Chapter 11

Functions of One Complex VariableIn the nineteenth century, complex analysis developed along with real analysis, the latterbeing the main topic of this book in which one considers functions of one real variable.However, many difficult real improper integrals can be best considered using contour in-tegrals so some introduction to this very important topic is useful. It is not intended to bea full course on complex analysis, just an introduction to some of the main ideas. Cauchywas the principal originator of the study of complex analysis in the early 1800’s. His-torically, the main theorems came from the Cauchy Riemann equations and a version ofGreen’s theorem. The Cauchy Riemann equations are considered later in a problem, butGreen’s theorem is part of multivariable calculus which is not being discussed in this book.However, using the ideas of Goursat (1858-1936) it is possible to present the main theoryin terms of functions of a single variable, this time a single complex variable.

11.1 Contour IntegralsThis is about contour integrals in C. First is the definition of an oriented C1 contour.Contours are sets of points in C. Smooth ones require the notion of the derivative of afunction of one real variable having values inC. There is also the concept of a differentiablefunction of a complex variable which is defined on an open set of C.

Definition 11.1.1 The derivative is defined as before. If γ : [a,b]→ C, then γ ′ (t)is said to exist exactly when γ (t +h)− γ (t) = ah+ o(h) for some a ∈ C and sufficientlysmall real h and in this case, a≡ γ ′ (t) . This is not any different than the earlier material onthe derivative other than γ having values in C. Derivatives from right and left are similarto before. Also, if f is defined on an open subset of C then it is differentiable at z meansf (z+h)− f (z) = ah+o(h) and a≡ f ′ (z) . Here h ∈ C since the derivative is on an opensubset of C rather than a subset of R. This has already been dealt with in Theorem 8.2.1on Page 157. f ′ (z)≡ limh→0

f (z+h)− f (z)h exactly as in the case of a real variable.

All properties of Theorem 7.5.1 continue to apply for the derivative just defined. Theproofs are exactly the same. In particular, the chain rule holds. You just have to use Crather than R for values of the function. The necessary complex arithmetic is in Section2.13.

Next is the idea of an oriented C1 curve.

Definition 11.1.2 A set of points γ∗ in C is called an oriented C1 curve or contourif the following conditions hold.

1. There exists γ a continuous function mapping some interval [a,b] to C such thatγ∗ ≡ γ ([a,b]).

2. This γ is one to one on [a,b) and the derivative γ ′ is continuous and exists on [a,b] ,being defined in terms of right or left derivatives at the end points. When γ (a) = γ (b)this is a closed curve.

3. If γ : [a,b] → γ∗ and γ̂ :[â, b̂]→ γ∗ are two parametrizations, then γ̂

−1 ◦ γ is acontinuous function which is one to one and so by Lemma 6.4.3 it is either strictlyincreasing or strictly decreasing on [a,b]. Two parametrizations γ̂,γ are said to

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