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

By Glen D. Anderson

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

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Conformal Invariants, Inequalities, and Quasiconformal Maps

A unified view of conformal invariants from the perspective of functions in geometric functionality idea and purposes and quasiconformal mappings within the airplane and in area.

Additional info for Conformal Invariants, Inequalities, and Quasiconformal Maps

Example text

Many special functions may be written in terms of the hypergeometric function. Extensive lists of such particular cases are given, for example, in [SO, pp. 1 57- 164) , [AS] , and [PBM] . Verify each of the following representations for appropriate values of the arguments: ( 1 ) (I - x)-a = F(a , 1 ; l ; x). (2) log( l + x) = x F( l , 1 ; 2; -x) . ; x ) . arctan x = x F ( l , ! ; -x ). arth x = x F( ! ; x ) . (3) arcsin x = x F( ! J"f+""x2) - X F(l ,· J. ,. J"f+""x2 ( I + x)- 2a 2 (8) - 2 (x X + ( I - :x) -2a = 2 F(a , !

58 (22)-(23), (28)-(29) deal with a special type of homeomorphism of subintervals of R that depend on the real parameter K and that reduce to the identity if K = 1. 44) where h : (0, 1) � (0, oo) is a homeomorphism and K > 0. 44 ). Certain perturbed identities play an important role in quasiconformal theory, and we will study these in Chapters 5, 10. Since the functions h and h - 1 are usually very complicated, a basic task is to find bounds for gK (r) in terms of elementary functions. The next result shows how, under the stated conditions, inequalities for h can be used to provide information about g K (r).

1 4) and ( 1 . 1 6) con verge absolutely and unifonnly on compact subintervals of ( - 1 , 1 ) . (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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