14.4 Derivatives of analytic functions 15.1 Complex series

Eugenia Malinnikova, NTNU

October 24 2017

1 Derivatives of analytic functions

Theorem If f is analytic in some domain D then it has derivatives of any order which are also analytic functions. Moreover,

1 I f (z) f (z0) = dz, 2πi C z − z0 1 I f (z) 0( ) = , f z0 2 dz 2πi C (z − z0) n! I f (z) (n)( ) = , f z0 n+1 dz 2πi C (z − z0) where C is a simple closed path in D that bounds some domain in D which contains z0.

2 Examples

— I z = π 2 dz 2 i C (z − z0) — I ez 2πiez0 = , ( z )000 = z 4 dz e e C (z − z0) 6 — I cos z −2πi cos π πi = = , ( )00 = − 3 dz cos z cos z C (z − π) 24 12

3 Cauchy’s inequality

Suppose that f is an analytic function in a disc of radius r around z0 and that |f (z)| ≤ M when |z − z0| = r. Then n!M |f (z)| ≤ M, and |f (n)(z )| ≤ , for |z − z | ≤ r. 0 r n 0

Let C = {z : |z − z0| = r}, we have n! I f (z) (n)( ) = f z0 n+1 dz 2πi C (z − z0) then taking the absolute values and applying ML-inequality, we get

n! M n!M |f (n)(z )| ≤ 2πr = 0 2π r n+1 r n

4 Applications

Theorem (Liouville) If a function is analytic in a whole complex plane and bounded in absolute value, then it is a constant Functions analytic in the whole plane are called entire functions. Theorem (Morera) If f is continuous in a simply connected domain D and I f (z)dz = 0 C for any closed curve C. Then f is analytic in D.

5 Series of complex numbers

∞ P∞ Given a sequence {wn}n=1 ∈ C consider the series n=1 wn The series is convergent if its partial sums have a limit:

N X SN = wn → S, as N → ∞. n=1 This limit is called the sum of the series. Otherwise the series is divergent.

Relation to series of real numbers: P Let wn = un + ivn. The series wn converges if and only if each of P P the real series un, vn converges.

6 Basic definitions and facts

P P — The series wn is absolutely convergent if |wn| is convergent. P — The series wn is convergent ⇒ wn → 0 as n → ∞

Some sufficient conditions for convergence PN P — Cauchy criterion: M wn → 0 as M, N → ∞ ⇔ wn converges P — Majorization: an > 0, n = 1, 2,..., |wn| ≤ an and an P converges ⇒ wn converges; P — Ratio test: |wn+1|/|wn| ≤ q < 1 n = 1, 2,... ⇒ wn converges; 1/n P — Root test: (|wn|) ≤ q < 1 n = 1, 2,... ⇒ wn converges

7 The most important example

n Geometric series: z ∈ C and wn = z

∞ ( X 1 , |z| < 1, zn = 1−z diverges, |z| ≥ 1. 0

Expansion of the Cauchy kernel: Fix z0 ∈ C and let ζ, z ∈ C be such that |z − z0| < |ζ − z0|. Then 1 1 1 1 = = z−z = ζ − z (ζ − z0) − (z − z0) ζ − z0 1 − 0 ζ−z0 ∞  n ∞ n 1 X z − z0 X (z − z0) = n+1 ζ − z0 ζ − z0 (ζ − z0) n=0 n=0 We are going to use this formula for making expansions of analytic functions into power series !

8 Further examples

P∞ (1+i)n 1. n=1 n! converges (ratio test) P∞ (1+i)2n 2n n 2. n=1 2n diverges , |(1 + i) /2 | = 1 6→ 0 P∞ n+i / P∞ 1 3. n=1 n2 diverges, the real parts are 1 n and n=1 n diverges n 4. P∞ √i converges, n=1 n

∞ X in i 1 i 1 √ = − √ − √ + √ + ... n 1 n=1 2 3 4 Separating real and imaginary parts we get two real alternating series, both of them converge and then the series n converge. It does not converge absolutely, P √i diverges. n n

9