1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 373

or

(^0 1) = 1/ 2 (A rgfz -Arg f z ) -^1 2 Arccos lf:zl lfzl +^1 27r
Similarly,
1 1 lf:zl 1

02 = 2"(Argfz-Argfz)+ 2Arccos lfzl + 271"

The angle 7f;1 can also be determined from

sm. ( w - ·'· 'f'l ) = If If: z I I

and the angle 'lj; 2 from

sm. (·'· '!-'2 -w ) = lfzl lf:zl

Clearly, 'lj; decreases when f~(z) describes the open arc T 2 BT 1 , and increases

when f~(z) describes the open arc T 1 AT 2 •

For J i- 0 we have

~ ( d'lj;) = -J 2p(dp/d0)

dO dO p4

and using (6.16-5), we get

d^2 'lj; = -J (G - E) sin20 + 2F cos 0

d02 p4

so

( G - E) sin 20 + 2F cos 20 = 0 (6.17-4)

is the condition for extrema of d'lj;/dO (regardless of the sign of J).

Equation (6.17-4) gives

2F

tan20 = --

E-G

(6.17-5)

whenever E i-G, and cos 20 = 0 if E = G, F i-0. Formula (6.17-5) is

the same as ( 6 .16-7). Hence the directions determined by ( 6 .1 7 -5) are the
principal directions, i.e., those corresponding to the endpoints A and B of
the diameter of the Kasner circle that contains the origin.