1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation

From the formulas for f z and fz in polar form we get

e i..PJ z + e -i'l/Jf z -- f r -- Ur + ZVr.

ir(ei'l/J fz -e-i'l/J fz) = f..p = u..p + iv..p

Also, from

z'(t) = lz'(t)ieiB = ei'l/J(t)[r'(t) + ir(t)1fi'(t)]

it follows that

ei(B-1/J) = r'(t) + ir(t) 1/i'(t)

lz'(t)i lz'(t)i

so that

r'(t)

cos(B-1/i) = iz'(t)i' sin(O -1/i) = r(t) l~:gjl

By using (6.10-15) and (6.10-16) in (6.10-14) we find that

349

(6.10-15)

(6.10-16)

f~(z) = e-iB[(ei'l/J fz + e-i'l/J fz) cos((} -1/i) + i(ei..P fz - e-i'l/J fz) sin(O -1/i)]

which simplifies to

(6.10-17)

6.11 TIIE KASNER CIRCLE

Letting ( = f~(z) in formula (6.10-2), or in (6.10-17), we have

., /' = f z + f ;:e -2i8 ' (6.11-1)
From this equation it is clear that for a fixed point z, the arc described by
( as (} varies in the interval [O, 27r] is an oriented circle described twice in
the negative (clockwise) direction (Fig. 6.6). This circle was introduced by
E. Kasner [65], who called it a clock, but it is usually known as the Kasner
circle (following E. R. Hedrick).

'·plane

)

--

Fig. 6.6