Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

We then find that

f′(z) = lim

∆z→ 0

[

(z+∆z)^2 −z^2

∆z

]

= lim

∆z→ 0

[

(∆z)^2 +2z∆z

∆z

]

=

(

lim

∆z→ 0

∆z

)

+2z=2z,

from which we see immediately that the limit both exists and is independent of

the way in which ∆z→0. Thus we have verified thatf(z)=z^2 is differentiable

for all (finite)z. We also note that the derivative is analogous to that found for

real variables.

Although the definition of a differentiable function clearly includes a wide

class of functions, the concept of differentiability is restrictive and, indeed, some

functions are not differentiable at any point in the complex plane.

Show that the functionf(z)=2y+ixis not differentiable anywhere in the complex plane.

In this casef(z) cannot be written simply in terms ofz, and so we must consider the
limit (24.1) in terms ofxandyexplicitly. Following the same procedure as in the previous
example we find

f(z+∆z)−f(z)

∆z

=

2 y+2∆y+ix+i∆x− 2 y−ix

∆x+i∆y

=

2∆y+i∆x

∆x+i∆y

.

In this case the limit will clearly depend on the direction from which ∆z→0. Suppose
∆z→0 along a line throughzof slopem,sothat∆y=m∆x,then

lim

∆z→ 0

[

f(z+∆z)−f(z)

∆z

]

= lim

∆x,∆y→ 0

[

2∆y+i∆x

∆x+i∆y

]

=

2 m+i

1+im

.

This limit is dependent onmand hence on the direction from which ∆z→0. Since this
conclusion is independent of the value ofz, and hence true for allz,f(z)=2y+ixis
nowhere differentiable.

A function that is single-valued and differentiable at all points of a domainR

is said to beanalytic(orregular)inR. A function may be analytic in a domain

except at a finite number of points (or an infinite number if the domain is

infinite); in this case it is said to be analytic except at these points, which are

called thesingularitiesoff(z). In our treatment we will not consider cases in

which an infinite number of singularities occur in a finite domain.