3. Elementary Results of Complex Analysis. Liouville's Theorem.

We assume the following results of advanced advanced calculus:

Integration of Complex Functions over Paths. If γ: [0, 1] → C is a rectifiable curve in the complex plane, we can write

γ(t) = α(t) + iβ(t)

where α, β are real functions of bounded variation. For a real valued function h defined on a region containing the image, C, of γ, we then define

C f dγ = ∫[0,1] h dα(t) + ∫[0,1] h dβ(t)

where the integrals on the right are Riemann-Stieltjes integrals. For a complex valued function f = u + iv, we set

C f dγ = ∫C u dγ + i ∫C v dγ

= [∫[0,1] u dα(t) - ∫[0,1] v dβ(t)] + i[∫[0.,1] v dα + ∫[0,1] u dβ(t)]

If γ is differentiable except on a finite number of points (piecewise differentiable), we then have

C f dγ = ∫[0,1] f (dγ/dt) dt

≤ sup |f| ∫[0,1] (dγ/dt) dt

≤ sup |f| ∫[0,1] |dγ/dt| dt

and we note that the last integral is actually the length of the arc γ.

An Important Integral.

eis 

ds = 2π     if |z| < 1.
0eis - z 

Proof. Let

φ (s,t) = eis   for 0 ≤ t ≤ 1, 0 ≤ s ≤ 2π

eis - tz

Then φ is continuously differentiable. Hence, if g(t)= ∫[0,1] φ(s,t) ds

g'(t) = (∂/∂t) ∫[0,2π]  φ(s,t) ds

=eisds


∂t0eis - tz
=z eisds

0(eis - tz)

by the theorem on differentiation under the integral sign. Since

eisds = i


(eis - tz)2eis - tz

[The constant of integration is omitted here.]

and eis = 1 for s = 0 or 2π, the definite integral vanishes, so g(t) is constant. Since g(0) = 2π, we must have g(t) = 2π for all t ∈ [0.1], giving the required result.

Proposition. Let G be a region (connected set with nonempty interior) in C, let f: G → C be analytic, and suppose |z - a| implies z ∈ G. If γ(t) = a + reit, 0 ≤ t ≤ 2π, then

f(z) = 1 f(w)dw


2πi γw - z

Proof. By considering G1 = {(z - a)/r : z ∈ G} and the function g(z) = f(a + rz), we see that, without loss of generality, we may take a = 0 and z = 1.

We want to show that

f(eis) eisds - 2πf(z) = /f(e)eis - f(z)\dz

|
|
0eis - z0\eis - z/

Put

φ (s, t) =  f(z + t(eis - z))eis - f(z)

eis - z

If g(t) = ∫[0,2π] φ(s,t) ds, then

g'(t) = (∂/∂t))∫[0,2π] φ(s,t) ds

= ∫[0,2π] eis f'(z + t(eis - z)) ds

by differentiation under the integral sign. But we have the indefinite integral

∫eis f'(z + t(eis - z)) ds = - it f(z + t(eis - z))

and noting that the right hand has the same value for s = 0 and s = 2π, g'(t) = 0 and g(t) is constant. Clearly

g(0) = /f(z) eis - f(z)\ds = 0
|
|
0\eis - z/

since we have established above that

eisds = 2π

0eis - z
Thus g(0) = g(1), proving the result.

Theorem (Taylor series expansion). If f is analytic for |z - a| < R, then f(z) = ∑n = 0, ... , ∞ an(z - a)n for all z in this region, where an = f(n)(a)/n!

Proof.If γ(t) = a + reit, 0 ≤ t ≤ 2π where 0 ≤ r ≤ R, then by the above proposition

f(z) = 1 f(w)dw     for |z - a| < r


2πi γ w - z

We have the geometric series expansion

1=1/z - a\n


|
|
w - z(w - a) n=0\w - a/

which converges uniformly for |z - a| < r. Thus we can write

f(z) = 1 f(w)(z - a)n
|

|
n=02πi γ (w - a)n + 1

Setting

an1 f(w)dw


2πi γ (w - a)n + 1

we get

f(z) = ∑n=0,...,∞ an(z - a)n

for |z - a| < r. Clearly an = f(n)(a)/n! Since r < R was chosen arbitrarily, the expansion holds for alll z with |z - a| < R.

Corollary.

f(n)(a) = n! f(w)dw


2πi γ (w - a)n + 1

where γ(t) = a + reit, o ≤ t ≤ 2π.

Cauchy's Estimate. Let f be analytic for |z - a| < r and suppose |f(z)| ≤ M for all such z. Then

|f(n)(a)| ≤ n!M

Rn

Proof. By the Corollary above

|f(n)(a)| ≤ |n!| |f(w)|d|w|
|
||
|
|2πi|γ|(w - a)n + 1|

 = (n!/2π).(M/rn+1).2πr = (n!M/rn)

The result follows since r < R is arbitrary.

Liouville's Theorem. If f is a bounded function analytic in the entire plane, then f is constant.

Proof. If |f(z)| ≤ M for all z ∈ C, then by Cauchy's estimate, since f is analytic in every disc {w: |w - z| < R}, we have |f'(z)| ≤ M/R. Since R can be made arbitrarily large, |f(z)| = 0, and the result follows.

Link. The treatment of the subject here is an extremely sketchy one, taken from John B. Conway: Functions of one Complex Variable. a detailed development of the subject is found at the online textbook

http://math.fullerton.edu/mathews/c2003/ComplexUndergradMod.html

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