1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions

which gives a+ 8 = 2k7r and ial^2 ± lbllci = 1. Thus we find that

/3 = k1r - (2q + 1 )'ff = (2k - 2q - 1 )'I!"

2 2

'Y = k1r + (2m + l)'ll" = (2k + 2m + l)'ll"

2 2

8 = - Cl:'+ 2k7r

and the transformation has one of the forms

iaieic>.z ± ilbl

w = ±ilciz + iaie-i<> =

az ± ilbl

±iiciz +a

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(5.12-9)

with ial^2 + lbllci = 1 or ial^2 - lbllci = 1, depending on whether like or unlike

signs are taken in (5.12-9). The circles of fixed points are given by

lz ± i 1 : 11 = l~I

where the sign correspond to those of ilci in (5.12-9).

A classification of the conjugate bilinear transformations can be made

based on the number of fixed points. The discussion is similar to that of

the ordinary bilinear transformations but somewhat less simple. For the

details, we refer the reader to J. McGinty Smith [9].

5.13 The General Bilinear Function

The function defined on C* (except possibly at certain points or lines) by

Az+Bz+C

w = Lz + Mz + N (5.13-1)

will be called the general bilinear function or the general 'bilinear trans-
formation. In dealing with (5.13-1) it is assumed that the matrix of the
coefficients

[

A B C]

L M N

is of rank 2, i.e., that the minors

are not all zeros. Otherwise, we would have A = kL, B = kM, C = kN for

some constant k, so the function will reduce to w = k, except possibly at

points where it is undefined. Clearly, (5.13-1) contains the bilinear function

and the conjugate bilinear function as particular cases.