1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 375

undefined. However, as the point P representing f 0 (z) for Bf:. B 0 +k7r moves

clockwise toward 0 along the circle, the chord OP tends to the semitangent

OT to the circle, and 7/J = Arg f 0 (z) has as a limit the argument 7/Jo of

OT (Fig. 6.17). Thus it seems convenient to complete the definition of

Argf 0 (z) by setting

(6.17-7)

and similarly,

(6.17-8)

We wish to investigate whether with those definitions equation (6.17-6)

remains valid. From

we get

-2B = arg ( iz) = Argfz -Argf-z +'Tr+ 2k7r

B =^1 / 2 (Argfz -Argfz)-1/ 2 7r - k1r =Bo+ k''Tr

by letting Bo=^1 / 2 (Argfz - Argfz) +%'Tr and k' = -(k + 1). Also,

or

Hence we obtain

7/Jo +Bo = 1/ 2 (Arg fz + Arg fz) = Arg(fz + fz)

Fig. 6.17