(a) For ρ1 < ρ < ρ2 , de¬ne

1 f (w)

f ± (z) = ±|z ’ z0 | > ±ρ.

dw,

w’z

2πi |w’z0 |=ρ

Show that f ± (z) is independent of ρ (provided that ±|z ’ z0 | >

±ρ remains valid). Conclude that f + , resp. f ’ , is analytic for

224 Appendix B. Functions with Values in Banach Spaces

|z ’ z0 | > ρ1 , resp. for |z ’ z0 | < ρ2 , and show

f (z) = f ’ (z) ’ f + (z), z ∈ R.

(b) Show that f can be expanded into a Laurent series

∞

fn (z ’ z0 )n , z ∈ R,

f (z) =

n=’∞

with coe¬cients fn ∈ E given by

1

f (w) (w ’ z0 )’n’1 dw.

fn =

2πi |w’z0 |=ρ

(c) In case of ρ1 = 0, de¬ne the notions of essential singularity,

resp. pole, resp. removable singularity as saying that, with fn as

de¬ned above, we have fn = 0 for in¬nitely many negative n,

resp. fn0 = 0 for some negative n0 and fn = 0 for n < n0 , resp.

fn = 0 for every negative n. Show that f (z) being bounded for

0 < |z ’ z0 | ¤ ρ < ρ2 implies that z0 is a removable singularity

of f .

3. Let G be a simply connected region, and let f be E -valued and holo-

morphic in G, except for singularities at points zn ∈ G, n ≥ 0, which

do not accumulate in G. De¬ne the residue of f at a point zn as

the coe¬cient f’1 in the Laurent expansion of f about zn . Show the

Residue Theorem:

The integral of f over any positively oriented Jordan curve in G which

avoids the points zn , n ≥ 0, equals 2πi times the sum of the residues

of f at those points zn which are in the interior region of the curve.

4. If G = G1 ∪G2 , for arbitrary regions Gk , and if fk ∈ H(Gk , E ) satisfy

f1 (z) = f2 (z) for every z ∈ G1 © G2 , then show that there is precisely

one f ∈ H(G, E ) with f (z) = fk (z) for z ∈ Gk .

5. For f ∈ H(C , E ), i.e., an entire E -valued function, show that if f is

bounded, then f is necessarily constant.

B.3 Holomorphic Continuation

Let f ∈ H(G, E ) and an arbitrary path γ with parameterization z(t), 0 ¤

t ¤ 1, beginning at some point z0 ∈ G but terminating at a point z1 ∈ G,

/

be given. Then we say that f can be holomorphically continued along γ,

provided that we can ¬nd a partitioning 0 = t0 < t1 < . . . < tm = 1 and

B.3 Holomorphic Continuation 225

some µ > 0, such that |z(t)’z(tk )| < µ for tk’1 ¤ t ¤ tk , 1 ¤ k ¤ m, and so

that f , by successive re-expansion of its power series about the points z(tk ),

can be de¬ned in the discs D(z(tk ), µ). Then, f obviously is holomorphic

in every one of these discs, but perhaps not in their union, since when the

path γ intersects with itself, f can in general not be unambiguously de¬ned

at intersection points.

Continuity of z(t) implies existence of a number δ, 0 < δ < µ, so that

|z(t) ’ z(tk )| ¤ δ for tk’1 ¤ t ¤ tk , 1 ¤ k ¤ m. This may be used

to show that f can also be analytically continued along every curve γ ˜

with parameterization z (t), 0 ¤ t ¤ 1, having the same endpoints as γ,

˜

provided that |z(t)’ z (t)| ¤ (µ’δ)/2 for every t. In particular, continuation

˜

along both curves leads to the same holomorphic function near z1 . This is

important in the proof of the following theorem:

Theorem 67 (Monodromy Theorem) Let G be an arbitrary region, let

D = D(z0 , ρ) ‚ G, and let f ∈ H(D, E ). Moreover, assume that f can be

holomorphically continued along every path γ, beginning at z0 and staying

inside G. Then the following holds true:

(a) If γ0 , γ1 are any two homotopic paths in G, both beginning at z0 and

ending, say, at z1 , then holomorphic continuation of f along either

path produces the same value f (z1 ).

(b) If G is simply connected, then f ∈ H(G, E ).

Proof: Clearly, (b) follows from (a), since in a simply connected region any

two paths with common endpoints are homotopic; hence by holomorphic

continuation we may de¬ne f (z) unambiguously at every point z ∈ G, and

then f is holomorphic in G. To show (a), choose any two paths γ0 , γ1 in

G, beginning at z0 and ending at a common point z1 . If the two paths

are homotopic, then by de¬nition there exists a continuous map z(s, t) of

the unit square R = [0, 1] — [0, 1] into G such that z(0, ·), resp. z(1, ·),

is a parameterization of γ0 , resp. γ1 , while for arbitrary s ∈ [0, 1] the

function z(s, ·) parameterizes some other path γs in G, connecting z0 and

z1 . By assumption f can be holomorphically continued along every one of

these paths, and we denote by fs (z1 ) the value of f at z1 , obtained by

holomorphic continuation along γs . For every ¬xed s, we conclude from the

de¬nition of holomorphic continuation existence of µs > 0 such that

1. the path γs is covered by ¬nitely many discs of radius µs ,

2. each disc contains the center of the following one (when moving along

the path),

3. f is holomorphic in each one of these discs.

Continuity of the function z(s, t) on R implies existence of δs > 0 so that

for s ∈ [0, 1] with |˜ ’ s| < δs the path γs remains within the union of these

˜ s ˜

226 Appendix B. Functions with Values in Banach Spaces

discs, which implies fs (z1 ) = fs (z1 ) for these values s, but this clearly

˜

˜

2

implies that fs (z1 ) has to be constant.

While we have seen that functions can be holomorphically continued by

successive re-expansion (provided that they are analytic at the points of

some path), this method has little practical value. We shall, however, see in

the following two subsections that other, more e¬ective ways of holomorphic

continuation are available in special cases. Moreover, both examples show

that continuation along paths that are not homotopic may, or may not,

produce the same value for the function at the common endpoint of the

paths.

• The Natural Logarithm

The (natural) logarithm of a nonzero complex number z is best de¬ned by

the integral

z

dw

= log |z| + i arg z.

log z = (B.7)

w

1

According to the Monodromy Theorem, this de¬nition is unambiguous in

any simply connected region containing 1 but not the origin, since then

it is irrelevant along which path integration is performed. However, since

w’1 dw = 2πi when integrating along positively oriented circles about the

origin, we see that it is not possible to give an unambiguous de¬nition of

log z in the full punctured complex plane (i.e., the complex plane with the

origin deleted). Instead, one can think of the logarithm as de¬ned in the

cut plane, i.e., the complex plane with the negative real numbers including

zero deleted, interpreting arg z to be in the open interval (’π, π), or what

is much better for our purposes, imagine log z as de¬ned on what is called

the Riemann surface of the logarithm: Think of a “spiraling staircase” with

in¬nitely many levels in both directions, “centered” at the origin. Points on

this staircase are uniquely characterized by their distance r > 0 from the

origin and the angle •, measured against some arbitrarily chosen direction.

To every point on the staircase, or simply every pair (r, •), there corre-

sponds the complex number z = r exp[2πi], called the projection. Instead

of a point, we can also consider the projection of subsets of the Riemann

surface. Consider a curve in the punctured complex plane, i.e., the plane

with the origin deleted, parameterized by z(t), 0 ¤ t ¤ 1. It can be “lifted”

to the Riemann surface by choosing a value for arg z(0) and requiring that

arg z(t) change continuously with respect to t. This implies that double

points on the curve in the plane may correspond to distinct points on the

surface. We shall use the same symbol z both for points in the punctured

plane and on the Riemann surface, but we emphasize that on the Riemann

surface the argument of a point is uniquely de¬ned. For example, the points

z and z exp[2πi], when considered on the Riemann surface, will not be the

same; instead, the second one sits directly above the ¬rst one when using

B.3 Holomorphic Continuation 227

the visualization as a staircase. Taking this into account, the function log z

can then be unambiguously de¬ned by (B.7). It will be very convenient

√

to view other functions z, or z ± , having branch points at the origin, as

being de¬ned on this Riemann surface, in order to avoid discussing which

one of their, in general in¬nitely many, branches one should take in a par-

ticular situation. For arbitrary functions f , de¬ned on some domain G

considered on the Riemann surface, we shall say that f is single-valued if

f (z) = f (z exp[2πi]) whenever both sides are de¬ned (which may very well

never happen). This is nothing else but saying that we may consider the

function, instead of the domain G on the Riemann surface, as de¬ned on

the corresponding projection in the plane. For example, the function z ± is

single-valued whenever ± ∈ Z.

• The Gamma Function

The following C -valued function plays a prominent role in this book as well

as in many other branches of mathematics:

The Gamma Function

For z in the right half-plane, the function

∞

tz’1 e’t dt

“(z) = (B.8)

0

(integrating along the positive real axis and de¬ning the, in gen-

eral multi-valued, function tz’1 according to the speci¬cation

of arg t = 0) is named the Gamma function.

It is easily seen that the integral in (B.8) converges absolutely and locally

uniformly for z in the right half-plane; hence the function is holomorphic

there, because the integrand is a holomorphic function of z. If, instead of

starting from zero, we integrate from one to in¬nity, the corresponding

function is entire, i.e., analytic in C . In the remaining integral, we expand

e’t into its power series and integrate termwise to obtain

∞ ∞

(’1)n

tz’1 e’t dt.

“(z) = +

n! (z + n) 1

n=0

Clearly, the series in this representation formula remains convergent for z

in the closed left half-plane except for the points 0, ’1, . . . where one of

the terms becomes in¬nite; thus, this formula provides the holomorphic

continuation of “(z) into the complex plane except for ¬rst-order poles at

the nonpositive integers. In particular, holomorphic continuation along ar-

bitrary paths avoiding these points is always possible and the value of the

function at the endpoint is independent of the path, because the function

228 Appendix B. Functions with Values in Banach Spaces

does not have any branch points. For completeness, and because these for-

mulas will play a role in the book, we mention the following two di¬erent

ways of obtaining the holomorphic continuation of “(z):

• For z in the right half-plane, integrating (B.8) by parts implies

“(1 + z) = z “(z). (B.9)

Solving for “(z), we see that we obtain holomorphic continuation, ¬rst

to values z = x + i y with x > ’1, except for a pole a the origin, then

to those with x > ’2, except for a pole at ’1, etc. It is worthwhile

to note that (B.9), together with “(1) = 1, implies “(1 + n) = n!,

n ∈ N.

• One can show (compare the exercises below) the following integral

representation for the reciprocal Gamma function, usually referred

to as Hankel™s formula:

1 1

w’z ew dw,

= (B.10)

“(z) 2πi γ

where γ is the path of integration from in¬nity along the ray arg w =

’π to the unit circle, then around the circle and back to in¬nity

along the ray arg w = π (note that both rays are the negative real

axis when pictured in the complex plane, but we better view them on

the Riemann surface of the logarithm and de¬ne the branch of the,

in general multivalued, function w’z according to the speci¬cations

of arg w). This integral can be shown to converge for every z ∈ C ,

so the reciprocal Gamma function is holomorphic in the full plane.

This again proves that “(z) can be analytically continued into the

left half-plane and has poles at all zeros of the reciprocal function “

however, we do not directly learn the location of these poles, resp.

zeros!

In this book, we shall frequently use the so-called Beta Integral: De¬ne

1

(1 ’ t)±’1 tβ’1 dt,

B(±, β) = Re ± > 0, Re β > 0.

0

To evaluate this integral, observe that by a simple change of variable we

obtain

u

(u ’ t)±’1 tβ’1 dt, u ∈ C.

u±+β’1 B(±, β) =

0

On the other hand, the identity

∞ u

’u

(u ’ t)±’1 tβ’1 dt du = “(±) “(β)

e

0 0

B.3 Holomorphic Continuation 229

follows by interchanging the order of integration and using the de¬nition of

the Gamma function. This, together with the integral for “(± + β), implies

“(±)“(β)

B(±, β) = , Re ± > 0, Re β > 0. (B.11)