Chapter 50

Riemann Stieltjes IntegralsIn the theory of functions of a complex variable, the most important results are those in-volving contour integration. I will base this on the notion of Riemann Stieltjes integralsas in [32], [95], and [65]. The Riemann Stieltjes integral is a generalization of the usualRiemann integral and requires the concept of a function of bounded variation.

Definition 50.0.1 Let γ : [a,b]→ C be a function. Then γ is of bounded variation if

sup

{n

∑i=1|γ (ti)− γ (ti−1)| : a = t0 < · · ·< tn = b

}≡V (γ, [a,b])< ∞

where the sums are taken over all possible lists, {a = t0 < · · ·< tn = b} . The set of pointsγ ([a,b]) will also be denoted by γ∗.

The idea is that it makes sense to talk of the length of the curve γ ([a,b]) , defined asV (γ, [a,b]) . For this reason, in the case that γ is continuous, such an image of a boundedvariation function is called a rectifiable curve.

Definition 50.0.2 Let γ : [a,b]→ C be of bounded variation and let f : γ∗ → X. LettingP ≡ {t0, · · · , tn} where a = t0 < t1 < · · ·< tn = b, define

||P|| ≡max{∣∣t j− t j−1

∣∣ : j = 1, · · · ,n}

and the Riemann Steiltjes sum by

S (P)≡n

∑j=1

f (γ (τ j))(γ (t j)− γ

(t j−1

))where τ j ∈

[t j−1, t j

]. (Note this notation is a little sloppy because it does not identify the

specific point, τ j used. It is understood that this point is arbitrary.) Define∫

γf dγ as the

unique number which satisfies the following condition. For all ε > 0 there exists a δ > 0such that if ||P|| ≤ δ , then ∣∣∣∣∫

γ

f dγ−S (P)

∣∣∣∣< ε.

Sometimes this is written as ∫γ

f dγ ≡ lim||P||→0

S (P) .

The set of points in the curve, γ ([a,b]) will be denoted sometimes by γ∗.

Then γ∗ is a set of points in C and as t moves from a to b, γ (t) moves from γ (a)to γ (b) . Thus γ∗ has a first point and a last point. If φ : [c,d]→ [a,b] is a continuousnondecreasing function, then γ ◦φ : [c,d]→ C is also of bounded variation and yields thesame set of points in C with the same first and last points.

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