Throughout ℂ stands for the set of all complex numbers. If z = x + ⅉy ∈ ℂ, its complex conjugate is denoted by z✶ = x − ⅉy, where ⅉ or j denotes the imaginary unit, so ⅉ² = −1.
Holomorphic functions
We consider functions of complex variable z ∈ ℂ defined on some domain Ω ⊂ ℂ. We also expect that f(z) will in general take values in &Co
pf; as well.
If f(z) = u + ⅉv, then the function u(x, y) is called the real part of f and
v(x, y) is called the imaginary part of f. Of course, it will not in general be possible to plot the graph of f(z), which will lie in ℂ², the set of ordered
pairs of complex numbers, that is the set {(z,w) ∈ ℂ²
: w = f(z)}. The graph can also be viewed as the subset of ℝ4
given by {(x, y, s, t): s = u(x, y), t = v(x, y)}. In particular, it lies in a four-dimensional space.
If z = x + ⅉy,
then a function f(z) is simply a function f(x, y) = u(x, y)+ ⅉv(x, y) of the two real variables x and y. As such, it is a function (mapping) from ℝ² to ℝ².
The usual operations on complex numbers extend to complex functions:
given a complex function f(z) = u + ⅉv, we can define functions Re f(z = u (the real part is also denoted by ℜ), Im f(z = v (the imaginary part is also denoted by ℑ), complex conjugate f✶(z = u − ⅉv, \( \displaystyle \quad |f(z)| = \sqrt{u^2 + v^2} . \quad \) Likewise, if g(z) is another complex-valued function, we can define f(z) g(z) and f(z)/g(z) provided g(z) ≠ 0.
Limits and continuity
The absolute value measures the distance between two complex numbers.
Thus, z₁ and z₂ are close when |z₁ − z₂| is small. We can then define the
limit of a complex-valued function f(z) as follows: we write
\[
\lim_{z\to c} \ f(z) = C ,
\]
where c and C are understood to be complex numbers, if the distance from f(z) to C, |f(z) − C|, is small whenever |z − c| is small. More precisely,
if we want |f(z) − C| to be less than some small specified positive real number ε, then there should exist a positive real number δ such that, if |z − c| < δ, then |f(z) − C| < ε. Note that, as with real functions, it does
not matter if f(c) = C or even that f(z) be defined at c. It is easy to see
that, if c =(c₁, c₂), C = 𝑎 + ⅉb and f(z) = u + ⅉv is written as real and imaginary parts, then \( \displaystyle \quad \lim_{z \to c} \ f(z) = C \quad \)
if and only if \( \displaystyle \quad \lim_{(x,y) \to (c_1 ,c_2 )}\ u(x,y) = a \quad\mbox{and} \quad \lim_{(x,y) \to (c_1 ,c_2 )}\ v(x,y) = b . \quad \) Thus, the story for limits of functions of a complex
variable is the same as the story for limits of real valued functions of the
variables x, y. However, a real variable x can approach a real number c only from above or below (or from the left or right, depending on your point of
view), whereas there are many ways for a complex variable to approach a
complex number c.
Sequences, limits of sequences, convergent series and power series can be
defined similarly.
As for functions of a real variable, a function f(z) is continuous at c if
\[
\lim_{z\to c} \ f(z) = f(c) .
\]
In other words, the limit exists, f(·) is defined at z = c, and its value at c is the limiting value. A function f(·) is continuous in some domain Ω ⊆ ℂ if it is continuous at all points z ∈ Ω. It means that a function f(z) = u + ⅉv
is continuous if and only if its real and imaginary parts are continuous; so
that the usual functions z, z✶, Rez, Imz, |z|, ez are continuous.
z
are continuous
if it exists. Here z = 𝑥 + ⅉy is a complex number and ⅉ or j is the imaginary unit in the complex plane ℂ, so ⅉ² = −1. In Δz = Δ𝑥 + ⅉΔy, Δ𝑥 and Δy are independent of each other.
Thus, approaching 0 along horizontal and vertical directions has given two
different expressions. Equating real and imaginary parts, we see that: if a function f(z) = u + ⅉv is complex differentiable, then its real and imaginary parts satisfy the Cauchy-Riemann equations:
Apostol, T.M., Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability, Wiley; 2nd edition, 1991; ISBN-13: 978-0471000075.
