51.3. CAUCHY’S FORMULA FOR A DISK 1621

If you want to consider φ having values in X , a complex Banach space a similar resultholds.

Corollary 51.3.8 Let φ : [a,b]× [c,d]→ X be continuous and let

g(t)≡∫ b

aφ (s, t)ds. (51.3.7)

Then g is continuous. If ∂φ

∂ t exists and is continuous on [a,b]× [c,d] , then

g′ (t) =∫ b

a

∂φ (s, t)∂ t

ds. (51.3.8)

Proof: Let Λ ∈ X ′. Then Λφ : [a,b]× [c,d]→ C is continuous and ∂Λφ

∂ t exists and iscontinuous on [a,b]× [c,d] . Therefore, from 51.3.8,

Λ(g′ (t)

)= (Λg)′ (t) =

∫ b

a

∂Λφ (s, t)∂ t

ds = Λ

∫ b

a

∂φ (s, t)∂ t

ds

and since X ′ separates the points, it follows 51.3.8 holds.You can give a different proof of this.

Theorem 51.3.9 Let φ : [a,b]× [c,d]→ X be continuous and suppose φ t is continuous.Then (∫ b

aφ (s, t)ds

),t=∫ b

a

∂φ

∂ t(s, t)ds

Here X is a complex Banach space.

Proof: Consider the following set P which is where the ordered pair (t,h) will be.

c d(t,h)

This is so that both t and t +h are in [a,b] . Then for such an ordered pair, consider

∆(s, t,h)≡{

φ(s,t+h)−φ(s,t)h if h ̸= 0

φ t (s, t) if h = 0

Claim: ∆ is continuous on the compact set [a,b]×P.Proof of claim: It is obvious unless h = 0. Therefore, consider the point (s, t,0) .

∥∥∆(s′, t ′,h

)−∆(s, t,0)

∥∥= ∥∥∥∥φ (s′, t ′+h)−φ (s′, t ′)h

−φ t (s, t)∥∥∥∥