1550251515-Classical_Complex_Analysis__Gonzalez_

Functions. Limits and Continuity. Arcs and Curves

y v

1.:-- - -

• ...-w2
  _....-f*
  ,,,,,.,:: .::: - - --f - - - - - -
  z~-=--.:-----:..-;;----w --..w 1

---:...---1

----------€> ............................ -""'-~____,f--------

0 x ...... "'::::-..................... ..Q

Fig. 3.4

f "-w -.....,_ W1

LI

Example If f(z) = i(z - 1) and D = C, we have

J(z) = - i(z - 1)

f*(z) = i(z -· 1)

]*(z) = - i(z - 1)

J1'(z) = i(z + 1)

Hence for z = 2 + i we have f(2 + i) = -1 + i, ](2 + i)

f*(2 + i) = 1 + i, ]*(2 + i) = 1 - i, and jl'(2 + i) = -1 + 3i.

Similarly, composition with H(z) = l/z gives

F =Hof (the reciprocal function)

G = f o H (the function of the reciprocal)

131

-1 - i,

M =Hof o H (the reciprocal of the function of the reciprocal)

The function Fis not defined for those values of z ED for which f(z) = 0

(the so-called zeros off), if any. G and M are defined only on a subset of

D (if there is one) that contains 1/z, and in the case of M, only at those

points for which J(l/z) =f. O, since

1

F(z) = f(z)

G(z) = (f o H)(z) = f ( ~)

1

M(z) =(Hof o H)(z) = f(l/z)