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2. Let f в€€ H(R, E ), R = {z : ПЃ1 < |z в€’ z0 | < ПЃ2 }.
(a) For ПЃ1 < ПЃ < ПЃ2 , deп¬Ѓne

1 f (w)
f В± (z) = В±|z в€’ z0 | > В±ПЃ.
dw,
wв€’z
2ПЂi |wв€’z0 |=ПЃ

Show that f В± (z) is independent of ПЃ (provided that В±|z в€’ z0 | >
В±ПЃ remains valid). Conclude that f + , resp. f в€’ , is analytic for
224 Appendix B. Functions with Values in Banach Spaces

|z в€’ z0 | > ПЃ1 , resp. for |z в€’ z0 | < ПЃ2 , and show

f (z) = f в€’ (z) в€’ f + (z), z в€€ R.

(b) Show that f can be expanded into a Laurent series
в€ћ
fn (z в€’ z0 )n , z в€€ R,
f (z) =
n=в€’в€ћ

with coeп¬ѓcients fn в€€ E given by
1
f (w) (w в€’ z0 )в€’nв€’1 dw.
fn =
2ПЂi |wв€’z0 |=ПЃ

(c) In case of ПЃ1 = 0, deп¬Ѓne the notions of essential singularity,
resp. pole, resp. removable singularity as saying that, with fn as
deп¬Ѓned above, we have fn = 0 for inп¬Ѓnitely 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 в€’ z0 | в‰¤ ПЃ < ПЃ2 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 в€€ G, n в‰Ґ 0, which
do not accumulate in G. Deп¬Ѓne the residue of f at a point zn as
the coeп¬ѓcient fв€’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 в‰Ґ 0, equals 2ПЂi 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 в€ЄG2 , for arbitrary regions Gk , and if fk в€€ H(Gk , E ) satisfy
f1 (z) = f2 (z) for every z в€€ G1 в€© G2 , then show that there is precisely
one f в€€ H(G, E ) with f (z) = fk (z) for z в€€ Gk .
5. For f в€€ 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 в€€ H(G, E ) and an arbitrary path Оі with parameterization z(t), 0 в‰¤
t в‰¤ 1, beginning at some point z0 в€€ G but terminating at a point z1 в€€ G,
/
be given. Then we say that f can be holomorphically continued along Оі,
provided that we can п¬Ѓnd a partitioning 0 = t0 < t1 < . . . < tm = 1 and
B.3 Holomorphic Continuation 225

some Оµ > 0, such that |z(t)в€’z(tk )| < Оµ for tkв€’1 в‰¤ t в‰¤ tk , 1 в‰¤ k в‰¤ m, and so
that f , by successive re-expansion of its power series about the points z(tk ),
can be deп¬Ѓned in the discs D(z(tk ), Оµ). Then, f obviously is holomorphic
in every one of these discs, but perhaps not in their union, since when the
path Оі intersects with itself, f can in general not be unambiguously deп¬Ѓned
at intersection points.
Continuity of z(t) implies existence of a number Оґ, 0 < Оґ < Оµ, so that
|z(t) в€’ z(tk )| в‰¤ Оґ for tkв€’1 в‰¤ t в‰¤ tk , 1 в‰¤ k в‰¤ m. This may be used
to show that f can also be analytically continued along every curve Оі Лњ
with parameterization z (t), 0 в‰¤ t в‰¤ 1, having the same endpoints as Оі,
Лњ
provided that |z(t)в€’ z (t)| в‰¤ (Оµв€’Оґ)/2 for every t. In particular, continuation
Лњ
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 , ПЃ) вЉ‚ G, and let f в€€ H(D, E ). Moreover, assume that f can be
holomorphically continued along every path Оі, beginning at z0 and staying
inside G. Then the following holds true:
(a) If Оі0 , Оі1 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 в€€ 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 deп¬Ѓne f (z) unambiguously at every point z в€€ G, and
then f is holomorphic in G. To show (a), choose any two paths Оі0 , Оі1 in
G, beginning at z0 and ending at a common point z1 . If the two paths
are homotopic, then by deп¬Ѓnition there exists a continuous map z(s, t) of
the unit square R = [0, 1] Г— [0, 1] into G such that z(0, В·), resp. z(1, В·),
is a parameterization of Оі0 , resp. Оі1 , while for arbitrary s в€€ [0, 1] the
function z(s, В·) parameterizes some other path Оіs 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 Оіs . For every п¬Ѓxed s, we conclude from the
deп¬Ѓnition of holomorphic continuation existence of Оµs > 0 such that
1. the path Оіs is covered by п¬Ѓnitely many discs of radius Оµs ,
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 Оґs > 0 so that
for s в€€ [0, 1] with |Лњ в€’ s| < Оґs the path Оіs remains within the union of these
Лњ s Лњ
226 Appendix B. Functions with Values in Banach Spaces

discs, which implies fs (z1 ) = fs (z1 ) for these values s, but this clearly
Лњ
Лњ
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 eп¬Ђective 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.

вЂў The Natural Logarithm
The (natural) logarithm of a nonzero complex number z is best deп¬Ѓned by
the integral
z
dw
= log |z| + i arg z.
log z = (B.7)
w
1
According to the Monodromy Theorem, this deп¬Ѓnition 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
wв€’1 dw = 2ПЂi when integrating along positively oriented circles about the
origin, we see that it is not possible to give an unambiguous deп¬Ѓnition 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 deп¬Ѓned 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 (в€’ПЂ, ПЂ), or what
is much better for our purposes, imagine log z as deп¬Ѓned on what is called
the Riemann surface of the logarithm: Think of a вЂњspiraling staircaseвЂќ with
inп¬Ѓnitely many levels in both directions, вЂњcenteredвЂќ at the origin. Points on
this staircase are uniquely characterized by their distance r > 0 from the
origin and the angle П•, measured against some arbitrarily chosen direction.
To every point on the staircase, or simply every pair (r, П•), there corre-
sponds the complex number z = r exp[2ПЂi], 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 в‰¤ t в‰¤ 1. It can be вЂњliftedвЂќ
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 deп¬Ѓned. For example, the points
z and z exp[2ПЂi], when considered on the Riemann surface, will not be the
same; instead, the second one sits directly above the п¬Ѓrst 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 deп¬Ѓned by (B.7). It will be very convenient
в€љ
to view other functions z, or z О± , having branch points at the origin, as
being deп¬Ѓned on this Riemann surface, in order to avoid discussing which
one of their, in general inп¬Ѓnitely many, branches one should take in a par-
ticular situation. For arbitrary functions f , deп¬Ѓned on some domain G
considered on the Riemann surface, we shall say that f is single-valued if
f (z) = f (z exp[2ПЂi]) whenever both sides are deп¬Ѓned (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 deп¬Ѓned on
the corresponding projection in the plane. For example, the function z О± is
single-valued whenever О± в€€ Z.

вЂў 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
в€ћ
tzв€’1 eв€’t dt
О“(z) = (B.8)
0

(integrating along the positive real axis and deп¬Ѓning the, in gen-
eral multi-valued, function tzв€’1 according to the speciп¬Ѓcation
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 inп¬Ѓnity, the corresponding
function is entire, i.e., analytic in C . In the remaining integral, we expand
eв€’t into its power series and integrate termwise to obtain
в€ћ в€ћ
(в€’1)n
tzв€’1 eв€’t dt.
О“(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, в€’1, . . . where one of
the terms becomes inп¬Ѓnite; thus, this formula provides the holomorphic
continuation of О“(z) into the complex plane except for п¬Ѓrst-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 diп¬Ђerent
ways of obtaining the holomorphic continuation of О“(z):
вЂў For z in the right half-plane, integrating (B.8) by parts implies

О“(1 + z) = z О“(z). (B.9)

Solving for О“(z), we see that we obtain holomorphic continuation, п¬Ѓrst
to values z = x + i y with x > в€’1, except for a pole a the origin, then
to those with x > в€’2, except for a pole at в€’1, etc. It is worthwhile
to note that (B.9), together with О“(1) = 1, implies О“(1 + n) = n!,
n в€€ N.
вЂў One can show (compare the exercises below) the following integral
representation for the reciprocal Gamma function, usually referred
to as HankelвЂ™s formula:
1 1
wв€’z ew dw,
= (B.10)
О“(z) 2ПЂi Оі

where Оі is the path of integration from inп¬Ѓnity along the ray arg w =
в€’ПЂ to the unit circle, then around the circle and back to inп¬Ѓnity
along the ray arg w = ПЂ (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 deп¬Ѓne the branch of the,
in general multivalued, function wв€’z according to the speciп¬Ѓcations
of arg w). This integral can be shown to converge for every z в€€ C ,
so the reciprocal Gamma function is holomorphic in the full plane.
This again proves that О“(z) can be analytically continued into the
left half-plane and has poles at all zeros of the reciprocal function вЂ“
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: Deп¬Ѓne
1
(1 в€’ t)О±в€’1 tОІв€’1 dt,
B(О±, ОІ) = Re О± > 0, Re ОІ > 0.
0

To evaluate this integral, observe that by a simple change of variable we
obtain
u
(u в€’ t)О±в€’1 tОІв€’1 dt, u в€€ C.
uО±+ОІв€’1 B(О±, ОІ) =
0
On the other hand, the identity
в€ћ u
в€’u
(u в€’ t)О±в€’1 tОІв€’1 dt du = О“(О±) О“(ОІ)
e
0 0
B.3 Holomorphic Continuation 229

follows by interchanging the order of integration and using the deп¬Ѓnition of
the Gamma function. This, together with the integral for О“(О± + ОІ), implies

О“(О±)О“(ОІ)
B(О±, ОІ) = , Re О± > 0, Re ОІ > 0. (B.11)
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