94 CHAPTER. 3 • ANALYTIC AND HARMONIC FUNCTIONS
If we let w = J (z) and t:.w = J (z) - J (zo), then we can use the Leibniz
notation ~~ for the derivat ive:
, ( ) dw. t:.wf zo = -d z = t>hm z-0 ~u z · (3-3)
• EXAMPLE 3.1 If J (z) = z^3 , show that J' (z) = 3z^2 •
Solution Using Equation (3-1), we have
3 - 3J' (zo) = Jim z zo
z-r-0 z - zo. (z - zo) (z^2 + zoz + zZ)
= hm
z~ro z - zo
= lim (z^2 + zoz + zZ)
z-zo
-32 - Zo·We can drop the subscript on zo to obtain f' (z) = 3z^2 as a general formula.
Pay careful attention to the complex value t:.z in Equation (3-3); the value
of the limit must be independent of the manner in which t:.z -+ 0. If we can find
two curves that end at zo along which i~ approaches distinct values, then %;
does not have a limit as t:.z -> 0 and f does not have a derivative at zo. T he
same observation applies to the limits in Equations (3-2) and (3-1).
• EXAMPLE 3.2 Show that the function w = f (z) = z = x - iy is nowhere
different iable.Solution We choose two approaches to the point zo = x 0 + iy 0 and compute
limits of t he difference quotients. First, we approach zo = xo + iyo along a line
parallel to the x-axis by forcing z to be of the form z = x + iyo.
Jim f (z) - f (zo) = lim f (x + iyo) - f (xo + iyo)
•-•o z -zo (x +iyo)-(xo+iyo) (x + iyo) - (xo + iyo)
r (x -iyo) - (xo -iyo)
(x+iyo)~xo+iyo) (x - xo) + i (yo - yo)= lim
x - xo
(x +iyo)-.(zo+iyo) X - Xo
= 1.Next, we approach zo along a line parallel to the y-axis by forcing z to be of the