By Elias M. Stein

This booklet has as its topic the boundary price conception of holomorphic services in different advanced variables, an issue that's simply now coming to the vanguard of mathematical research. For one variable, the subject is classical and very well understood. in different variables, the required realizing of holomorphic services through partial differential equations has a up to date foundation, and Professor Stein's ebook, which emphasizes the potential-theoretic facets of the boundary worth challenge, should still turn into the traditional paintings within the field.

Originally released in 1972.

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**Additional resources for Boundary Behavior of Holomorphic Functions of Several Complex Variables**

**Sample 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(!