Theorem 2.1 A continuous function on a compact set Ω is bounded and attains a maximum and minimum on Ω
2.2 Holomorphic functions
We now present a notion that is central to complex analysis, and in distinction to our previous discussion we introduce a definition that is genuinely complex in nature.
Let Ω be an open set inC and f a complex-valued function on Ω. The function f is holomorphic at the point z0∈ Ω if the quotient
(1) f (z0+ h)− f(z0)
h
converges to a limit when h→ 0. Here h ∈ C and h = 0 with z0+ h∈ Ω, so that the quotient is well defined. The limit of the quotient, when it exists, is denoted by f(z0), and is called the derivative of f at z0:
f(z0) = lim
h→0
f (z0+ h)− f(z0)
h .
2. Functions on the complex plane 9 It should be emphasized that in the above limit, h is a complex number that may approach 0 from any direction.
The function f is said to be holomorphic on Ω if f is holomorphic at every point of Ω. If C is a closed subset of C, we say that f is holomorphic on C if f is holomorphic in some open set containing C.
Finally, if f is holomorphic in all ofC we say that f is entire.
Sometimes the terms regular or complex differentiable are used in-stead of holomorphic. The latter is natural in view of (1) which mimics the usual definition of the derivative of a function of one real variable.
But despite this resemblance, a holomorphic function of one complex variable will satisfy much stronger properties than a differentiable func-tion of one real variable. For example, a holomorphic funcfunc-tion will actu-ally be infinitely many times complex differentiable, that is, the existence of the first derivative will guarantee the existence of derivatives of any order. This is in contrast with functions of one real variable, since there are differentiable functions that do not have two derivatives. In fact more is true: every holomorphic function is analytic, in the sense that it has a power series expansion near every point (power series will be discussed in the next section), and for this reason we also use the term analytic as a synonym for holomorphic. Again, this is in contrast with the fact that there are indefinitely differentiable functions of one real variable that cannot be expanded in a power series. (See Exercise 23.)
Example 1. The function f (z) = z is holomorphic on any open set in C, and f(z) = 1. In fact, any polynomial
p(z) = a0+ a1z +· · · + anzn is holomorphic in the entire complex plane and
p(z) = a1+· · · + nanzn−1. This follows from Proposition 2.2 below.
Example2. The function 1/z is holomorphic on any open set inC that does not contain the origin, and f(z) =−1/z2.
Example3. The function f (z) = z is not holomorphic. Indeed, we have f (z0+ h)− f(z0)
h = h
h
which has no limit as h→ 0, as one can see by first taking h real and then h purely imaginary.
10 Chapter 1. PRELIMINARIES TO COMPLEX ANALYSIS
An important family of examples of holomorphic functions, which we discuss later, are the power series. They contain functions such as ez, sin z, or cos z, and in fact power series play a crucial role in the theory of holomorphic functions, as we already mentioned in the last paragraph.
Some other examples of holomorphic functions that will make their ap-pearance in later chapters were given in the introduction to this book.
It is clear from (1) above that a function f is holomorphic at z0 ∈ Ω if and only if there exists a complex number a such that
(2) f (z0+ h)− f(z0)− ah = hψ(h),
where ψ is a function defined for all small h and limh→0ψ(h) = 0. Of course, we have a = f(z0). From this formulation, it is clear that f is continuous wherever it is holomorphic. Arguing as in the case of one real variable, using formulation (2) in the case of the chain rule (for exam-ple), one proves easily the following desirable properties of holomorphic functions.
Proposition 2.2 If f and g are holomorphic in Ω, then:
(i) f + g is holomorphic in Ω and (f + g)= f+ g. (ii) f g is holomorphic in Ω and (f g)= fg + f g. (iii) If g(z0)= 0, then f/g is holomorphic at z0 and
(f /g)= fg− fg g2 .
Moreover, if f : Ω→ U and g : U → C are holomorphic, the chain rule holds
(g◦ f)(z) = g(f (z))f(z) for all z∈ Ω.
Complex-valued functions as mappings
We now clarify the relationship between the complex and real deriva-tives. In fact, the third example above should convince the reader that the notion of complex differentiability differs significantly from the usual notion of real differentiability of a function of two real variables. Indeed, in terms of real variables, the function f (z) = z corresponds to the map F : (x, y) → (x, −y), which is differentiable in the real sense. Its deriva-tive at a point is the linear map given by its Jacobian, the 2× 2 matrix of partial derivatives of the coordinate functions. In fact, F is linear and
2. Functions on the complex plane 11 is therefore equal to its derivative at every point. This implies that F is actually indefinitely differentiable. In particular the existence of the real derivative need not guarantee that f is holomorphic.
This example leads us to associate more generally to each complex-valued function f = u + iv the mapping F (x, y) = (u(x, y), v(x, y)) from R2 to R2.
Recall that a function F (x, y) = (u(x, y), v(x, y)) is said to be differ-entiable at a point P0= (x0, y0) if there exists a linear transformation J :R2 → R2 such that
(3) |F (P0+ H)− F (P0)− J(H)|
|H| → 0 as|H| → 0, H ∈ R2. Equivalently, we can write
F (P0+ H)− F (P0) = J(H) +|H|Ψ(H) ,
with|Ψ(H)| → 0 as |H| → 0. The linear transformation J is unique and is called the derivative of F at P0. If F is differentiable, the partial derivatives of u and v exist, and the linear transformation J is described in the standard basis ofR2 by the Jacobian matrix of F
J = JF(x, y) =
∂u/∂x ∂u/∂y
∂v/∂x ∂v/∂y
.
In the case of complex differentiation the derivative is a complex number f(z0), while in the case of real derivatives, it is a matrix. There is, however, a connection between these two notions, which is given in terms of special relations that are satisfied by the entries of the Jacobian matrix, that is, the partials of u and v. To find these relations, consider the limit in (1) when h is first real, say h = h1+ ih2 with h2 = 0. Then, if we write z = x + iy, z0= x0+ iy0, and f (z) = f (x, y), we find that
f(z0) = lim
h1→0
f (x0+ h1, y0)− f(x0, y0) h1
= ∂f
∂x(z0),
where ∂/∂x denotes the usual partial derivative in the x variable. (We fix y0 and think of f as a complex-valued function of the single real variable x.) Now taking h purely imaginary, say h = ih2, a similar argument yields
12 Chapter 1. PRELIMINARIES TO COMPLEX ANALYSIS
where ∂/∂y is partial differentiation in the y variable. Therefore, if f is holomorphic we have shown that
∂f
∂x = 1 i
∂f
∂y.
Writing f = u + iv, we find after separating real and imaginary parts and using 1/i =−i, that the partials of u and v exist, and they satisfy the following non-trivial relations
∂u
∂x = ∂v
∂y and ∂u
∂y =−∂v
∂x.
These are the Cauchy-Riemann equations, which link real and complex analysis.
We can clarify the situation further by defining two differential oper-ators Proposition 2.3 If f is holomorphic at z0, then
∂f
Proof. Taking real and imaginary parts, it is easy to see that the Cauchy-Riemann equations are equivalent to ∂f /∂z = 0. Moreover, by our earlier observation
2. Functions on the complex plane 13 and the Cauchy-Riemann equations give ∂f /∂z = 2∂u/∂z. To prove that F is differentiable it suffices to observe that if H = (h1, h2) and h = h1+ ih2, then the Cauchy-Riemann equations imply
JF(x0, y0)(H) =
where we have identified a complex number with the pair of real and imaginary parts. After a final application of the Cauchy-Riemann equa-tions, the above results imply that
(4)
So far, we have assumed that f is holomorphic and deduced relations satisfied by its real and imaginary parts. The next theorem contains an important converse, which completes the circle of ideas presented here.
Theorem 2.4 Suppose f = u + iv is a complex-valued function defined on an open set Ω. If u and v are continuously differentiable and satisfy the Cauchy-Riemann equations on Ω, then f is holomorphic on Ω and f(z) = ∂f /∂z. the Cauchy-Riemann equations we find that
f (z + h)− f(z) =
14 Chapter 1. PRELIMINARIES TO COMPLEX ANALYSIS