1 ( p 0) p - Example 0.3 cf . p 1 1 0.7 , 1.3 0.7 . 0.3 1.3 0.3 1 p 1 as p 0. p - Using the above relation, 1) Gamma(p= negative integers) infinite. 2) For p < 0, the sign changes alternatively in the intervals between negative integers 4 5. Some important formulas involving gamma functions 1/ 2 (prove) 1 / 2 0 1 / 2 2 p 1 p 1 1 t y2 y2 e dt e 2 ydy 2 e dy. 0 y 0 t 4 0 0 e x2 y2 dxdy 4 / 2 r 2 0 0 e rdrd . . sin p 5 6. Beta functions B p, q x p 1 1 x dx, 1 q 1 0 y i ) B p, q 0 a a ii) B p, q 2 /2 0 iii ) B p, q 0 p 1 p 0, q 0. y 1 a q 1 a dy 1 q 1 p q 1 y p 1 a y dy. x y / a 0 a a sin 2 p 1 cos 2 q 1 d . y p 1dy . pq 1 y cf . B p, q Bq, p x sin 2 x y / 1 y 6 7. Beta functions in terms of gamma functions p q p q B p, q P rove) p t 0 e dt 2 y 0 p q 4 0 4 0 0 2 p 1 y 2 e x 2 q 1 y 2 p 1e x /2 2 2 p 2 q 1 r 2 e dr /2 0 q 2 x 2 q 1e x dx dy, y2 2 0 dxdy 2 q 1 2 p 1 r r cos r sin e 0 4 r 0 p 1 t 2 rdrd cos 2 q 1 sin 2 p 1 d 1 p q 1 B p, q . 2 2 7 - Example I 0 x3dx 1 x 5 cf . B p, q 0 y p 1dy . pq 1 y p q 5, p 1 3 p 4, q 1. 41 3! 1 . 5 4! 4 8 8. The simple pendulum T 2 1 2 1 m v m l 2 2 V m glcos L T V 1 2 2 m l m glcos 2 d m l2 2 m glsin 0 dt g l sin . - Example 1 For small vibration, sin g l T 1 2 l / g . 9 - Example 2 g g g sin sin or d sin d : l l l 1 2 g cos const. 2 l cf. elliptic integral In case of 180 swings (-90 to +90) 90 0 const. 0. 1 2 g cos , 2 l /2 0 d 2g cos , dt l d 2g dt. l cos d 2g T / 4 2g T dt . 0 l l 4 cos T 4 l / 2 d , 0 2g cos Beta function!! Using computer, T 7.42 l / g . 10 9. The error function (useful in probability theory) - Error function: erf x 2 0 e t dt. 2 - Standard model or Gaussian cumulative distribution function 1 x t 2 / 2 1 1 x e dt erf x / 2 2 2 2 x 1 1 2 2 x et 2 /2 1 dt erf x / 2 . 2 - Complementary error function 2 t 2 / 2 erfcx e dt 1 erf x / 2 , x 2 x erfc 2 x et 2 /2 dt. - in terms of the standard normal cumulative distribution function erf x 2 x 2 1. 11 - Several useful facts erf x erf x erf 2 0 e t dt 2 2 1 1 2 1 1. 2 2 2 t4 2 x3 x5 2 . x 1 erf x e dt 1 t dt x 0 0 2! 3 5 2! 2 x t 2 2 x - Imaginary error function:erfix 2 x 0 t2 e dt. erf ix ierfix 12 10. Asymptotic series erfcx 1 erf x 2 x t 2 e dt. 2 1 2 1 2 1 d 1 t 2 Using e t te t te t e , t t t dt 2 x e t dt 2 x 1 d 1 t 2 1 1 t 2 1 t 2 1 e dt e x e 2 dt t dt 2 t 2 x 2 t 1 x 2 1 1 t 2 e 2 e dt. 2x 2 x t 13 Using 1 / t 2 e t 1 / t 3 d / dt 12 e t x 2 2 d 1 t 2 1 1 t 2 1 t 2 3 1 x 2 3 1 t 2 e dt 3 e x e 4 dt 3 e x 4 e dt. dt 2 t 2 2x 2 t x 2 t e 1 1 3 1 3 5 erfcx 1 erf x ~ 1 2 . x 1 2 3 2 2 x 2x 2x 2x x2 - This series diverges for every x because of the factors in the numerator. For large enough x, the higher terms are fairly small and then negligible. For this reason, the first few terms give a good approximation. (asymptotic series) 14 11. Stirling’s formula - Stirling’s formula n! ~ n ne n 2n p 1 p e p p 1 1 p p 2p 1 ~ p e 2p . 2 12 p 288 p 15 11. Elliptic integrals and functions - Legendre forms: - First kind : F , k d 1 k sin 2 0 - Second kind : E , k 0 2 0 k 1, , 1 k 2 sin 2 d , 0 k 1. dt 0 k 1, - Jacobi forms: t sin , x sin F , k d x 0 1 k sin 1 k sin d E , k 0 2 2 2 2 1 t 0 x 0 2 1 k t 2 2 1 k 2t 2 1 t 2 , dt. 16 - Complete Elliptic integrals (=/2, x=sin=1): 1 d dt /2 K or K k F , k , 2 2 2 2 t 0 0 2 1 k sin 1 t 1 k t 2 t 1 1 k t /2 2 2 E or E k E , k 1 k sin d dt. 2 0 0 2 1 t - Example 1 /3 0 or 1 1 / 2 sin 2 d E , k E / 3, 1 2 ~ 0.964951 2 E x, k E 3 / 2,1 / k or E , sin 1 k E / 3, / 4 17 - Example 2 /3 0 cf . 16 8 sin d 4 2 /3 0 2 1 1/ 2 sin 2 d 2 1 k 2 sin 2 d E 1 , k E 2 , k 1 cf . F n , k 2nK F , k E n , k 2nE E , k . 18 - Example 4. Find arc length of an ellipse. x a sin , y b cos ds2 dx2 dy2 a 2 cos2 b 2 sin 2 d 2 . ds a a b sin d a 2 2 2 2 a 2 b2 2 1 sin d . 2 a a 2 b2 2 2 elliptical integral of the second kind, k e : eccentricity of ellipse 2 a (using computer or tables) 19 - Example 5. Pendulum swing through large angles. 2 2g cos const. l 2 2g cos cos . l 0 d 2 g T 2 K sin elliptic integral l 4 2 cos cos T 4 l l 2 K sin 4 2 K sin 2g 2 g 2 20 For α not too large sin 2 /2 12 , approximation by series. 2 2 2 l 1 1 3 l 2 4 1 T 4 1 sin sin 2 g 2 2 2 2 4 2 g 16 for small , sin / 2 ~ / 2 - For =30, this pendulum would get exactly out of phase with one of very small amplitude in about 32 periods. 21 - Elliptic Functions u x u x 0 0 dt 1 t2 sin 1 x dt 1 t 2 1 k 2t 2 sn 1 x (elliptic function) x sn u . cnu 1 x 2 dn u 1 k 2 x 2 d sn u cnu dn u du 22