Functions of a complex variable

24.1 FUNCTIONS OF A COMPLEX VARIABLE
students are more at ease with the former type of statement, despite its lack of
precision, whilst others, those who would contemplate only the latter, are usually
well able to supply it for themselves.
24.1 Functions of a complex variable
The quantity f(z) is said to be a function of the complex variable z if to every value
of z in a certain domain R (a region of the Argand diagram) there corresponds
one or more values of f(z). Stated like this f(z) could be any function consisting
of a real and an imaginary part, each of which is, in general, itself a function of x
and y. If we denote the real and imaginary parts of f(z) by u and v, respectively,
then
f(z) = u(x, y) + iv(x, y).
In this chapter, however, we will be primarily concerned with functions that are
single-valued, so that to each value of z there corresponds just one value of f(z),
and are differentiable in a particular sense, which we now discuss.
A function f(z) that is single-valued in some domain R is differentiable at the
point z in R if the derivative
f(z + ∆z) − f(z)
(24.1)
f (z) = lim
∆z→0
∆z
exists and is unique, in that its value does not depend upon the direction in the
Argand diagram from which ∆z tends to zero.
Show that the function f(z) = x2 − y 2 + i2xy is differentiable for all values of z.
Considering the definition (24.1), and taking ∆z = ∆x + i∆y, we have
f(z + ∆z) − f(z)
∆z
(x + ∆x)2 − (y + ∆y)2 + 2i(x + ∆x)(y + ∆y) − x2 + y 2 − 2ixy
=
∆x + i∆y
2x∆x + (∆x)2 − 2y∆y − (∆y)2 + 2i(x∆y + y∆x + ∆x∆y)
=
∆x + i∆y
(∆x)2 − (∆y)2 + 2i∆x∆y
= 2x + i2y +
.
∆x + i∆y
Now, in whatever way ∆x and ∆y are allowed to tend to zero (e.g. taking ∆y = 0 and
letting ∆x → 0 or vice versa), the last term on the RHS will tend to zero and the unique
limit 2x + i2y will be obtained. Since z was arbitrary, f(z) with u = x2 − y 2 and v = 2xy
is differentiable at all points in the (finite) complex plane. We note that the above working can be considerably reduced by recognising
that, since z = x + iy, we can write f(z) as
f(z) = x2 − y 2 + 2ixy = (x + iy)2 = z 2 .
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COMPLEX VARIABLES
We then find that
f (z) = lim
∆z→0
(z + ∆z)2 − z 2
(∆z)2 + 2z∆z
= lim
∆z→0
∆z
∆z
= lim ∆z + 2z = 2z,
∆z→0
from which we see immediately that the limit both exists and is independent of
the way in which ∆z → 0. Thus we have verified that f(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 function f(z) = 2y + ix is not differentiable anywhere in the complex plane.
In this case f(z) cannot be written simply in terms of z, and so we must consider the
limit (24.1) in terms of x and y explicitly. Following the same procedure as in the previous
example we find
2y + 2∆y + ix + i∆x − 2y − ix
f(z + ∆z) − f(z)
=
∆z
∆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 through z of slope m, so that ∆y = m∆x, then
lim
∆z→0
2m + i
f(z + ∆z) − f(z)
2∆y + i∆x
= lim
=
.
∆x, ∆y→0
∆z
∆x + i∆y
1 + im
This limit is dependent on m and hence on the direction from which ∆z → 0. Since this
conclusion is independent of the value of z, and hence true for all z, f(z) = 2y + ix is
nowhere differentiable. A function that is single-valued and differentiable at all points of a domain R
is said to be analytic (or regular) in R. 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 the singularities of f(z). In our treatment we will not consider cases in
which an infinite number of singularities occur in a finite domain.
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