Kenneth Kuttler

360 CHAPTER 14. FUNDAMENTALS

and

lim supn→∞

(n(n−1) · · ·(n− k+1)(n− k)((n− k)−1) ·

∥an∥(

1+ |h||z−z0|

)n−k

)1/n

= lim supn→∞

∥an∥1/n |z− z0|(

1+|h||z− z0|

)< 1 if h is small enough.

Thus, by the root test, the infinite series which multiplies h converges for small h and isdecreasing in h and so that entire term converges to 0 as h→ 0. This leaves

f (k+1) (z) =∞

∑n=k+1

n(n−1) · · ·(n− k+1)(n− k)an (z− z0)n−(k+1) ■

Corollary 14.8.4 Suppose f is continuous on ∂B(z0,r) and suppose that for all z ∈B(z0,r) ,

f (z)≡ 12πi

∫γ

f (w)w− z

dw,

where γ (t) ≡ z0 + reit , t ∈ [0,2π] . Then f is analytic on B(z0,r) and in fact has infinitelymany derivatives on B(z0,r) .

Proof: This is just a repeat of the above arguments. You show that f (z) is given by apower series for |z− z0|< r and from this, the result follows. ■

Also, the following illustrates a difference from what is expected in real analysis. Itsays that uniform convergence tends to take with it differentiability.

Lemma 14.8.5 Let γ (t) = z0+reit , for t ∈ [0,2π]. Let fn→ f uniformly on B(z0,r) andsuppose fn (z) = 1

2πi∫

fn(w)w−z dw for z ∈ B(z0,r) . Then it follows that f (z) = 1

2πi∫

f (w)w−z dw,

implying that f is analytic on B(z0,r) .

Proof: From the formula for fn and the uniform convergence of fn to f on γ∗, theintegrals converge to 1

2πi∫

f (w)w−z dw. Therefore, f (z) = 1

2πi∫

f (w)w−z dw. ■

Note that this shows you can’t expect anything like the Weierstrass approximation the-orem to hold. That theorem allows the uniform appoximation of a continuous nowheredifferentiable function with polynomials.

Because of Theorem 14.8.3, from now on, the term analytic will be used interchange-ably with “has a derivative”. This has shown that if the function has one derivative on anopen set, then it has all of them. Now here is another version of Morera’s theorem.

Corollary 14.8.6 Let Ω be an open set. Suppose that whenever γ (z1,z2,z3,z1) is aclosed curve bounding a triangle T, which is contained in Ω, and f is a continuous functiondefined on Ω, it follows that

∫γ(z1,z2,z3,z1)

f (z)dz = 0, then f is analytic on Ω.

Proof: As in the proof of Morera’s theorem, let B(z0,r)⊆Ω and use the given conditionto construct a primitive, F for f on B(z0,r) . Then F is analytic and so by Theorem 14.8.3,it follows that F and hence f have infinitely many derivatives, implying that f is analyticon B(z0,r) . Since z0 is arbitrary, this shows f is analytic on Ω. ■

The following observation is useful to keep in mind.

360 CHAPTER 14. FUNDAMENTALSandlim supn—yoon(n 1) (nk 1) (WK) (nb) 1): I/nanil (1+ A)h= lim sup |lan||!/"|z—zo| (14 a ) < 1 if his small enough.no Iz —zo|Thus, by the root test, the infinite series which multiplies h converges for small h and isdecreasing in h and so that entire term converges to 0 as h — 0. This leavescoFEV) = Yo n(n= 1) (n=k+ 1) (n= Wan (2-20)n=k+1Corollary 14.8.4 Suppose f is continuous on OB(z,r) and suppose that for all z €B(zo,1),° _ 1 ffl),amo W,where y(t) =z +re" ,t € [0,27]. Then f is analytic on B(z9,r) and in fact has infinitelymany derivatives on B(zo,Tr).Proof: This is just a repeat of the above arguments. You show that f(z) is given by apower series for |z—zo| < r and from this, the result follows. IlAlso, the following illustrates a difference from what is expected in real analysis. Itsays that uniform convergence tends to take with it differentiability.Lemma 14.8.5 Let y(t) =z +re", fort € [0,27]. Let fy — f uniformly on B(zo-r) r) andsuppose fn (Z) = se ae dw for z € B(zo,r). Then it follows that f (z) = +5 f& — d dw,implying that f is analytic on B(zo,r).Proof: From the formula for Fn and the uniform convergence of f, to f on y*, theintegrals converge to 54; I,4 w) +dw. Therefore, f(z) = 35; A ~ £0) aw, aNote that this shows you ca can’t expect anything like the Weierstrass approximation the-orem to hold. That theorem allows the uniform appoximation of a continuous nowheredifferentiable function with polynomials.Because of Theorem 14.8.3, from now on, the term analytic will be used interchange-ably with “has a derivative”. This has shown that if the function has one derivative on anopen set, then it has all of them. Now here is another version of Morera’s theorem.Corollary 14.8.6 Let Q be an open set. Suppose that whenever Y(z,22,23,21) is aclosed curve bounding a triangle T, which is contained in Q, and f is a continuous functiondefined on Q, it follows that Sy(zy.20,23.21) f (z)dz=0, then f is analytic on Q.Proof: As in the proof of Morera’s theorem, let B (zo,r) C Q and use the given conditionto construct a primitive, F for f on B(zo,r). Then F is analytic and so by Theorem 14.8.3,it follows that F and hence f have infinitely many derivatives, implying that f is analyticon B(zo,r). Since zo is arbitrary, this shows f is analytic on Q. HlThe following observation is useful to keep in mind.