MAS 381:
Mathematics III
University of Sheffield, 2020
1 Complex functions
1.1 Introduction
The term complex analysis refers to the calculus of complex-valued functions,
f(z), depending on a single complex variable z. To an un-initiated, it may
seem that this subject should merely be a simple reworking of standard real
variable theory that you learned in first and second year calculus. However, this
naive first impression could not be further from the truth! Complex analysis is
the culmination of a deep and far-ranging study of the fundamental notions
of complex differentiation and integration, and has an elegance and beauty
not found in the real domain. For instance, complex functions are necessarily
analytic, meaning that they can be represented by convergent power series, and
hence are infinitely differentiable. Thus, difficulties with degree of smoothness,
strange discontinuities, subtle convergence phenomena, and other pathological
properties of real functions never arise in the complex world.
Complex functions have a very wide applicability in electrical engineering.
The calculation of harmonically varying currents in circuits consisting of re-
sistors, capacitors, induction, coils, etc. require functions of complex variable.
Secondly, functions of complex variables are used for solving Laplace and Pois-
son equations describing electrostatic potential.
The driving force behind many of the applications of complex analysis is the
remarkable connection between complex functions and harmonic functions of
two variables, a.k.a. solutions of the planar Laplace equation. To wit, the real
and imaginary parts of any complex analytic function are automatically har-
monic. In this manner, complex functions provide a rich spectrum of additional
solutions to the two-dimensional Laplace equation, which can be exploited in a
wide range of physical, mathematical, and engineering applications.
One of the most useful consequences stems from the elementary observation
that the composition of two complex functions is also a complex function. We
re-interpret this operation as a complex change of variables, producing a confor-
mal mapping that preserves (signed) angles in the Euclidean plane. Conformal
mappings can be effectively used for constructing solutions to the Laplace equa-
tion on complicated planar domains that appear in a wide range of physical
problems, including fluid mechanics, aerodynamics, thermomechanics, electro-
statics, and elasticity. In particular, conformal mapping is a widespread math-
ematical technique to approach the electromagnetic study of e
lectric machines,
e.g. energy conservation in magnetostatic and time-harmonic problems; current
source assignment in the transformed domain; transformation of the flux density
components through conformal mapping, etc.
1.2 Properties of complex functions
Our principal objects of study are complex-valued functions, f(z), depending
on the complex variable z = x + iy and they are defined over of the complex
plain, C.
2
Any complex function f(z) can be uniquely written as a complex combina-
tion
f(z) = f(x+ iy) = u(x, y) + iv(x, y), (1)
of two real functions both depending on the two real variables, x and y with
Re(f) = u(x, y) and Im(f) = u(x, y). For example, the function f(z) = z2 can
be expanded and written as
z2 = (x+ iy)2 = x2 y2 + 2ixy
so
Re(f) = x2 y2, Im(f) = 2xy
Many of the well-known functions appearing in real-variable calculus (polyno-
mials, rational functions, exponentials, trigonometric functions, logarithms, and
many more) have natural complex extensions. For example, complex polynomi-
als
p(z) = anz
n + an 1zn 1 + · · ·+ a1z + a0 (2)
are complex linear combinations (meaning that the coefficients ak are allowed
to be complex numbers) of the basic monomial functions zk = (x+ iy)k.
As we will see the notion of harmonic function is very important for our
module. By definition a function is called harmonic if it is at least twice
differentiable and it satisfies the well-known Laplace equation. In 2D this partial
differential equation takes the form
2f
x2
+
2f
y2
= 0.
Before going deeper into the many remarkable properties of complex func-
tions, let us look at some of the most basic examples. In each case, you can
directly check that the harmonic functions provided by the real and imagi-
nary parts of the complex function are indeed solutions to the two-dimensional
Laplace equation.
Examples of complex functions
(a) Harmonic Polynomials: As noted above, any complex polynomial is a linear
combination, as in (2), of the basic complex monomials
zn = (x+ iy)n = un(x, y) + ivn(x, y). (3)
Their real and imaginary parts, un(x, y), vn(x, y), are harmonic polynomials.
(b) Rational functions: Ratios of the form
f(z) =
p(z)
q(z)
, (4)
3
where p(z) and q(z) are complex polynomials provide a large variety of harmonic
functions. The simplest case is
1
z
=
1
x+ iy
=
x iy
(x+ iy)(x iy) =
x iy
x2 + y2
=
x
x2 + y2
i y
x2 + y2
. (5)
Apart from an interesting behaviour at the origin (x = 0, y = 0), these functions
are harmonic everywhere. It is easy to see that they are twice differentiable and
they satisfy the 2D Laplace equation. For the real part