1608 CHAPTER 50. RIEMANN STIELTJES INTEGRALS

where here

X[a,b] (s)≡{

1 if s ∈ [p,q]0 if s /∈ [p,q] .

Also, ∫ b

af (γ (s))γ

′ (s)ds =∫ b

a

n

∑j=1

f (γ (s))X[t j−1,t j ] (s)γ′ (s)ds

and thanks to 50.0.13,∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣

=∑nj=1 f(γ(τ j))(γ(t j)−γ(t j−1))︷ ︸︸ ︷∫ b

a

n

∑j=1

f (γ (τ j))X[t j−1,t j ] (s)γ′ (s)ds−

=∫ b

a f (γ(s))γ ′(s)ds︷ ︸︸ ︷∫ b

a

n

∑j=1

f (γ (s))X[t j−1,t j ] (s)γ′ (s)ds

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣≤

n

∑j=1

∫ t j

t j−1

∣∣∣∣ f (γ (τ j))− f (γ (s))∣∣∣∣ ∣∣γ ′ (s)∣∣ds≤

∣∣∣∣γ ′∣∣∣∣∞ ∑

jε(t j− t j−1

)= ε

∣∣∣∣γ ′∣∣∣∣∞(b−a) .

It follows that∣∣∣∣∣∣∣∣∫γ

f dγ−∫ b

af (γ (s))γ

′ (s)ds∣∣∣∣∣∣∣∣≤

∣∣∣∣∣∣∣∣∣∣∫

γ

f dγ−n

∑j=1

f (γ (τ j))(γ (t j)− γ

(t j−1

))∣∣∣∣∣∣∣∣∣∣

+

∣∣∣∣∣∣∣∣∣∣ n

∑j=1

f (γ (τ j))(γ (t j)− γ

(t j−1

))−∫ b

af (γ (s))γ

′ (s)ds

∣∣∣∣∣∣∣∣∣∣≤ ε

∣∣∣∣γ ′∣∣∣∣∞(b−a)+ ε.

Since ε is arbitrary, this verifies 50.0.12.

Definition 50.0.10 Let Ω be an open subset of C and let γ : [a,b]→ Ω be a continuousfunction with bounded variation f : Ω→ X be a continuous function. Then the followingnotation is more customary. ∫

γ

f (z)dz≡∫

γ

f dγ.

The expression,∫

γf (z)dz, is called a contour integral and γ is referred to as the contour.

A function f : Ω→ X for Ω an open set in C has a primitive if there exists a function, F, theprimitive, such that F ′ (z) = f (z) . Thus F is just an antiderivative. Also if γk : [ak,bk]→Cis continuous and of bounded variation, for k = 1, · · · ,m and γk (bk) = γk+1 (ak) , define∫

∑mk=1 γk

f (z)dz≡m

∑k=1

∫γk

f (z)dz. (50.0.14)

In addition to this, for γ : [a,b]→ C,define −γ : [a,b]→ C by −γ (t) ≡ γ (b+a− t) . Thus γ simply traces out the points of

γ∗ in the opposite order.