Kenneth Kuttler

1610 CHAPTER 50. RIEMANN STIELTJES INTEGRALS

Corollary 50.0.14 If γ : [a,b]→C is continuous, has bounded variation, is a closed curve,γ (a) = γ (b) , and γ∗ ⊆Ω where Ω is an open set on which F ′ (z) = f (z) , then∫

f (z)dz = 0.

Another important result is a Fubini theorem for these contour integrals.

Theorem 50.0.15 Let γ i be continuous and bounded variation. Let f be continuous onγ∗1× γ∗2 having values in X a complex complete normed linear space. Then∫

γ1

∫γ2

f (z,w)dwdz =∫

γ2

∫γ1

f (z,w)dzdw

Proof: This follows quickly from the above lemma and the definition of the contour in-tegral. Say γ i is defined on [ai,bi]. Let a partition of [a1,b1] be denoted by {t0, t1, · · · , tn}=P1 and a partition of [a2,b2] be denoted by {s0,s1, · · · ,sm}= P2.∫

γ1

∫γ2

f (z,w)dwdz =n

∑i=1

∫γ1([ti−1,ti])

∫γ2

f (z,w)dwdz

∑i=1

∑j=1

∫γ1([ti−1,ti])

∫γ2([s j−1,s j])

f (z,w)dwdz

To save room, denote γ1 ([ti−1, ti]) by γ1i and γ2([

s j−1,s j])

by γ2 jThen if ∥Pi∥ , i = 1,2 issmall enough, ∥∥∥∥∥

∫γ1i

∫γ2 j

f (z,w)dwdz−∫

γ1i

∫γ2 j

f (γ1 (ti) ,γ2 (s j))dwdz

∥∥∥∥∥=

∥∥∥∥∥∫

γ1i

∫γ2 j

( f (z,w)− f (γ1 (ti) ,γ2 (s j)))dwdz

∥∥∥∥∥≤max

(∥∥∥∥∥∫

γ2 j

( f (z,w)− f (γ1 (ti) ,γ2 (s j)))dw

∥∥∥∥∥)

V (γ1, [ti−1, ti])

≤ εV(γ2,[s j−1,s j

])V (γ1, [ti−1, ti]) (50.0.15)

Also from this theorem,∥∥∥∥∥∫

γ2 j

∫γ1i

f (z,w)dzdw−∫

γ2 j

∫γ1i

f (γ1 (ti) ,γ2 (s j))dzdw

∥∥∥∥∥≤max

(∥∥∥∥∫γ1i

( f (z,w)− f (γ1 (ti) ,γ2 (s j)))dz∥∥∥∥)V

(γ2,[s j−1,s j

])≤ εV

(γ2,[s j−1,s j

])V (γ1, [ti−1, ti]) (50.0.16)

1610 CHAPTER 50. RIEMANN STIELTJES INTEGRALSCorollary 50.0.14 If: [a,b] > C is continuous, has bounded variation, is a closed curve,y(a) = y(b), and y* C Q where Q is an open set on which F' (z) = f (z), then[soae=0.Another important result is a Fubini theorem for these contour integrals.Theorem 50.0.15 Let y; be continuous and bounded variation. Let f be continuous onYi X Y> having values in X a complex complete normed linear space. Then/ f (Z,w) dwdz= | f (z,w) dzdwNY Y2 RINProof: This follows quickly from the above lemma and the definition of the contour in-tegral. Say y; is defined on [a;,b;]. Let a partition of [a,b1] be denoted by {t0,t1,--+ ,tu} =P, and a partition of [a2,b2] be denoted by {59,51,--- ,5m} = Po.f (z,w) dwdz= / f (z,w) dwdzI 12 (ew) d 11 ([i-1ti]) 2 ¥2 em)_ nom . dndLT can dat Fe ) meSj-18jTo save room, denote ¥; ([f-1,ti]) by Y,; and 7 ([sj-1,5;]) by Y2;Then if ||P;|| ,i = 1,2 issmall enough,/ f (z,w) dwdz— Js (% (ti) , Yo (sj)) dwdzNi Yj Ni Yj=| fu f(% (4); Yo (8j))) dwdz <Ni? Y2;[ (F (.w) — (1 (i) .%2 (s/))) awJ) V(%, [i-1,4i))max (Also from this theorem,< eV (79, [5j-1,8j]) V (M. [ti-1, ti) (50.0.15)/ f (zw) dedw— | [ Ff (% (ti) ¥2 (sj) dedwJ%2j FN: 8%25 YN< max ([ (rem) roy a rlsiy)ee] )V Ore bv)< eV (%, [5j-1,5j]) VN, [i-1.4i) (50.0.16)