By Petr Mandl
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Extra info for Analytical treatment of one-dimensional Markov processes
Bh # 1,2,3,.. ,k) ~ O, + I, + 2 ..... ,k; h = I .... ,k; j # h) h - bh ~ O, -I, -2 ..... ,n-o+l;h = I, .... k). J These conditions are assumed as they are involved during the proof but can be waived by an appeal to the considerations of continuity and following the techniques given in Chapter V. 9), we easily get Gm,p,nr p,q tz exp[(2k-£)~i]lla r] k = C. , .... _bq;(-l) p'm'n z). S. 3. 2. 9) and hence the result follows. 1a. 38) i. 4). 1b. Let m,n,p and q be integers such that I < n < p < q, 2 < m < q, P the numbers a.
A function f(z) is said to be dominant compared with another function g(z) if the leading term of the asymptotic expansion of g(z) is of an order less than the error term of the asymptotic expansion of f(z). For example, consider Z fl(z) ~ e f3 (z ) z -- r % air z , r=o 1 ~ the asymptotic expansions -- ¢o 2 Z z5 f2(z) ~ -- r Z a2r z , r=o co a3r -z- r , f4 (z) ~ , f6(z) ~ e iz r=o -z fb(z) ~ e -- r ~ abr z r=o --2z e -z- 2 2 r=o a4r -z- --r ~ a6r z r=o r , , where z is positive and it is assumed that none of Then evidently a.
And Related 1 Erf(x) -t 2n -~ = 2 e Functions. 2x dt = ~ 7 F 1 3 i i(~ ' ~; -x2) " o I Erfc(x) 2~ = f e -t 2 dt l-Erf(x) x 1 2 -- X (~x) - 2 e " 2 - - W_I i (x2) . 4'4 1 C(x) = I (2~) -2 t cos t dt. O S(x) i I (2~) -2 ~ t -~ sin t dt. 4f. 3,4g. Coulomb Wave Functions. L+I FL(~,~) CL(~)~ 2Le - ? Ce(q) L+l-iq e l~ ; 2i~ ) , IFI(2L+2 IP(L+l+iq) l = (2L+I)~ By virtue of Kummer's transformation IFI(~; ~,z) = ez FL(~,a ) is real it follows that, we usually take L to be a positive namely, IFI($-~; if ~,q and L are real.
Analytical treatment of one-dimensional Markov processes by Petr Mandl