Download The Confluent Hypergeometric Function: with Special Emphasis by Professor Dr.-Ing. Herbert Buchholz (auth.) PDF

By Professor Dr.-Ing. Herbert Buchholz (auth.)

The topic of this ebook is the better transcendental functionality referred to as the confluent hypergeometric functionality. within the final twenty years this functionality has taken on an ever expanding importance due to its use within the program of arithmetic to actual and technical difficulties. there isn't any doubt that this pattern will proceed until eventually the overall concept of confluent hypergeometric features turns into established to nearly all of physicists in a lot an identical method because the cylinder services, that have been formerly much less popular, at the moment are utilized in many engineering and actual difficulties. This publication is meant to extra this improvement. the real functional value of the services that are taken care of hardly ever calls for an concerned dialogue considering they contain, as particular circumstances, a couple of easier precise capabilities that have lengthy been the standard software of the physicist. it's adequate to say that those contain, between others, the logarithmic crucial, the quintessential sine and cosine, the mistake critical, the Fresnel indispensable, the cylinder services and the cylinder functionality in parabolic cylindrical coordinates. For a person who places forth the trouble to check the confluent hypergeometric functionality in additional aspect there's the inestimable good thing about with the ability to comprehend the homes of different features derivable from it. This gen­ eral standpoint is especially worthwhile in reference to sequence ex­ pansions legitimate for values of the argument close to 0 or infinity and in reference to some of the imperative representations.

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Extra info for The Confluent Hypergeometric Function: with Special Emphasis on its Applications

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Il} ~/ 2 (l-p. 2 )/ 4} · Y(z) =0 (3b) Section 3. Related Differential Equations 33 with the solution (3a) We now consider a number of particular applications of these formulae. I. = 0 and A= 1 in Eq. (3a), then we see directly that the function y(z) = P,,I'12 (Az) satisfies the differential equation 1-p2} "A A2 y"(z)+ { --+-+~4z2 ·y(z)=O. 2'la, t ·o/1 + 4c (2 V~· z) · Y (z) (4{J) If A= 1 in equation (4a) then the resulting differential equation is 1 4 y"(z)+ { - - 1X -J12 ++- 4z z 2 } · y(z)=O.

The formulae showing the connection between these various particular solutions of The Various Forms of the Differential Equation 40 Eq. (22) could be derived directly from the differential equation. Our approach, however, is to derive these relationships from those already shown for the various solutions of Whittaker's differential equation. ,,. 7 to be always linearly independent, correspond to the functions D,(z) and D_,_ 1 (± iz) in the case now under consideration. In fact, we have the exact correspondence as given in Eq.

1- s2) -2-. G+ :r ds -1 J 1+1' ,. = 2-l'z~ e•i2·cosrp. sin~' 0 · 0 The two Hankel integrals, Eqs. )r(1+p-A) z 1 jx'--1 ·(1-x)ll-'-· 1F 1 0 (14) 1+1' . -2-. ) > -1 Section 2. 12 (z}. In particular the following relation follows from Eq. Jt. 1 (z) x,p2 = 1/1 n (1+p ) r - 2- - x r(2x) . 2"-p/2 . zl12+x . e-z/2 (14a) 1 -jxPf2-x . (1- x)2x-l. /2-x(~28). e"z/2 . dx. Re(x) > 0, Re(,u- 2x) > -1. 1. and x which severely restrict the validity of Eq.

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