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Complex differentiation

and solving non-linear equations

COMPLEX DIFFERENTIATION

Part I

Three different ways to specify a

complex function

• In terms of z

w =z^2

• In terms of x and y

u = x^2 – y^2

v = 2xy

• In terms of r and 

R = r^2

 = 2 

The function z

as transformation

2 2.5 3 3.5 4 4.5 5

2

3

4

5

6

7

x

y

-30 -20 -10 0 10

0

10

20

30

40

50

60

70

u

v

Terminologies

• A complex-valued function f is said to be complex

differentiable z 0 if we can find a complex number c, such that for any small complex number h, we have

f(z 0 + h)  f(z 0 )+ c h.

• The complex scaling factor c is called the derivative of f(z) at z 0 , and is

usually denoted by f’(z 0 ).

• The derivative of f as a function of z is often denoted by df/dz.
• A complex differentiable function is sometime called

holomorphic, or analytic.

• If f(z) is differentiable at all points in the complex plane, than

f(z) is called an entire function.

Non-example: Complex

conjugation

Domain z-plane

Image w-plane

z 0

conj(z 0 )

f(z) = conj(z)

For any z 0 , we cannot find a complex constant c such that f(z 0 + h) – f(z 0 )  c h. The complex conjugate function is not holomorphic at any point.

kshum ENGG2012B

Formal definition of complex

differentiation

• In terms of limit, the approximation

f(z 0 + h)  f(z 0 )+ c h means

– In terms of epsilon-delta, it says: for any arbitrarily

small and positive real number , there exists a

positive real number , such that

|(f(z 0 + h) – f(z 0 ))/h – c|  for all h with |h|  .

• If we can find a complex number c satisfying the

above, then the function f is holomorphic at z 0.

Example: z

n

is an entire function

• Let n be a positive integer.

• We first guess that nzn-1^ is the derivative of zn,

and then prove that it is correct.

The usual rules for calculus apply

• Derivative of zn: (for integer n0)

• Derivative of exp and log:

• Derivative of sine and cosine

• Sum rule:

• Product rule:

• Quotient rule:

• Chain rule

Example

• Find the derivative of z^2 +z+3 at z=j.

(z^2 +z+3)’

= (z^2 )’ +(z )’ +(3)’

= 2z + 1 + 0

Evaluate at z=j  the derivative at z=j is 1+2j.

Cauchy and Riemann

• Augustin-Louis Cauchy

• French mathematican

• Bernhard Riemann

• German mathematician

Review: 

• The symbol  in calculus stands for partial

differentiation of a multi-variable function.

– “” is pronounced as “partial” or “round”.

• For a function g (x,y) of two variables, we

sometime write

– g x(x,y) for the partial derivative with respect to x, and

– g y(x,y) for the partial derivative with respect to y.

Approaching z horizontally

• When h is restricted to real numbers,

z (^) z+h = z+ x

Approaching z vertically

• If h is restricted to purely imaginary numbers,

z

z+h = z+ j y