Conformal Invariants, Inequalities, and Quasiconformal Maps by Glen D. Anderson

By Glen D. Anderson

A unified view of conformal invariants from the perspective of functions in geometric functionality thought and functions and quasiconformal mappings within the aircraft and in area.

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For a table of infinite products see [Hans] . 13. The hypergeometric function. For a real number a we denote the shifted factorial by the Appell symbol (a, n) = a(a + I ) · · · (a + n - I) if n is a 6 Basic Fu nctions positive integer, and set (a , 0) = 1 . The (Gaussian) hypergeometric series is defined by (1 . J -, (c, n ) n ! n=O z E C , provided that no (c, n ) is zero. For nonnegative integers p , q the generalized hypergeometric function P Fq is defined by (1 . 1 5) provided that (bk , n) =I= 0 for all k, n .

The infinite product given £ > 0 , there exists No such that TI: 1 (1 + ak) is con vergent if and only if, for all n 2:: No and p 2:: I . For a proof we refer the reader to [Hi, pp. 222-223) . The infinite product n: 1 0 +ak) is said to be absolutely con vergent if l ak I) is convergent. n:. 7. only if Hypergeometric Functions 5 Theorem. The product fl: 1 (1 + ak) is absolutely convergent if and I::: 1 ak is absolutely convergent. Proof. 5. Assume that 1 + D ak =fa O for k � 1. Then the product ak) is absolutely convergent if and only if I:::, Log (1 + ak) is ab­ :1 (1 +convergent, flsolutely where Log denotes the principal branch of the logarithm.

2) For C > b > 0 , F(a b· c · x) ' ' ' = B (b, 1 b) C- f t b - 1 ( 1 - t) c - b - l (l - x t)-adt . (3) F(a , b; c; x) satisfies the hypergeometric differential equation ( 1 . 1 7) . (4) If c > a + b , and if none of c, c - a , c - b is zero or a negative integer, then F(a , b; c; 1 ) f(c)r(c - a - b) · f(c - a)f(c - b) = (5) If a , b , c are neither O nor a negative integer and if a + b > c, then F(a , b; c; x) is asymptotic to D ( I - x)-

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