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680 CHAPTER 19 Complex Numbers and Functions

19.2.2 The Cauchy-Riemann Equations
We will derive a pair of partial differential equations that are intimately tied to differentiability
for complex functions. These equations also play a role in potential theory and in treatments of
the Dirichlet problem.
If f is a complex function and z = x + iy, then we can always write
f (z) = f (x + iy) = u(x, y) + iv(x, y)

where u and v are real-valued functions of two real variables:
u(x, y) = Re( f (z)) and v(x, y) = Im( f (z)).

EXAMPLE 19.9

2
Let f (z) = z . Then
2
2
2
f (z) = (x + iy) = x − y + 2ixy = u(x, y) + iv(x, y).
2
For this function, u(x, y) = x − y and v(x, y) = 2xy.
2
If g(z) = 1/z for z = 0, then
1 x y
g(z) = = − i = u(x, y) + iv(x, y).
2
2
x + iy x + y 2 x + y 2
For g(z),
x y
u(x, y) = and v(x, y) =− .
2
2
x + y 2 x + y 2
For f to be differentiable, partial derivatives of u and v must be related in a special way.
THEOREM 19.4 Cauchy-Riemann Equations

Let f (x + iy) = u(x, y) + iv(x, y) be differentiable at z = x + iy. Then, at (x, y),
∂u ∂v ∂v ∂u
= and =− .
∂x ∂y ∂x ∂y

These are the Cauchy-Riemann equations for the real and imaginary parts of f . In deriving

these equations, we will also obtain expressions for f (z).
If f is differentiable at z, then
f (z + h) − f (z)

f (z) = lim .
h→0 h
This difference quotient approaches f (z 0 ) regardless of path of approach of h to 0. Focus on two

specific paths.
Path 1 Along the Real Axis.
Now h is real and moves left or right toward 0. Since h is real, z + h = x + h + iy and
f (z + h) − f (z)

f (z) = lim
h→0 h
u(x + h, y) + iv(x + h, y) − u(x, y) − iv(x, y)
= lim
h→0 h

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October 15, 2010 18:5 THM/NEIL Page-680 27410_19_ch19_p667-694