u = \frac{( b - c ) ( z - a )}{( b - a ) ( z - c )}

a, b, c

0, 1, \infty

, respectively. Because a Möbius map is locally biholomorphic away from its pole, it carries each exponent pair with its singular point and gives

P ⎩ ⎨ ⎧ 0 α α^{'} 1 β β^{'} \infty γ γ^{'}; u ⎭ ⎬ ⎫ .

y (u) = u^{α} (1 - u)^{β} v (u),

so that

v = u^{- α} (1 - u)^{- β} y

α

from both exponents at zero, subtracts

β

from both exponents at one, and adds

α + β

to both exponents at infinity. Hence

v

P ⎩ ⎨ ⎧ 00 α^{'} - α 10 β^{'} - β \infty γ + α + β γ^{'} + α + β; u ⎭ ⎬ ⎫ .

A = γ + α + β, B = γ^{'} + α + β, C = 1 + α - α^{'} .

α^{'} - α = 1 - C

and

β^{'} - β = C - A - B

. After renaming

u

z

, the symbol is therefore

P ⎩ ⎨ ⎧ 00 1 - C 10 C - A - B \infty A B; z ⎭ ⎬ ⎫,

A, B, C

a, b, c

gives exactly the displayed symbol in the question.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.