w±’1

1 w

E± (z) = e dw, (B.19)

w± ’ z

2πi γ

¬rst for |z| < 1 and then, using Cauchy™s theorem to change the path

of integration, one can show a corresponding representation for arbitrary

values of z. It is shown in the exercises below that E± is an entire function

of exponential order k = 1/± and type „ = 1.

As for the classical case of E = C , order and type of entire functions can

be expressed in terms of their power series coe¬cients:

Theorem 69 Let f ∈ H(C , E ) have the power series expansion f (z) =

fn z n , z ∈ C . Then the order k and, in case of 0 < k < ∞, the type „

of f are given by

n log n

k = lim sup , (B.20)

n’∞ log(1/ fn )

1 k/n

„= lim sup n fn . (B.21)

ek n’∞

Proof: Assume f (z) ¤ C exp[c|z|κ ], for some C, c, κ ≥ 0. Then we con-

clude from Proposition 27 (p. 222), with z0 = 0 and arbitrary ρ > 0, that

234 Appendix B. Functions with Values in Banach Spaces

fn ¤ C ρ’n exp[cρκ ], n ≥ 1. The right-hand side, as a function of ρ, be-

comes minimal for cρκ = n/κ; hence fn ¤ C (cκ/n)n/κ exp[n/κ], n ≥ 1

follows. Conversely, such an estimate for fn implies, using Stirling™s formula,

that fn ¤ C (Kc1/κ )n /“(1 + n/κ), n ≥ 1, for every K > 1 and C su¬-

˜ ˜

ciently large. Hence, according to the above discussion of Mittag-Le¬„er™s

˜˜κ

function, we obtain f (z) ¤ C E1/κ (Kc1/κ |z|) ¤ C ec|z| , for every c > c.

˜ ˜

2

Using this, one can easily complete the proof.

Exercises: In the following exercises, let ± > 0 and consider Mittag-

Le¬„er™s function E± (z).

¯

1. Assume ± < 2, and let S = {z : | arg z| ¤ ±π/2. Use (B.19) and

¯

Cauchy™s integral theorem to show for s ∈ S and |z| > 1 that

w±’1

1

1/±

E± (z) = ±’1 ez ew

+ dw,

w± ’ z

2πi γ

and the integral tends to zero as z ’ ∞.

¯

2. Let ± and S be as in the previous exercise. Use (B.19) and Cauchy™s

integral theorem to show that E± (z) remains bounded in arbitrary

¯

closed subsectors that do not intersect with S. Use this and the pre-

vious exercise to show that E± (z) is of order k = 1/± and type „ = 1.

3. Compute order and type of E± (z), for arbitrary ± > 0.

B.5 The Phragm´n-Lindel¨f Principle

e o

In the literature one ¬nds a number of theorems, all bearing the name

Phragm´n-Lindel¨f and being variants, resp. consequences, of the well-

e o

known Maximum Modulus Principle, which easily generalizes to Banach

space valued functions:

Proposition 29 (Maximum Modulus Principle) Let G be any region

in C , and assume that some z0 and ρ > 0 with D(z0 , ρ) ‚ G exist for

which we have

f (z) ¤ f (z0 ) , z ∈ D(z0 , ρ).

Then f (z) is constant.

∞

Proof: Without loss of generality, let z0 = 0. Expanding f (z) = 0 fn z n ,

|z| < ρ, we wish to show fn = 0 for n ≥ 1. Suppose we did so for

1 ¤ n ¤ m ’ 1, which is an empty assumption for m = 1. Accord-

ing to Hahn-Banach™s theorem, there exists φ ∈ E — with φ = 1 and

B.5 The Phragm´n-Lindel¨f Principle

e o 235

φ(f0 ) = f0 , |φ(fm )| = fm . By continuity of φ we conclude φ(f (z)) =

∞

φ(f0 ) + n=m φ(fn ) z n , |z| < ρ. The assumptions on f and φ imply

|φ(f (z))| ¤ f (z) ¤ f (0) = φ(f0 ). Taking z so that z m φ(fm ) and

∞

φ(f0 ) have the same argument, and |z| so small that | m+1 φ(fn ) z n | ¤

|z m φ(fm )|/2, we obtain

|φ(f (z)| ≥ |φ(f0 ) + z m φ(fm )| ’ |z m φ(fm )|/2 = |φ(f0 )| + |z m φ(fm )|/2,

2

implying φ(fm ) = 0, hence fm = 0.

Another way of expressing the above result is by saying that for arbitrary

f ∈ H(G, E ) with f (z) ≡ 0 the function F (z) = f (z) cannot have a

local maximum in G. Using this, we now prove the following important

result:

Theorem 70 (Phragmen-Lindelof Principle) For k > 0, let S =

´ ¨

{ z : |z| > ρ0 , ± < arg z < β }, with ρ0 > 0, 0 < β ’ ± < π/k. For some

f ∈ H(S, E ), assume f (z) ¤ c exp[K|z|k ] in S, for su¬ciently large

c, K > 0. Moreover, assume that f is continuous up to the boundary of S

and is bounded by a constant C there. Then

f (z) ¤ C, z ∈ S.

Proof: Without loss of generality, let the positive real axis bisect the sector

S. Then for κ larger than, but su¬ciently close to, k and arbitrary µ > 0,

κ κ

the function e’µz decreases throughout S, and therefore F (z) = e’µz f (z)

tends to zero as z ’ ∞ in S. For every su¬ciently large ρ > 0 we conclude

that F (z) is bounded by C, for z on the boundary of the region S ©

D(0, ρ). The Maximum Modulus Principle then implies the same estimate

inside the region. Letting µ ’ 0 and ρ ’ ∞, one can complete the proof.

2

One should note that we have tacitly allowed that S may be a region on

the Riemann surface of the logarithm, i.e., β ’ ± may be larger than 2π.

The theorem admits various generalizations; for a very useful cohomological

version, see Sibuya [251].

Exercises:

1. Show that the above theorem does not hold in general if the sector

S has opening larger than or equal to π/k.

2. Let f be an E -valued entire function that is bounded in a closed

¯

sector S = {z : ± ¤ arg z ¤ β}. What can be said about its order?

3. Compare Exercise 7 on p. 63 to see that entire functions of in¬nite

order exist that are bounded in closed sectors of openings 2π ’µ, with

arbitrary small µ > 0.

Appendix C

Functions of a Matrix

There is a beautiful classical theory of what is called a function of a matrix,

to be distinguished from the term matrix function. In this context, one is

concerned with interpreting f (A), for a suitable class of C -valued functions

f and square matrices A. More generally, one can instead of matrices choose

A in a Banach algebra, but we shall not need this here. In the simplest

m

case, f will be a polynomial, say, f (z) = 0 fn z n , and then we simply set

m

f (A) = 0 fn An . Similarly, when f is an entire function, one can use the

power series expansion of f to de¬ne f (A), and we shall do this in the case

of f (z) = ez in the next section. More interesting are the cases when f is

holomorphic in a region G = C . It can then be seen that it is crucial to

restrict ourselves to matrices with spectrum in G, and then one can de¬ne

f (A) very elegantly by a generalization of Cauchy™s integral formula. A

similar representation holds even for multivalued holomorphic functions on

a Riemann surface.

For applications in the theory of ODE, it is su¬cient to understand,

aside from the exponential function, the natural logarithm of a matrix and

the closely related matrix powers of a complex variable z. In the following

sections we shall present some basic facts concerning these concepts. In par-

ticular, we shall show that every invertible matrix has a matrix logarithm

which, however, is not uniquely de¬ned. For matrices A that are close to

the unit matrix, i.e., I ’ A is small, one can de¬ne the main branch of

log A by means of the power series expansion of the logarithm, and the

same is possible whenever I ’ A is nilpotent. For general A, however, we

have to ¬nd a di¬erent de¬nition of log A; this will be done in Section C.2,

where we shall also show that we can select a unique branch of log A by

238 Appendix C. Functions of a Matrix

requiring its eigenvalues to have imaginary parts in a ¬xed half-open inter-

val of length 2π. This corresponds to selecting a ¬xed branch of log z by

restricting the argument of z accordingly.

C.1 Exponential of a Matrix

Given a square matrix A, we de¬ne

∞

An

A

e = exp[A] = ,

n!

n=0

where A0 = I and An = A · . . . · A, n times repeated, for n ≥ 1. From

µ µ

m A /(n!) ¤

n n

m A /(n!) one can easily conclude convergence of

the series, thus the exponential of a matrix is always de¬ned. Manipulating

with the in¬nite matrix power series, one can show the following rules:

1. For A, T ∈ C ν—ν , T invertible, we have

exp[T ’1 A T ] = T ’1 exp[A] T.

2. For Λ = diag [»1 , . . . , »ν ] we have

exp[Λ] = diag [e»1 , . . . , e»ν ].

3. For a triangularly blocked matrix A with diagonal blocks Ak , the

matrix exp[A] is likewise blocked and has diagonal blocks exp[Ak ], in

the same order.

4. For matrices A, B ∈ C ν—ν with AB = BA, we have

exp[A + B] = exp[A] exp[B] ( = exp[B] exp[A] ),

but this rule does not hold in general, as is shown in one of the

exercises below.

5. For every square matrix A, the inverse of exp[A] exists and equals

exp[’A].

We conclude from the above set of rules that exp[A] can theoretically be

computed by ¬rst transforming A into Jordan form J = Λ + N with com-

muting terms Λ (diagonal) and N (nilpotent), and then computing exp[Λ]

and exp[N ]; for the second note that the de¬ning series terminates.

Closely related to the exponential of the matrix are powers z A , z ∈

C \ {0}, A ∈ C ν—ν , since we set

z A = exp[A log z].

C.1 Exponential of a Matrix 239

To make this de¬nition unambiguous, we have to select a branch of the mul-

tivalued function log z, e.g., by specifying a value for arg z. Whether or not

z A is indeed a multivalued function, and how many branches this function

has, depends on the eigenvalues of A as well as its nilpotent part; e.g., if A

is diagonalizable and all eigenvalues are integers, then z A is single-valued,

while for A being a nilpotent nonzero matrix the function has in¬nitely

many branches. By termwise di¬erentiation of the de¬ning series one can

show that

dA

z = A z A’I = z A’I A,

dz

independent of the selected branch of the function.

Exercises: If nothing else is said, let

A11 O

A=

A21 A22

be a constant matrix, lower triangularly blocked as indicated, with square

diagonal blocks.

1. Assume that A11 and A22 have disjoint spectra, i.e., do not have

a common eigenvalue. Show existence and uniqueness of a constant

matrix T so that

z A11 O

zA = .

T z A11 ’ z A22 T A22

z

2. For a nilpotent Jordan block N , compute z N .

3. Show: det exp[A] = exp[ trace A ].

4. For arbitrary ν — ν matrices A, B, show that

exp[(A + B)z] = exp[Az] exp[Bz]

holds for every z ∈ C if and only if A and B commute.

5. Let A(z) be a square matrix whose entries all are holomorphic in

some region G ‚ C , so that A (z) A(z) = A(z) A (z) for every z ∈ G.

Show

d A(z)

= A (z) eA(z) = eA(z) A (z), z ∈ G.

e

dz

240 Appendix C. Functions of a Matrix

C.2 Logarithms of a Matrix

Every reasonable interpretation for log A should be a matrix X solving