Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

whereiandjare the unit vectors along thex-andy-axes, respectively. A similar

expression exists for∇v, the normal to the curvev(x, y) = constant. Taking the

scalar product of these two normal vectors, we obtain

∇u·∇v=

∂u

∂x

∂v

∂x

+

∂u

∂y

∂v

∂y

=−

∂u

∂x

∂u

∂y

+

∂u

∂y

∂u

∂x

=0,

where in the last line we have used the Cauchy–Riemann relations to rewrite

the partial derivatives ofvas partial derivatives ofu. Since the scalar product

of the normal vectors is zero, they must be orthogonal, and the curvesu(x, y)=

constant andv(x, y) = constant must therefore intersect atright angles.

Use the Cauchy–Riemann relations to show that, for any analytic functionf=u+iv,the

relation|∇u|=|∇v|must hold.

From (24.9) we have

|∇u|^2 =∇u·∇u=

(

∂u

∂x

) 2

+

(

∂u

∂y

) 2

Using the Cauchy–Riemann relations to write the partial derivatives ofuin terms of those
ofv,weobtain

|∇u|^2 =

(

∂v

∂y

) 2

+

(

∂v

∂x

) 2

=|∇v|^2 ,

from which the result|∇u|=|∇v|follows immediately.

24.3 Power series in a complex variable

The theory of power series in a real variable was considered in chapter 4, which

also contained a brief discussion of the natural extension of this theory to a series

such as

f(z)=

∑∞

n=0

anzn, (24.10)

wherezis a complex variable and theanare, in general, complex. We now

consider complex power series in more detail.

Expression (24.10) is a power series about the origin and may be used for

general discussion, since a power series about any other pointz 0 can be obtained

by a change of variable fromztoz−z 0 .Ifzwere written in its modulus and

argument form,z=rexpiθ, expression (24.10) would become

f(z)=

∑∞

n=0

anrnexp(inθ). (24.11)