Page 21 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Another Example

Let f(z) = zz. Then,



0
0
lim f(z   z) f(z ) = lim (z   z)(z   z) z 0 0 z
0
0
 z 0  z  z 0  z

= lim z  z)  0 z z   z z
0
 z 0  z
  z 

= lim z   z z 0 

0
 z 0   z 
Suppose this limit exists, and choose z = (x, 0). Then,
  z    x 


lim  0 z   z z 0  = lim z   x z 0 

0
 z 0   z   x 0   z 
= z  z 0
0
Now, choose z = (0, y). Then,
  z   i y 




lim  0 z   z z 0  = lim z  i x z 0 

0

 z 0   z   x 0  i y 
= z – z 0
0
Thus, we must have z  z  0 z  z , or z = 0. In other words, there is no chance of this limit’s
0
0
0
0
existing, except possibly at z = 0. So, this function does not have a derivative at most places.
0
Now, take another look at the first of these two examples. It looks exactly like what you did in
Mrs. Turner’s 3rd grade calculus class for plain old real-valued functions. Meditate on this and
you will be convinced that all the ”usual” results for real-valued functions also hold for these
new complex functions: the derivative of a constant is zero, the derivative of the sum of two
functions is the sum of the derivatives, the ”product” and ”quotient” rules for derivatives are
valid, the chain rule for the composition of functions holds, etc., For proofs, you need only go
back to your elementary calculus book and change x’s to z’s.
A bit of jargon is in order. If f has a derivative at z , we say that f is differentiable at z . If f is
0
0
differentiable at every point of a neighborhood of z , we say that f is analytic at z . (A set S is a
0
0
neighborhood of z if there is a disk D = {z : |z – z | < r, r > 0} so that D  S. If f is analytic at every
0
0
point of some set S, we say that f is analytic on S. A function that is analytic on the set of all
complex numbers is said to be an entire function.
2.3 Derivatives
Suppose the function f given by f(z) = u(x, y) + iv(x, y) has a derivative at z = z = (x , y ). We know
0
0
0
this means there is a number f’(z ) so that
0

0
0
f'(z )  lim f(z   z) f(z )
0
 z 0  z
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