14.4. CONTOUR INTEGRALS 347

+ f (γ (σ∗))(γ (tp)− γ (t∗))+n

∑j=p+1

f (γ (σ j))(γ (t j)− γ

(t j−1

)),

S (P)≡p−1

∑j=1

f (γ (τ j))(γ (t j)− γ

(t j−1

))+

= f(γ(τ p))(γ(tp)−γ(tp−1))︷ ︸︸ ︷f (γ (τ p))(γ (t∗)− γ (tp−1))+ f (γ (τ p))(γ (tp)− γ (t∗))

+n

∑j=p+1

f (γ (τ j))(γ (t j)− γ

(t j−1

)).

Therefore,

∥S (P)−S (Q)∥ ≤p−1

∑j=1

1m

∣∣γ (t j)− γ(t j−1

)∣∣+ 1m

∣∣γ (t∗)− γ (tp−1)∣∣+

1m

∣∣γ (tp)− γ (t∗)∣∣+ n

∑j=p+1

1m

∣∣γ (t j)− γ(t j−1

)∣∣≤ 1m

V (γ, [a,b]) . (14.10)

Clearly the extreme inequalities would be valid in 14.10 if Q had more than one extra point.You simply do the above trick more than one time. Let S (P) and S (Q) be Riemann Steiltjessums for which ∥P∥ and ∥Q∥ are less than δ m and let R≡ P∪Q. Then from what was justobserved,

∥S (P)−S (Q)∥ ≤ ∥S (P)−S (R)∥+∥S (R)−S (Q)∥ ≤ 2m

V (γ, [a,b]) .

and this shows 14.9 which proves 14.8. Therefore, there exists a unique point, I ∈ ∩∞m=1Fm

which satisfies the definition of∫

γf dz.

Now consider the claim about C1 contours. First, why is γ of bounded variation if it isC1? For P≡ {t0, t1, ..., tn} a partition of [a,b] ,

n

∑k=1|γ (tk)− γ (tk−1)| =

n

∑k=1

∣∣∣∣∫ tk

tk−1

γ′ (s)ds

∣∣∣∣≤ n

∑k=1

∫ tk

tk−1

∣∣γ ′ (s)∣∣ds

≤ Mn

∑k=1

(tk− tk−1) = M (b−a)

where M ≥max{|γ ′ (t)| : t ∈ [a,b]}. Thus∫

γf dz exists.

Let P = {t0, · · · , tn} . Let γ = γ1 + iγ2, γ j real. Then using the mean value theorem,

S (P) =n

∑k=1

f (γ (σ k))(γ (tk)− γ (tk−1))

=n

∑k=1

f (γ (σ k))(γ1 (tk)− γ1 (tk−1))+ in

∑k=1

f (γ (σ k))(γ2 (tk)− γ2 (tk−1))

=n

∑k=1

f (γ (σ k))γ′1 (τk)(tk− tk−1)+ i

n

∑k=1

f (γ (σ k))γ′2 (ξ k)(tk− tk−1)