Page 24 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 2: Complex Functions

  u  v  stuff Notes
=   x   i x      x i y .


Here,
stuff = x( + i ) + y( + i ).
4
3
1
2
It’s easy to show that
stuff
lim  0,
 z 0  z
and


0
0
lim f(z   z) f(z )   u  i  v .
 z 0  z  x  x
In particular we have, as promised, shown that f is differentiable at z .
0
Example 3:
Let’s find all points at which the function f given by f(z) = x – i(1 – y) is differentiable. Here we
3
3
have u = x and v = – (1 – y) . The Cauchy-Riemann equations, thus, look like
3
3
3x = 3(1 – y) , and
2
2
0 = 0.
The partial derivatives of u and v are nice and continuous everywhere, so f will be differentiable
everywhere the C-R equations are satisfied. That is, everywhere
x = (1 – y) ; that is, where
2
2
x = 1 – y, or x = –1 + y.
This is simply the set of all points on the cross formed by the two straight lines

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