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2. Let f ∈ H(R, E ), R = {z : ρ1 < |z ’ z0 | < ρ2 }.
(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)

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( 61 .)



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