Cauchy-Riemann Equations 191say that a function is analytic if it is analytic somewhere. An entire
function is a function that is analytic everywhere in the complex plane.
Remark 4.1 Sometimes, the word holomorphic is used instead of ana-
lytic.
EXERCISE 4.2.4? (a) Convince yourself that a function is analytic at a point
if and only if the function is analytic in some neighborhood of the point.
Hint: a neighborhood of a point is an open set; see page 186. (b) Verify that
a polynomial is an entire function, (c) Verify that a rational function
f(z) = P(z)/Q(z), where P,Q are polynomials, is analytic everywhere ex-
cept at the roots ofQ(z).
4.2.2 Cauchy-Riemann Equations
Let / = f(z), z = x + iy, be a function of a complex variable. Being a
complex number itself, f(z) can be written as
f(z)=u(x,y) + iv(x,y) (4.2.1)for some real functions u, v of two real variables x, y. FOR EXAMPLE, if
f(z) = z^2 , then/(,z) = (x+iy)^2 = (x^2 ~y^2 ) + 2ixy, so that u(x,y) = x^2 -y^2
and v(x, y) = 2xy.
EXERCISE 4.2.5.C Find the functions u,v if f(z) = z^3.
The objective of this section is to investigate the connection between
differentiability of the complex function / and differentiability of the func-
tions u, v. The motivation for this investigation comes from the following
exercise, showing that differentiability of a function of a complex variable
requires more than mere differentiability of the real and imaginary parts.
EXERCISE 4.2.6.C (a) Show that the function f(z) = u(x,y) + iv(x,y) is
continuous at the point ZQ = XQ + iyo, in the sense of Definition 4-1, if and
only if both functions u,v are continuous at (xo,yo), as real functions of
two variables, (b) Show that if the function f = f(z) is differentiable, then
the corresponding functions u, v are also differentiable, as functions of x
and y. (c) Show that the function f{z) — x is not differentiable anywhere,
in the sense of Definition 4-1-
To understand what is going on, let us assume that / = f(z) is dif-
ferentiable at ZQ SO that, according to Definition 4.1, we have f(zo +
z) - /(*>) = */'(*>) + e(z), where \e(z)\/\z\ -• 0, \z\ -• 0. Writing