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By Glen D. Anderson
A unified view of conformal invariants from the viewpoint of functions in geometric functionality conception and purposes and quasiconformal mappings within the aircraft and in area.
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Extra resources for Conformal Invariants, Inequalities, and Quasiconformal Maps
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2) B (p, q) = f t P- 1 (1 - t )q- l dt , I 0 Re p > 0, Re q > O. 3) B (p, q) =2 j :n:/2 0 sin2p - l cp cos2q - l ({) dcp, Re p > 0, Re q > 0. Integrals like these arise in a number of applications, for example, in probability theory and in some geometric problems. 4. Lemma. For Re z > 0 and any positive integer n, (1) f (z + 1 ) (2) (3) = zf (z) . f (z + n) = (z + n - l )(z + n - 2 ) · · · (z + 1) z f (z) = (z, n)f (z) . f (n + 1 ) = n ! 4 ( 1) see, for example, [Ap2, p. 278] or [Leb, p. 3]; parts (2) and (3) follow by induction.
L + x. ) K + ( 1 - x ) K ( 1 + x') K + ( 1 - x') K , 2x K x' = �. JI - 4 1 - K a } . (b) max { ( ; ) , 2 1 - 4 1 - ( I / K ) b } < th ( � arth r ) < min {r, v'f=b} . = (25) Show that, for fixed K > l , f(x) x 1 f K ch((l / K )arch( l /x)) is increas K 0 / ing from (0, I) onto (2 H , I) and that g (x) = x K ch(K arch( l /x)) is decreasing from (0, 1 ) onto ( 1 , 2 K - 1 ) . Deduce that (a) 2 0 / K ) - I :::: x 1 f K ch(( l /K ) arch( l /x)) :::: 1 . (b) 1 :::: X K ch(K arch( l /x)) :::: 2 K - I .
1 Remarks. 21. Hypergeometric Fu nctions 9 (1) Theorem 1 . , c,. 1 ) . B (b , c - b) Two examples of generalized hypergeometric functions are (2) 3 F ( 1 , 3, 3; 2, 2; x) = i O - x ) - 3 (4 - 3x +x 2 ) [PBM, p. 523, Formula 448], 2 arcsin2 ,Jx • • (3) 3 F2 ( 1 , l , 1 , 23 , 2, x) _ [PBM, p. 5 1 8, Formula 353] . X (4) We are interested mainly in the properties of the hypergeometric function as a function of the variable x . In its dependence on its upper and lower parameters, p Fq satisfies important difference equations (see [AS, Ch.