706 CHAPTER 35. ANALYTIC FUNCTIONS

Proof: Denote by Γr the insider circle in the above picture having radius r oriented asshown. Then f (w) = 1

w−z has a derivative which is

−1

(w− z)2

a continuous function, and so its real and imaginary parts are continuous for w ̸= z. There-fore, the function is analytic near Ur the open set bounded by the two curves Γr and Γ.It follows from the Cauchy theorem that for γ an orientation on Γ as shown and γ̂r anorientation as shown on Γr, ∫

γ

1w− z

dw+∫

γ̂r

1w− z

dw = 0

Therefore, orienting Γr in the usual direction, a parametrization for this circle is

x = a+ r cos t,y = b+ r sin t, t ∈ [0,2π]

Deote this parametrization by γr. Then∫γ

1w− z

dw =∫

γr

1w− z

dw

and using the definition of the contour integral, the right side reduces to 2πi. Thus thewinding number is 1. Therefore, if Γ were oriented the opposite direction, you would get−1 for the winding number. If z /∈U ∪Γ, the function is analytic near U and so the Cauchyintegral theorem implies right away that the winding number is 0. ■

The expression 12πi∫

γ1

w−z dw ≡ n(γ,z) is called the winding number. As explained, itis either 1 or −1 depending on how the curve Γ is oriented. The winding number can bedefined with much more generality for any closed curve, simple or not. However, I will notdo so, choosing instead to emphasize the most basic ideas. The greater generality is neededhowever, when you consider general versions of the Cauchy integral formula, a marvelousrepresentation theorem for an analytic function.

Definition 35.4.3 Given Γ a simple closed curve, the orientation is said to be positive ifthe winding number is 1 and negative if the winding number is −1.

35.5 Primitives and Cauchy Goursat TheoremIn beginning calculus, the notion of an antiderivative was very important. It is similar forfunctions of complex variables. The role of a primitive is also a lot like a potential incomputing line integrals.

Definition 35.5.1 A function F such that F ′ = f is called a primitive of f .

The following theorem shows that the primitive acts just like a potential, the differencebeing that a primitive has complex, not real values. In calculus, in the context of a functionof one real variable, this is often called an antiderivative and every continuous function hasone thanks to the fundamental theorem of calculus. However, it will be shown below thatthe situation is not at all the same for functions of a complex variable.

So what if a function has a primitive? Say F ′ (z) = f (z) where f is continuous.