Kenneth Kuttler

352 CHAPTER 14. FUNDAMENTALS

Theorem 14.5.1 Suppose γ is continuous and also of bounded variation, and that γ

is a parametrization of Γ where t ∈ [a,b] and the orientation is from t = a to t = b. Supposef : Γ→ X is continuous and has a primitive F. Thus F ′ (z) = f (z) for some open Ω ⊇ Γ.Then

∫γ

f (z)dz = F (γ (b))−F (γ (a)) .

Proof: First suppose f has values in C. Then the desired conclusion is from Proposi-tion 14.4.4. It holds in general because if φ ∈ X ′, then φ (F)′ = φ ( f ) and so from Theo-rem 14.4.3, φ (F (b)−F (a)) = φ (F (b))−φ (F (a)) =

∫γ

φ ( f (z))dz = φ

(∫γ

f (z)dz)

Proposition 12.3.9 about how the elements of X ′ separate points, F (b)−F (a) =∫

γf (z)dz

as claimed. ■Probably the most fundamental result in the subject is Cauchy’s theorem which says

that the contour integral of an analytic function over a simple closed curve equals 0. Thefollowing is like what was first done by Cauchy back in the early 1800’s. Recall that f ′

continuous is part of the definition of “analytic”.

Theorem 14.5.2 Let γ∗ be a simple closed curve with parametrization γ (t) havingfinite length, (γ has finite total variation). Letting U be the inside, assume U is a regionfor which Green’s theorem holds. Let f be analytic near U ∪ γ∗ and have complex values.Then

∫γ

f dz = 0. The same conclusion holds if f has values in a complex Banach space.

Proof: From Observation 14.4.4,∫γ

f dz =∫

C(u(x,y) ,−v(x,y)) ·dr+ i

∫C(v(x,y) ,u(x,y)) ·dr

where C is the oriented simple closed curve in the plane resulting from γ∗. Now by Green’stheorem, this equals

∫U ((−v)x−uy)dm2 + i

∫U (ux− vy)dm2 = 0 because of the Cauchy

Riemann equations, Proposition 14.2.2. To obtain the last claim, by Theorem 14.4.3,φ

(∫γ

f dz)=∫

γφ ( f )dz = 0. By Proposition 12.3.9 about how the elements of X ′ sepa-

rate points, it follows that∫

γf dz = 0. ■

The following picture illustrates the theorem which follows in which we punch holes inU above. Assume the closures of these holes do not intersect, as illustrated in the picture.

γ1z2γ2

γ3z3

Ωγ

Assume the curves γk illustrated above bound ellipses or circular disks or more gener-ally regions for which Green’s theorem holds. The main interest is in circular disks. Orientthese clockwise as shown. Then using Corollary 10.8.4, it follows that for U the inside ofγ1 and U j the inside of γ j, U \

(∪n

j=1U j

)also satisfies Green’s theorem. In this picture

n = 3. The following is a generalization of the above theorem.

Theorem 14.5.3 Let γ∗ be a simple closed curve with parametrization γ (t) havingcounter clockwise orientation and finite length, (γ has finite total variation). Letting U bethe inside, assume U is a region for which Green’s theorem holds with the orientation of γ .Let the U j be as described above for j = 1,2, ...,n and suppose f is analytic near the closed

352 CHAPTER 14. FUNDAMENTALSTheorem 14.5.1 Suppose ¥ is continuous and also of bounded variation, and that yis a parametrization of V where t € [a,b] and the orientation is from t = a to t = b. Supposef : TX < is continuous and has a primitive F. Thus F' (z) = f (z) for some openQ DT.Then J, f (z)dz= F (y(b))—F (y(a))-Proof: First suppose f has values in C. Then the desired conclusion is from Proposi-tion 14.4.4. It holds in general because if @ € X’, then @ (F)’ = @(f) and so from Theo-rem 14.4.3, 6 (F (b) —F (a)) = 0 (F (b)) — 0 (F (a) = fy O(f @)) dz = 6 (yf (az) ByProposition 12.3.9 about how the elements of X’ separate points, F (b) — F (a) = J, f (z)dzas claimed.Probably the most fundamental result in the subject is Cauchy’s theorem which saysthat the contour integral of an analytic function over a simple closed curve equals 0. Thefollowing is like what was first done by Cauchy back in the early 1800’s. Recall that f’continuous is part of the definition of “analytic”.Theorem 14.5.2 Ler Y¥ be a simple closed curve with parametrization y(t) havingfinite length, (y has finite total variation). Letting U be the inside, assume U is a regionfor which Green’s theorem holds. Let f be analytic near UU Y* and have complex values.Then Sy fdz=0. The same conclusion holds if f has values in a complex Banach space.Proof: From Observation 14.4.4,[fae= [ (ul), -v(e9))-ae +i [ (o(e9) (9) aewhere C is the oriented simple closed curve in the plane resulting from y*. Now by Green’stheorem, this equals fy ((—v),—uy)dmz +i fy (Ux — vy) dmz = 0 because of the CauchyRiemann equations, Proposition 14.2.2. To obtain the last claim, by Theorem 14.4.3,“) (J, faz) = J, 9 (f)dz = 0. By Proposition 12.3.9 about how the elements of X’ sepa-rate points, it follows that J, fdz =0. iThe following picture illustrates the theorem which follows in which we punch holes inU above. Assume the closures of these holes do not intersect, as illustrated in the picture.Assume the curves ; illustrated above bound ellipses or circular disks or more gener-ally regions for which Green’s theorem holds. The main interest is in circular disks. Orientthese clockwise as shown. Then using Corollary 10.8.4, it follows that for U the inside ofy, and U; the inside of 7;, U \ (Ti) also satisfies Green’s theorem. In this picturen= 3. The following is a generalization of the above theorem.Theorem 14.5.3 Lez Y* be a simple closed curve with parametrization y(t) havingcounter clockwise orientation and finite length, (y has finite total variation). Letting U bethe inside, assume U is a region for which Green’s theorem holds with the orientation of Y.Let the U; be as described above for j = 1,2,...,n and suppose f is analytic near the closed