Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.2 THE CAUCHY–RIEMANN RELATIONS

Sincexandyare related tozand its complex conjugatez∗by

x=

1

2

(z+z∗)andy=

1

2 i

(z−z∗), (24.6)

we may formally regard any functionf=u+ivas a function ofzandz∗, rather

thanxandy. If we do this and examine∂f/∂z∗we obtain

∂f

∂z∗

=

∂f

∂x

∂x

∂z∗

+

∂f

∂y

∂y

∂z∗

=

(

∂u

∂x

+i

∂v

∂x

)(

1

2

)

+

(

∂u

∂y

+i

∂v

∂y

)(

−

1

2 i

)

=

1

2

(

∂u

∂x

−

∂v

∂y

)

+

i

2

(

∂v

∂x

+

∂u

∂y

)

. (24.7)

Now, iffis analytic then the Cauchy–Riemann relations (24.5) must be satisfied,

and these immediately give that∂f/∂z∗is identically zero. Thus we conclude that

iffis analytic thenfcannot be a function ofz∗and any expression representing

an analytic function ofzcan containxandyonly in the combinationx+iy,not

in the combinationx−iy.

We conclude this section by discussing some properties of analytic functions

that are of great practical importance in theoretical physics. These can be obtained

simply from the requirement that the Cauchy–Riemann relations must be satisfied

by the real and imaginary parts of an analytic function.

The most important of these results can be obtained by differentiating the

first Cauchy–Riemann relation with respect to one independent variable, and the

second with respect to the other independent variable, to obtain the two chains

of equalities

∂

∂x

(

∂u

∂x

)

=

∂

∂x

(

∂v

∂y

)

=

∂

∂y

(

∂v

∂x

)

=−

∂

∂y

(

∂u

∂y

)

,

∂

∂x

(

∂v

∂x

)

=−

∂

∂x

(

∂u

∂y

)

=−

∂

∂y

(

∂u

∂x

)

=−

∂

∂y

(

∂v

∂y

)

.

Thus bothuandvareseparatelysolutions of Laplace’s equation in two dimen-

sions, i.e.

∂^2 u

∂x^2

+

∂^2 u

∂y^2

= 0 and

∂^2 v

∂x^2

+

∂^2 v

∂y^2

=0. (24.8)

We will make significant use of this result in the next chapter.

A further useful result concerns the two families of curvesu(x, y) = constant

andv(x, y) = constant, whereuandvare the real and imaginary parts of any

analytic functionf=u+iv. As discussed in chapter 10, the vector normal to the

curveu(x, y) = constant is given by

∇u=

∂u

∂x

i+

∂u

∂y

j, (24.9)