In this chapter we will discover the incredible difference between the analysis of functions of a single complex variable as opposed to functions of a single real variable. Up to this point, in some sense, we have treated them as being quite similar subjects, whereas in fact they are extremely different in character. Indeed, if f is a differentiable function of a complex variable on an open set U⊆C, then we will see that f is actually expandable in a Taylor series around every point in U. In particular, a function fof a complex variable is guaranteed to have infinitely many derivatives on U if it merely has the first one on U. This is in marked contrast with functions of a real variable. See part (3) of Theorem 4.17..
The main points of this chapter are:
The Cauchy-Riemann Equations (Theorem 7.1.),
Cauchy's Theorem (Theorem 7.3.),
Cauchy Integral Formula (Theorem 7.4.),
A complex-valued function that is differentiable on an open set is expandable in a Taylor series around each point of the set (Theorem 7.5.),
The Identity Theorem (Theorem 7.6.),
The Fundamental Theorem of Algebra (Theorem 7.7.),
Liouville's Theorem (Theorem 7.8.),
The Maximum Modulus Principle (corollary to Corollary 7.4.),
The Open Mapping Theorem (Theorem 7.10.),
The uniform limit of analytic functions is analytic (Theorem 7.12.), and
The Residue Theorem (Theorem 7.17.).
We begin with a simple observation connecting differentiability of a function of a complex variable to a relation among of partial derivatives of the real and imaginary parts of the function. Actually, we have already visited this point in Exercise 8..
Let f=u+iv be a complex-valued function of a complex variable z=x+iy≡(x,y), and suppose f is differentiable, as a function of a complex variable, at the point c=(a,b). Then the following two partial differential equations, known as the Cauchy-Riemann Equations, hold:
and
We know that
and this limit is taken as the complex number h approaches 0. We simply examine this limit for real h's approaching 0 and then for purely imaginary h's approaching 0. For real h's, we have
For purely imaginary h's, which we write as h=ik, we have
Equating the real and imaginary parts of these two equivalent expressions for f'(c) gives the Cauchy-Riemann equations.
As an immediate corollary of this theorem, together with Green's Theorem (Theorem 6.15.), we get the following result, which is a special case of what is known as Cauchy's Theorem.
Let S be a piecewise smooth geometric set whose boundary CS has finite length.
Suppose f is a complex-valued function that is continuous on S
and differentiable at each point of the interior S0 of S.
Then the contour integral 
Prove the preceding corollary. See Theorem 6.12..
Suppose f=u+iv is a differentiable, complex-valued function on an open disk Br(c) in C, and assume that the real part u is a constant function. Prove that f is a constant function. Derive the same result assuming that v is a constant function.
Suppose f and g are two differentiable, complex-valued functions on an open disk Br(c) in C. Show that, if the real part of f is equal to the real part of g, then there exists a constant k such that f(z)=g(z)+k, for all z∈Br(c).
For future computational purposes, we give the following implications of the Cauchy-Riemann equations. As with Theorem 7.1., this next theorem mixes the notions of differentiability of a function of a complex variable and the partial derivatives of its real and imaginary parts.
Let f=u+iv be a complex-valued function of a complex variable, and suppose that f is differentiable at the point c=(a,b). Let A be the 2×2 matrix
Then:
| f' ( c ) | 2 = det ( A ) .
The two vectors
are linearly independent vectors in R2 if and only if f'(c)≠0.
The vectors
are linearly independent vectors in R2 if and only if f'(c)≠0.
Using the Cauchy-Riemann equations, we see that the determinant of the matrix A is given by
proving part (1).
The vectors 
and 
are the columns of the matrix A,
and so, from elementary linear algebra, we see that they are linearly independent if and only if the
determinant of A is nonzero.
Hence, part (2) follows from part (1).
Similarly, part (3) is a consequence of part (1).
It may come as no surprise that the contour integral of a function f around the boundary of a geometric set S is not necessarily 0 if the function f is not differentiable at each point in the interior of S. However, it is exactly these kinds of contour integrals that will occupy our attention in the rest of this chapter, and we shouldn't jump to any conclusions.
Let c be a point in C, and let S be the geometric set that is a closed disk 
Let φ be the parameterization of the boundary Cr of S given by
φ(t)=c+reit for t∈[0,2π].
For each integer n∈Z, define fn(z)=(z–c)n.
Show that 
for all n≠–1.
Show that
There is a remarkable result about contour integrals of certain functions that aren't differentiable everywhere within a geometric set, and it is what has been called the Fundamental Theorem of Analysis, or Cauchy's Theorem. This theorem has many general statements, but we present one here that is quite broad and certainly adequate for our purposes.
Let S be a piecewise smooth geometric set whose boundary CS has finite length, and
let 
be a piecewise smooth geometric set, whose boundary 
also is of finite length.
Suppose f is continuous on 
i.e., at
every point z that is in S but not in 
and assume that f is differentiable on 
i.e., at every point z in S0 but not in 
(We think of these sets as being the points “between” the boundary curves of these geometric sets.)
Then the two contour integrals
and 
are equal.
Let the geometric set S be determined by the interval [a,b] and the
two bounding functions u and l, and let the geometric set
be determined by the subinterval 
of [a,b] and the two bounding functions
and 
Because 
we know that 
and 
for all 
We define four geometric sets S1,...,S4 as follows:
S1 is determined by the interval 
and the two bounding functions u and l restricted to that interval.
S2 is determined by the interval 
and the two bounding functions
u and 
restricted to that interval.
S3 is determined by the interval 
and the two bounding functions 
and l
restricted to that interval.
S4 is determined by the interval 
and the two bounding
functions u and l restricted to that interval.
Observe that the five sets 
constitute a partition of the geometric set S.
The corollary to Theorem 7.1. applies to each of the four geometric sets S1,...,S4.
Hence, the contour integral of f around each of the four boundaries of these geometric sets is 0.
So, by Exercise 20.,
as desired.