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By Harold Jeffreys

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Example text

The Hypcrgcomclric Equation. Tn certain problems it is possible to reduce the solution to lhnl of solving the second order linear differential equation rr-y tt(l- x) d:r:~ in which 0:, + {y - (1 dy + a. ,8y = O. 1) ,8 nnd y nrc constnnts. For instancc, thc Schro- 24 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § a dinger equation for n symmetricnl-top molecule, whieh is of importnnce in the theory of molecular spectra,l) CUll, by simple transformations, be reduced to this typc. Irises in the study of the flow of compressible fluids.

1: I < I, Le. 8) nrc convergent ill (0,2). 8). t ing x = 1 we have 2/,'I(ex, pj y; 1) = A, if we assume that I > Y> 0 p. 2) we sec that A _ F(y)F(y - 0 - P) - T(y o)F(y P)' and that + 1- _I T(o+P-y+1)T(1-y) - . F(P y+ 1 )F(o y+ 1) +n r(y-o-p+1 )1'(1-y) 1'(1 P)r(! 0) , so that n _ T(y)F(o+ P- y) 1'(o)F(P) , whence we find that 2 PI (0:, T(y)F(y-o-P) ex)r(y fJ) 2 F I (rx, p; y; x) - rey p; IX +fJ - y+ 1 : 1 -;1:) F (y-O( ,,_R. 2:) + r(i')r(o:+{J-Y)(I_X)l'_"_~ r(o)F(P) , . -x ill O

IX, we put 28 THE SPECIAL fUNCTIONS Of PHYSICS AND CHEMISTRY §9 which gives .. • • .... e. a'I P+) 1, ',. Tnking Co = 1 we obtnin the solulion x-":P1(a. a- y + 1; 1 a-p+ 1; -). 0) 9. Linear Relations between the Solutions of the Equation. 1: I < I, Le. 8) nrc convergent ill (0,2). 8). t ing x = 1 we have 2/,'I(ex, pj y; 1) = A, if we assume that I > Y> 0 p. 2) we sec that A _ F(y)F(y - 0 - P) - T(y o)F(y P)' and that + 1- _I T(o+P-y+1)T(1-y) - . F(P y+ 1 )F(o y+ 1) +n r(y-o-p+1 )1'(1-y) 1'(1 P)r(!

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