1Analytic functions

IB Complex Methods

1.3 Harmonic functions

This is the last easy section of the course.

Definition

(Harmonic conjugates)

.

Two functions

u, v

satisfying the Cauchy-

Riemann equations are called harmonic conjugates.

If we know one, then we can find the other up to a constant. For example, if

u(x, y) = x

2

− y

2

, then v must satisfy

∂v

∂y

=

∂u

∂x

= 2x.

So we must have

v

= 2

xy

+

g

(

x

) for some function

g

(

x

). The other Cauchy-

Riemann equation gives

−2y =

∂u

∂y

= −

∂v

∂x

= −2y − g

0

(x).

This tells us

g

0

(

x

) = 0. So

g

must be a genuine constant, say

α

. The corresponding

analytic function whose real part is u is therefore

f(z) = x

2

− y

2

+ 2ixy + iα = (x + iy)

2

+ iα = z

2

+ iα.

Note that in an exam, if we were asked to find the analytic function

f

with real

part

u

(where

u

is given), then we must express it in terms of

z

, and not

x

and

y, or else it is not clear this is indeed analytic.

On the other hand, if we are given that

f

(

z

) =

u

+

iv

is analytic, then we

can compute

∂

2

u

∂x

2

=

∂

∂x

∂u

∂x

=

∂

∂x

∂v

∂y

=

∂

∂y

∂v

∂x

=

∂

∂y

−

∂u

∂y

= −

∂

2

u

∂y

2

.

So u satisfies Laplace’s equation in two dimensions, i.e.

∇

2

u =

∂

2

u

∂x

2

+

∂

2

u

∂y

2

= 0.

Similarly, so does v.

Definition

(Harmonic function)

.

A function satisfying Laplace’s equation equa-

tion in an open set is said to be harmonic.

Thus we have shown the following:

Proposition.

The real and imaginary parts of any analytic function are har-

monic.