1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions

Combining (5.7-3) with (5.7-2), we get

μ - Z1 = M2 z - Z1

μ-Z2 Z - Z2

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Similarly, after n applications of the transformation we have, with ( = Ln z,

(-z1 _ n Z - Z1

---M --

(-z2 z-z2

(5.7-4)

The multiplier M is related to the coefficients a and d of the

transformation by the formula

1

M + M =(a+ d)^2 - 2 = H + 2

In fact, we have

1 a - CZ1 a - CZ2

M+-=--+ -

M a - cz2 a - cz1

But

a-d

Z1 + Z2 =

c

= ---:,----~-~-:----::-"--~ 2a^2 - 2ac(z^1 + z2) + c^2 (z~ + z~)

a^2 - ac(z1 + z2) + c^2 z1z2

b

Z1Z2 = - - ,

c

2 2 a2 + d2 - 2

Z1 + Z2 = c 2

(5.7-5)

(5.7-6)

since z 1 and z 2 are the roots of (5.6-1) and ad - be= 1. Introducing these
values into (5.7-6), equation (5.7-5) results.
Since equation (5. 7-2) can be written in the form
w-z2 1 z-Z2
w -Z1 = M z-Z1

it follows that 1/M could also be regarded as the multiplier of the original
transformation. This follows as well from the symmetry of ( 5. 7-5). Whether

we consider M or 1/ M as the multiplier depends on which fixed point is

called z1.
Next, consider the case c = 0, H 'f. 0. Then the transformation has
the linear form

a b

w = -z + -

d d

with ad= 1, and has the finite fixed point z 1 = b/(d - a). Hence

W - Z1 = ~ Z + ~ - b = ~ (z -b)
d d d-a d d-a
or