AN ABSTRACT OF THE THESIS OF
DOROTHY MARGUERITE RIVERA for the DOCTOR OF PHILOSOPHY
(Degree)
(Name)
MATHEMATICS
in
presented on
May 24, 1972
(Date)
(Major)
Title:
ON A GENERALIZATION OF THE LERCH ZETA FUNCTION
Signature redacted for privacy.
Abstract approved:
Fritz 0 erhettinger
Application of a Mellin transform to a series which
represents a generalization of the Lerch zeta function
yields a transformation series.
One obtains the asymp-
totic behavior of the series, together with some associated expansions and limit relations
and moreover,
specialization of parameters yields several classical
results.
® 1972
DOROTHY MARGUERITE RIVERA
ALL RIGHTS RESERVED
ON A GENERALIZATION OF THE LERCH ZETA FUNCTION
by
DOROTHY MARGUERITE RIVERA
A THESIS
submitted to
OREGON STATE UNIVERSITY
in partial fulfillment of
the requirements for the
degree of
DOCTOR OF PHILOSOPHY
June 1973
APPROVED:
Signature redacted for privacy.
C))/L/e\J?
Professor of Mathematics
in charge of major
Signature redacted for privacy.
Acta.
hairman of Department of Mathematics
Signature redacted for privacy.
Dean of Graduate School
Date thesis is presented
May 24, 1972
Typed by Linda Knuth for
Dorothy Marguerite Rivera
ACKNOWLEDGMENT
I wish to express my sincere gratitude to Professor
Fritz Oberhettinger for his advice, encouragement, and
guidance in the preparation of this thesis.
Research for this thesis was supported in part by
a National Science Foundation Science Faculty Fellowship.
TABLE OF CONTENTS
Page
Chapter
INTRODUCTION
1
A SUMMATION FORMULA INVOLVING THE
MELLIN TRANSFORM
7
A GENERALIZATION OF LERCH'S
ZETA FUNCTION
15
HURWITZ'S SERIES AND LERCHI
TRANSFORMATION FORMULA
24
BIBLIOGRAPHY
29
APPENDICES
Appendix I
32
Appendix II
Appendix III
36
39
ON A GENERALIZATION OF THE LERCH ZETA FUNCTION
CHAPTER I
INTRODUCTION
The transcendental function defined by the power
series
00
a) =
0(z, s
(1)
(a + n)
z
n
n=
valid for
IzI
1
0, -1, -2, ...
a
and
arbitrary
s
has been the subject of a large number of investigations.
For
z = 1
Re s > 1
the restriction
is necessary and
CO
(2)
(n + a)-s =
0(1, s, a) =
(s, a)
Res > 1
n=
is the Hurwitz zeta function.
(3)
0(z, s, a) = zm 0(z, s
The difference equation
(a + n -szn,
m + a) +
n=
m =1, 2,
which is a consequence of (1)
restrict the parameter
main properties of (1)
a
3,
...
makes it sufficient t
such that
0 < Re a 1
1.
The
(for a short condensation see [7,
2
vol. 1, p. 27]) were investigated by Lipschitz [16],
Lerch [13]
[la]
Hardy
and Barnes [3].
Important results
of the above mentioned contributions are:
admits the transfor-
(set z =e-T)
The series (1)
mation into a Fourier series [13]
Co
(4)0(eT
,
s, a) =
>
(a + n)se
-nT
n=0
CO
i2ffna
i2Trn)s-1
T
or Res < 0, 0 < a < 1
Re s < 1, 0< a < 1
which leads to the Lerch transformation formula
(5)
(e_T, s, a) - eaTF(l
-
s)T s-1
= -±(210s-lr (1 - s)eaT
i(27ra +
s)
.
0(el2Tra
-
- s, 1-+
2Tra.
-e
or, if we set
+
T=
It
)
log
0 (e-i2Tra,
1
(7)
1 _ s, 1 - 2;1
(6)
0(z, s, a) = i(27)5- r(1
(e-i2Tra ,
-e
iff(2 + 2a)
s
-a
z
st log
27Ti
,
0(ei2Tra,
z
- Sp 1 - log
2Tri
Hardy's relation
(7)
lim[0(z, s a) - z-ar(l - s)(log
j
z+1
Os a)
at z = 1,
gives the character of the singularity of
which is the only singular point in the finite part of
the z-plane.
4)
Barnes's result.
The function
0(z, s, a) -
z1-a cD(z,
s, 1)
has no singularities in the finite part of the z-plane,
except the singularity due to z 1-a at the origin, and
admits the expansion
(6)
(1)(z, s, a
1-a
z)k
= z -a > ' (log
lc!
0(z, s,
[(s
- k, 1)1,
k, a)
k=0
convergent for
'log zi
<
2ff.
4
In all these results the analysis employed is based on
the conversion of the infinite series (1) into a loop
integral, a subsequent deformation of the loop, and
finally the evaluation of the integral by the residue
theorem.
In more recent contributions a different
Apostol [1] proved (6) by a
approach has been chosen.
method based on the transformation theory of the theta
functions and in a further paper [2] it was shown that
Hurwitz's series for
C(s, a),
CO
ru -
Os, a) =-2(270
(9)
s-1sin(2ffna +
2
s),
n=1
0 <a <1, Re s <1 or
0 < a < 1, Res <0
This procedure, however, made
can be obtained from (6).
a rearrangement of (1) necessary, since (9) cannot be
obtained from (6) by putting
z = 1.
It was shown later
in a paper by Oberhettinger [20] that the transition
z
1
in (6), or
T
0
in (5),
Hardy's limit relation (7).
In
can be made utilizing
addition, a new proof of
(5) and (7) was given applying the summation formulas of
Poisson and Plana respectively.
in (1) leads to a function
The special case
a = 1
5
n
F(z, s) = z0(z, s, 1) =
-s n
z ,
1
zl
<
1,
n=1
which is also known as Jonqufere's function [17
p. 33],
whose properties have been extensively investigated [11,
12, 19, 21, 23, 26, 27].
It has been widely overlooked,
however, that as a special case of
cerning
F(z
s)
0
many results con-
are simple consequences of Lerch's
transformation formula (6).
Furthermore, a large number
of functions occurring in various branches of physics,
for example the Fermi-Dirac functions [4, 18, 22], the
Bose-Einstein functions [5], the Hahn functions [9], and
a number of others [14] are likewise special cases of (1).
In view of the fact that the various results
0
concerning
have been obtained by rather different methods, an
attempt is made in this thesis to present a More direct
and unified approach to this important transcendental
function.
In addition a
generalization
of (1) is obtained.
This is achieved by the use of a particular summation formula which is based on the Mellin transform and its asymptotic properties.
These general considerations are the
subject of Chapter II.
In Chapter IlIwe apply these
results to a simple special case.
This leads to the
transformation of a function defined by a generalization
of (1) containing four parameters.
The result is that
6
CO
(a +
T(T, v, a, 6) =
n)-v
e-T(n+a)6
n=0
cS
>0
Re
T>
can be transformed into the convergent or asymptotic
(depending on
6) series
1 r/1 - V
v-1
- 6k, a)Tk
(-1)k
.
k=0
Since for
6 = 1,
Co
(12)
T(T, v, a, 1) =
(a
+ n)
-v eT (n+a)
n=0
-Ta-T
(1)(e.
,
v, a),
the series (11) includes a generalization of Barnes's
series (8) and Hardy's relation (7).
In fact all the
relations (4) - (9) can be derived from (11) upon specializing parameters.
This is carried out in Chapter IV.
7
CHAPTER II
A SUMMATION FORMULA INVOLVING THE MELLIN TRANSFORM
Consider the two-sided Laplace transform of a function
and its inversion formula
F(t)
00
g(s) =
-
fF(t).est dt
-co
cf+ico
F(t) -
With
x = e
-t
and
g(s)esds
47T1
one has the equiva-
F(-log x) = f(x)
lent Mellin transform of a function
.
f(x)
and its inver-
sion formula
co
g(s)
jr
f(x)xS-1 x
0
cr+1.03
f(x) =
1
jr
g(s)x s
a-ico
The theory of this type of an integral transform is well
developed [6].
We note here a few simple properties:
8
A sufficient condition for the existence of (15),
is that the integral (13) be absolutely convergent.
function of
and
S
represents an analytic
g(s)
The transform function
in a strip
al < Re s < a2,
al
are the abscissas of ordinary convergence of
a2
The strip of analyticity may extend
the integral (15).
Re s.
to infinity in one or both directions of
latter case,
al = a2,
only.
where
represents an entire function.
g(s)
then
In the
If
is defined on the line Re s = al = a2
g(s)
We denote the abscissas of absolute conver-
gence of the integral (15) by
constant
such that
01, 82 (al
<
82).
The
in the inversion integral (16) is chosen
a
(31
<q
<
a
2
that is, the path of integra-
.
'
tion in (16), consisting of a straight line parallel to
the imaginary s-axis at a distance
a,
lies in the strip
of absolute convergence.
We consider now a function
x >
0
f(x)
such that (15) and (16) hold.
we form the series
s=
f(n + a).
n=
Then by (16)
of a real variable
With this function
9
1
S =
g(s)(n + a
sds,
2'rri
0-i00
Max(1,
<
<2
'
co
where
and
Again,
jrf(x)xs-1 x.
g(s)
are
0
the abscissas of absolute convergence.
Interchanging
the order of summation and integration, we have the
n + a
series
which is Hurwitz's zeta function
[7, vol. 1, p. 24]
00
(n + a
C(s, a) =
(18)
This series is absolutely convergent for
Moreover,
c(s
-s
Re s > 1.
1
a) - -----
is an entire function of
f(n + a) -
1
s-
1
We have then
c+ico
(19)
S =
C(s, a)g(s)ds
c-ico
c+1.00
=
3.
h(s)ds
77 717
c-i00
where
s.
10
h(s) =
(20)
s, a)g(s).
Formula (19) represents the transformation of the left
side series into a complex integral, the path of integration being of
the type in (16).
It may be noted that
(19) is a generalization of a formula given by Ferrar
[8] (see also [25, p. 63]).
Of special interest is the
case when the Mellin transform
of the function
g(s)
involved in the summation is such that it is analy-
f(x)
tic in a half-plane
to the left of the path
Re s < c
of integration except for a finite or infinite number of
singularities of one-valued character.
holds then for the function
h(s).
The same behavior
This suggests the
possibility of completing the path of integration in (19)
to a closed path by adding a suitable simple contour
such that all or part of the singularities of
Re s < c
lie inside the closed contour.
h(s)
C
in
The application
of the residue theorem combined with a suitable behavior
of
h(s)
may then effect the transformation of the left
side series in (19) into a so-called residue series for
the integral at the right side.
This residue series may
turn out to be finite, in which case a summation of the
series
f(n
4-
a)
has been effected.
In the case of
a convergent infinite series, this procedure leads to a
11
transformation of two infinite series into each other.
Of special importance is the case in which
a parameter
z,
h(s)
say, such that
contains
h(s)
as given by (20)
is of the form
h(s) =
where
cp(s)
-scp(s),
is independent of
z.
Then (under certain
conditions) an asymptotic series of the sum
for small positive
$
in (19)
can be given and this series, if
z
convergent, represents the exact result.
This is expressed
in the following theorem [6, vol. 2, p. 115]:
Theorem.
Let
c+i=
(21)
(z)
-
1
=
fzs cp(s)ds,
c-i=
such that
(a)
cp(s)
is analytic in a left half-plane
Re s < c
except for singular points of
one-valued character (poles or essential
singularities)
X0,X1,X2,...;C
> Re X0 > Re Xi >
-
The principal part of the Laurent expansion
of
=P(s)
at
Xv
has the form
12
(22)
-
b(v)
1
X
+ b
-1
(v)
(v)-r
(s
X
-2
(s -
+ b2
)
+
+
+
)
rv
14)(x + iy)
as
0
1Y1
03
Between two singular points
xv+1
Re X
there is a
v+1
<
(3.
v
< c,
co <
In each strip of finite width
uniformly in
Xv
and
(real) with
(3v
(v = 0, 1, 2, ...)
< Re Xv,
such that the integral
00
ji
+ iy)dy
z-i17(4)(13
converges uniformly for
0 < z < Zv
.
[This is the case, for example, if
cp(s) = cp(x + iY) =
x
0(1 11
for fixed
with Rea < O.]
Then
c+ioo
1
771 Jr
z-s:P(s)ds = (1)(z)
c-iconverges for
0 < z < Z
and
x.
13
(v)
(23)
b1
=
(1)(z)
+
1!
+
+
(-log z)
-A
r.-1
+
(-log z
-.4
+
f
21-1
as
+ico
-s
z
y(s)ds
(through positive values),
0
where the last term is
o(z
We have then the
).
following asymptotic expansion for
00
(24)
(D(z):
b(v)
b(v) +
(D(z)
1
2
1.
(-log z) +
+
v=
b(v)
r,-1
rv
1
r
as
z
0
(-log z)
+
-xv
(through positive values).
14
s-plane
Figure 1.
The first term
in the series (24) is
(v = 0)
due to a shift to the left of the line of integration
from
c - jco
pass between
residue of
to
c +
ico
X
and
X
0
z-scP(s)
at
parallel to itself such as to
under consideration of the
1
s = X0.
The following terms
result from a repeated operation of the same kind.
A
similar theorem holds for the asymptotics of an integral
of the same kind as (21) for the case
the singularities of
Re s > c
cp(s)
z + 00
provided
are in a right half-plane
[6, vol. 2, p. 115].
15
CHAPTER III
A GENERALIZATION OF LERCH'S ZETA FUNCTION
The analysis outlined in Chapter II is now applied
to a special case.
This procedure will lead to expansion
formulas concerning a function which represents a generalSpecialization of
ization of the Lerch zeta function.
certain parameters will yield previously known results.
We apply now (19) to the function
6
-V
-TX
f(x) =x e
(25)
Here
6
> 0
real and positive.
T
t
and we choose, for the time being
The analytic continuation to complex
will be obvious later [See formula (34)].
op
1
tT
g(s)
s-v)-1
e
T
Then from
x6 = t
(15) and (25), with the substitution
-it
dt
.
> 0
.
This is the gamma function integral and
1
7(v-s)
(26)
to be
g(s) =
1
u
6
Res > Re V,
r
(s
T
6
v-------
> 0,
16
= Re v,
The abscissas of absolute convergence are
12,
2
= 00.
(25) and (26) we have
By (19),
-T(n+a)
71(n + a
(27)
0
c+ico
v
s
s 2fficS
by
s
Ss,
a)ds,
Ciw
c >
Replacing
V)(
(l, Re v )
.
one obtains
.00
-T(n+a)
(n + a
(28)
n=0
v
T
T
= 77TE
c+ic.
icig. T5 c(6s,
c >
1
a)r(s
T) s,
Max(1, Re v)
This integral is of the form (21).
uate the integrals (27) and (28).
We proceed now to evalFirst we make use of (24)
in order to establish the asymptotic behavior of (28) for
small
T.
Having done this, we use (27) for a more de-
tailed investigation of the possible convergence of the
asymptotic expansion obtained by the previously mentioned
theorem concerning an integral of the form (21).
(28) and (21), we have
z=T
and
Comparing
17
(29)
= TT
cp(s)
a)r(s -
(6s,
.
6
It is easily verified that all conditions for
[see (3),
(9),
(4),
arities of
hold
The only singular-
(10) Appendix I].
in the finite s-plane are poles of the
cp(s)
first order.
cp(s)
Hence only the coefficients
(v)
are
bI
different from zero; that is, in the expansion (24) only
the first term inside the summation sign gives a contribution.
c(z
Since
and since
r(z)
(k = 0, 1, 2, ...),
a) -
is an entire function of
(-1)k
1
k!
+ k
z
is analytic near
z =
it follows that for the set of sin-
gularities (all of them simple poles)
r(
T
b(0)
T
1
V
k, blk)
=
=
7(v -
v)
OsS, a,
[due to
1,k
Sk, a)'-''
lc!
[due to
-
-1-')]
Therefore, by (24) and (28)
CO
(31)
-ve-T(1+a)6
(n
n=0
v-1
T
r1
- v
-6
)T
c(v
+
k=0
Sk,
a)Tk
18
Now the series on the left side of (31) represents a
function of
analytic in
T
Re
T
The possible
> 0.
convergence of the power series in
on the right side
T
depends on the behavior of its coefficients for large
indices
We find from (5) and (1), Appendix I, that
k.
for large
k
and fixed
1
k!
(32)
- 6k, a) =
v, 6, a
k
0[(27')-6ke-k(1-6)log
the following result:
We obtain then
The function
00
(33)
T(T, v, a, 6) =
(n + a)
-v
e
-T(n+a)6
n=
6
0, -1, 2,
> 0, a
is an analytic function of
T
in
.
Re
T
By (32)
> 0.
one has
V-1
(34)
T(T, V,
a, 6) - 1r(1
00
(-1)k
- 6k, a Tk
k!
k=0
wherewhere the series is convergent for all
convergent for
ITI < 2,
if
6 = 1.
T
If
if
6
6
and
< 1
> 1,
the
series is divergent and in this case the series on the
right in (34) represents an asymptotic expansion of the
19
function of
T
Putting
0.
T
we can write (34) as follows.
= log(.),
Z(n
(35)
on the left as
T
v-1
6
+ a)-vz(n+a)
1(log 11)
r(1
-
=
6
n=0
ov -
(log z
kl
6k, a)
k=0
The series on the right is valid for all
0< 6
For
< 1.
tically as
6
> 1,
For
z + 1.
in (35) converges for
if
0
z
the relation (35) holds asympto6 = 1,
'log z
<
the series on the right
27r
Thus, we obtain
.
the limit relations
v-l(36)
(n + a)-ve-T(n+a)6
lim
T+0
OV,
(37)
1 6
a)
n + a -vz(n+a)
lim
-
1
(v
,
-T-
r 16
z+1
=
),
vaLW,og
a)
we have
For
6 = 1,
'P(-r,
v, a, 1) =
(n + a -ve-T(n+a) = e
-T4
e
-T
v, a)
,
20
CO
T(log
1
,
>,(n
v, a, 1)
+ a)-vzn+a = za 4)(z, V, a)
.
Consequently,
(38)
(I)(e-T
r(l -
v, a) - e4T
v)Tv-1 =
00
(-1)x
Ta
a) Tk
--
--FTk.0
iTI
(I)(z,
(39)
27T
<
a
v, a) -
-
ou
1og v1=
03
(log z)
= z
a)
k!
k=0
flog zl
<
2Tr
(The domain of convergence for this series is
discussed in Appendix III.)
and
(40)
lim
n + a)
-ve-nT
rti -
v, a)
,
T+0
(41)
lim
Z(n
(n
+ a)-vzn - z-ar(1 - \)) (log 1)v-1 =
v, s).
The last two equations represent Hardy's limit relation
(7), while (38) and (39) lead to Barnes's expansion (8).
21
It is worthwhile to note that the integral occurring
in (27) can also be evaluated directly without recourse
to the general theorem, as reflected by equations (21) (24).
The integrand in (27)
_s
h(s)
(42)
=
t6
s = 1
has poles of the first order at
(k = 0, 1, 2,
...).
a)
(s
F
and
s = v - Sk,
Their respective residues are equal
to
T
(-1)
k!
1 t-----)
S
k-
k
6T
at s=l
ov _ kS
and
a)
s = v
.
We consider now the integral
(43)
jrh(s)ds
taken over the closed contour as indicated in Figure 2,
Cc > Max(1, Re v)].
22
s -plane
c+iRN
v-(N+1)6
v-26 v-6
V -N6
v
Figure 2.
The radius
of the half-circle in the left half-plane
RN
is chosen such that it separates the poles
and
s = v
to infinity.
(N
1)6
of
h(s).
s = v - N6
We then let
RN
tend
The contributions to the integral along
the horizontal lines vanish as
This follows
23
from the behavior of
due to (3),
h(s)
(10),
(4),
Appendix I.
On the circumference of the half-circle
S = Re iT
We have for large
R
by (1),
(7),
(8),
Appendix I,
_v
expER(
ih(s)1 = 0 {R
(44)
1)cos T log R +
1
1
+ R cos T(log 2rr - T log T)
/1
1)sin T -
RT(T
T log 6)]}
- R cos Tk- - 1
7T
<
CP
<
2
cos T < 0,
In this interval
exp[11(- - 1)cos T log 11]
6
sin T > 0
is the dominant one provided
That is to say, the contribution to the integral
1.
(43) along the half-circle vanishes as
of
T
and the term
if
6
< 1.
If
half-circle vanishes if
6 = 1,
T < 2ff
RN 4-
> 1
(this is the condition
leads to an exponential increase of
half-circle as
R
increases.
regardless
the contribution along the
for the convergence of the residue series).
6
co
Thus
The case
h(s)
on the
the residue theorem
leads exactly to our formerly obtained result (34), where
in addition the asymptotic character of the residue series
in the case
6
> 1
was stated.
24
CHAPTER IV
HURWITZ'S SERIES AND LERCH'S TRANSFORMATION FORMULA
Let us apply now the results obtained in Chapter III
to two special cases.
series (9).
a = 1,
and
First we wish to derive Hurwitz's
In (34) we set
(5 = 1.
T =
< 1),
(0 <
i27
Then
11-7N
eT2TriS(n+1)(n + 1)-v =
+
-
n=0
,n
(-1)n
e
----- (2113)
2
- n)
n!
.
n=
This relation is valid for
0 <
Re v > 1,
0
< 1.
Multi-
++
plying the above relation first by
2
e
,
then by
and adding the results, we obtain
CO
(n + 1)-vcos 2[(n + 1)S - i v]
n=0
(21-a)v-1sinorami-v) +
Re v > 1,
(-1)
(2,T0)11(v-n)cos[ (v+h)],
0<
< 1
nt-
.
25
If the series on the right is split into even and odd
terms, it follows (with
sin(nv)F(1 - v) = r7rv
co
n vcos(271-n8 -
7n v) -
)
that
v-1
ff(27113)
F(v)
n=1
00
+ cos(
v)
(-un
(2n)!
-
(2")2n C(V
2n) +
n=
00
+ sin(
2
(-1)n
(2n + I)!
v)
(2nfi)2n+1C(v - 2n - 1),
n=
Re v > 1,
If
v
is replaced by
0 <
1 - s,
< 1
.
we obtain by means of (5),
Appendix II,
00
s-1
nsin(21110
(45)
1 (2101-S C(S,
r(1 -
2
s)
which is Hurwitz's series (9).
We proceed now to derive Lerch's transformation
formula (5).
mula (38).
For this purpose we use the expansion forWe divide (38) by
r(l -
v)
and replace the
Hurwitz zeta function in the series at the right by the
just derived expression (45).
We have
26
-k
(-1)k
a)
ru - v + k) (-1)
mr(1 - v)
2(2ff)v-1(27T)-k
r(1- v)
kl
00
nv-1n-k sin(2Trna +
2
v-
2
k) .
We observe that (binomial coefficients)
(-1) k
r(1 - v + k)
k!F(1 - v)
Then (interchanging the order of summation),
7
k, a) Tk ..2(21)v-1
(-
ru - v)
k=0
- 1)
[k=0
n=1
2Trn
.
sin(27rna +
v-
k)
Collecting the even and odd terms of this sum,
00
(-1)k
k!
k=0-
- k a) k
ru
v)
T
00
=
v-1 sin(2Trna + 7IT v)
.
2(21.)v-
k=0
n=1
(\27<l )
)
2Trn
(T
00
n v-1 cos(2Trna + 7
Tr
n=1
(2k:1)(-1)k(7k)2k4.1
2k
27
We use now the elementary formulas
00
f_ilkta \z2k
1
k2k)
2
[(1
+ iz)a + (1- iz)a]
k=0
co
,
(-1)k(2k+1)z2k+1
i[(1 + iz)a - (1 - iz)a]
=
k=0
from which it follows that
1
2
.
1(1 -
)a e
i6
+
1
2
i(1 + iz)ae-i6 =
00
= sin
2
(k)
(-1)
k
a
z2k - cos
2k+1
a.
k=0
k=0
so that
Co
cc
(-1)
k!
k=0
=
Therefore,
t (.\) - k, a)
r(1 - v)
k
T
(-1)kz2k+1
28
(I)(e
-T
,
v, a) - r(1-V)e Ta Tv-1
,
00
e_2)
i(2ffnairv)
e
1-v
n=1
(n +
-i(21)v-1eTar(1-v)
(n
,....
lir)
lr-v
which is (5) written in explicit form (see also [20,
equation 11]).
Re v < 0,
The conditions given in [20],
0 < a < 1,
0 < Im
T
<
2ff
are sufficient to justify interchanging the order of
summation carried out previously,
29
BIBLIOGRAPHY
On the Lerch zeta function. Pacific
Apostol, T.M.
Journal of Mathematics 1:161-167. 1951.
Remark on the Hurwitz zeta function.
Apostol, T.M.
Proceedings of the American Mathematical Soc1951.
iety 2:690-693.
On certain functions defined by Taylor
Barnes, E.W.
series of finite radius of convergence. Proceedings of the London Mathematical Society
1906.
4:284-316.
Applied
The Fermi-Dirac integral.
Dingle, R.B.
Scientific Research 6:225-239. 1957.
Dingle, R.B.
The Bose-Einstein integral. Applied
Scientific Research 6:240-244. 1957.
Handbuch der Laplace Transformation.
Doetsch, G.
Basel, Birkhauser, 1960-1965. 3 vols.
Erdelyi, A. et al. Higher transcendental functions.
3 vols.
New York, McGraw-Hill, 1953-1954.
Summation formulae and their relations
Ferrar, W.L.
Compositio mathematica
to Dirichlet series.
1935.
1:344-360.
Hahn, W.C. A new method for the calculation of cavity
Journal of Applied Physics 12:62resonators.
65.
1941.
Hardy, G.H. Method for determining the behavior of
certain classes of power series near a singular
point on the circle of convergence. Proceedings
of the London Mathematical Society 3:381-389.
1905.
Kuttner, B. On discontinuous Riesz means of type n.
Journal of the London Mathematical Society 37:
354-364.
1962.
12.
Lawden, D.F.
nrzn
The function
and associated
n=1
polynomials.
Proceedings of the Cambridge
30
1951.
Philosophical Society 47:309-314.
13.
Lerch
M.
Note sur,la fonction
ei2Trkx
.
k=
K(w, x, s) =
Acta Mathematica 11:19-24. 1887.
(w+k)s
Dilogarithms and associated functions.
Lewin, L.
353 p.
London, MacDonald, 1955.
LindelOf, E.
Le calcul des R6sidus.
Paris, 1905. 141p.
Lipschitz, R. Untersuchung einer aus vier Elementen
gebildeten Reihe. Journal fiir die Reine und
1857.
Angewandte Mathematik 54:313-328.
Magnus W., Oberhettinger, F. and R.P; Soni. Formulas
and theorems for the special functions of mathe508 p.
matical physics. Berlin, Springer, 1966.
McDougall, J. and E.C. Stoner. The computation of
Philosophical TransacFermi-Dirac functions.
tions of the Royal Society of London ser. A,
237:67-104.
1939.
Miesner, W. and E. Wirsing.
On the zeros of
00
(n + 1)k zn .
Journal of the London Mathe-
matical Society 40421-424.
1965.
Oberhettinger, F. Note on the Lerch zeta function.
1956.
Pacific Journal of Mathematics 6:117-120.
On the modulus of power series of
Peyerimhoff, A.
a certain type.
Journal of the London Mathe1965.
matical Society 40:260-261.
Rhodes, P. Fermi-Dirac functions of integral order.
Proceedings of the Royal Society of London ser.
1950.
A, 204:396-405.
Sur l'equivalence de certaines melthodes
Riesz, M.
de sommation. Proceedings of the London Mathe1924.
matical Society 22:412-419.
31
The theory of the Riemann zeta
Titchmarsh, B.C.
346 p.
Oxford, Clarendon, 1951.
function.
Introduction to the theory of
Titchmarsh, E.C.
Fourier integrals. 2nd ed., Oxford, 1948.
394 p.
On a function which occurs in the
Truesdell, C.A.
theory of the structure of polymers. Annals of
1945.
Mathematics 46:144-157.
Wirtinger, W. Ober eine besondere Dirichletsche
Journal fiir die Reine und Angewandte
Reihe.
1905.
Mathematik 129:214-219.
APPENDICES
32
APPENDIX I
Asymptotic Behavior of
and
r(s)
The asymptotic behavior of certain analytic functions
is needed for the evaluation of the integral in (27) or
This concerns mainly the
(28) as outlined in Chapter III.
functions
that if
g(s)
g(s), r(s)
g(s)
0
as
First we notice
Os, a).
and
Im s
Re s
for fixed
:boo
Os, a)
transition as
Let
R -4-
00
Isl
(in the strip
With regard to
Lebesgue lemma for Fourier integrals.
and
then
This follows by the Riemann-
of absolute convergence).
F(s)
f(x),
is the Mellin tramsform of
we consider two possibilities of the
c°-
s = x +iy = Re',
arg s,
and let
a < arg s <
in a given sector
Let s= x + iy
and let
y
for fixed
±-=
x.
We have (Stirling's formula [7, vol. 1, p. 47])
1
(1)
r(s) =
r(s + a)
(2)
2n7
s
e
-s
a B
es
log
s[1
[1 + O(!.)],
+
(2s--.)]
;
fixed)
(a,
.
+ (3)
Both formulas are valid as
± 00
in
-Tr
< arg s
<7
.
33
Also
1
(3)
iy)I =
Ir(x
as
1T
0(1y1X-2- e-71y1\
y + 0°
for fixed
x.
C(s, a)
As for the asymptotic behavior of
large
s
for
(0 < a < 1), it is clear (for instance from the
integral expression [17, p, 23] or from Hurwitz's series
(9)) that the dependency on the parameter
gible importance.
of
C(s, a)
a
is of negli-
That is to say, the asymptotic behavior
for large
can be replaced by that of
s
CO
C(s, 1)
= a(s).
From the definition of
C(s) =
In-s
1
1
(Re s > 1) as a Dirichlet series, it is obvious that
Ic(s)1
= 0(1)
Re s > 1
.
Using (4) and the well-known functional equation
C(s) = 2(27)s-i51n(i s)r(1 - s)(1 - s)
ff(27r)s-lc(1
-
s)
cos(i s)r(s)
with
s = x + iy,
1r(s)C(s)1
the result is
ilYI
ex log(21)
,
Re s <
34
[The equal sign in (4) and consequently in (6)
follows by
continuity
if one considers (10)].
Applying (1) to (5), we obtain
1
'--
R
ev(R,cP)
where,
= - R
v(R,
log
R cos cp + R cos CP
log(27) - (i - cOR sin cp
+ R cos cp
as
R
(the validity
in
0.
in
Z
<
cp
<7
and
<
'*-7
CP
<
the latter interval follows by Schwartz's
reflection principle).
Similarly,
x)(9)
(3) applied to (6) gives
k(s) I
as
Finally, in
1(x ± iy)I = 0(IyI2
=
1
y
0.
x <0.
and
the critical strip
0 < x < 1
p. 82], we have
/
(10)
(x
as
iy)
÷ 00
= 0
and
11
IY17-r)
0 < x < 1.
[24,
35
It is not difficult to show (see the integral
expression [7, vol. 1, p. 26] or use Hurwitz's series for
(s, a)) that the same asymptotic relations hold also for
Os,
a).
That is to say, in the asymptotic sense the
terms involving
a
in
Os,
a)
are of a lower order
compared with the dominant part which is independent
of
a.
36
APPENDIX II
Os,
A Taylor Series Expansion for
a)
It follows from
00
Os, a) =
Re s > 1
(n + a)-5
that
n
Os,
(-1)n r(n + s)
a)
r (s)
r(n + s)
.
-'a=1
Hence, by Taylor's theorem
CO
(1)
(-1 n (a - 1)nr(n + s)(n + s)
)
r(s)C(s, a) =
n!
1
-
if
< 1,
1
s
or, with Riemann's functional equation
s - n)
(210s+n c(1
r(n + s)On + s)
22=
cos[(s
+ n)]
2
(2)
r(s)(s, a) =
Co
1.(21)s-1 y (-1)n [27(a - Inn
n!
cos[-(s + n)]
2
0
-
11
<1,
s
1
.
c(1 - s - n)
37
If (2) is split into even and odd terms in the summation,
the result is
r(s)c(s,
(3)
a) =
(-1)n [2ff(a - 1)
= Tr(21.)
(2n)!
COS
CO
(-1)n
+
,
la - 11
(72- s)
(-
<1,
1
s
S
2n)
.
we obtain
sin(i s) cos(- s),
c(s
(4)
s - 2n) +
sin (-2 s)
0
Upon multiplying by
(1
Tf
1)]2n+1
[27(a -
(2n + 1)!
]2n
a) =
03
2(27r)1r
(-1)n
sin (12- s
1-s
1
(2n)!
[2ff(a-1)]2 C(1-s-2n) +
00
(-1)fl
+ cos (i s)/
1
(2n+1)!
[2ff(a-1)_]
2n+1
C(-s-2n)
0
la - 11
or, with
a = 1 + a
and
< 1,
c(s,
S
+ a
1
;
= Os,
a)
-
a-s
38
Os, a)
(5)
- a-s
= Os, 1 + a) =
CO
= 2 (2Tr)
s-1
(1-s
sin(- s
((-2n1))11!
(27m) 2n (1-s-2n) +
00
(-1)n
+ cos
s
(2n+1) !
a
2n+1
(2Tra)
< 1,
s
1
(-s-2
.
39
APPENDIX III
Some Mapping Properties
The transition from (34) to (35) represents a
mapping of the-T.-plane into the z-plane by means of the
mapping function
z = e-
Let
T
+ iy,
= a + it,
x = e-a cos t,
T
then
sin t
Y =
.
in the T-plane is mapped into the
The strip 0< t < 27
entire z-plane-and the half-strip
0 < a
<
is mapped into-the interior of the circle
0 < t < 27
co
I
= 1.
z
Of
particular importance is the mapping of the T.-domain
(ITI
<
27) of the power series in (38) into that
(Ilog zl
< 27) of the corresponding series
T=
pe
1,4)
z =
eicp
(39).
Setting
= rel.°
one obtains
(3)
cos
IzI = r = e-p
=--p sin cp
.
It is sufficient to investigate the mapping of the
upper half-circle
171
<2n
into the z-plane, since the
40
same considerations apply to the lower half-circle.
this purpose we observe the image point
z
as
T
For
tra-
verses the arc ABC of the upper half-circle (Figure 1).
'r-plane
Figure 1.
Then by (3),
= r =
e-271. cos eP
6 = arg z = -2ff sin eP
.
41
It is obvious from (5) that
from
6 = 0
to
0 = -2Tr
decreases monotonically
e
as
describes the arc AB,
T
as
T
Thus, for each
ABC.
ferent values of
such that r> r'
Tr
-27
r.
<
0
0 = 0
increases
r
e2ff
along the entire arc
< 0
there exist two dif-
Denote these values by
and
r'
r = r'
if
r
(the smaller is encountered if
the larger, if
0 < P < 7 ;
P =
to
e
back to
while by (4),
describes the arc BC;
monotonically from
2ff
e =
and increases monotonically from
7 <
cp
<
ir
,
and
Therefore from (4) and (5),
).
(6)
r=e 47r2-02
(7)
r' =e -1/4
<
1
cp< ff)
Tr
(0 < cp <).
These are the polar equations of two curves
in the z-plane.
C
We observe that
r = r'
dr,
dr'
de - au
for
= 0
e = -2ii;
for
0 = 0
.
Furthermore,
dr
de
and
de
+'
as
-27
.
and
C'
42
C
The last observation means that the slope of both
and
C'
is zero at
-2 It.
e
A few numerical examples will serve to illustrate
the situation.
In the following table, we give the cor-
respondence of a point
T
on the arc ABC with its image
z-point as expressed by (5),
and
r'
to
(6),
(7)
(r
belongs to
C
C').
4)
sin cf)
0
(in degrees)
1
7.2
T
14.5
1
4
---1T
1
--Tr
2
3
'Tiff
T
I/175
e2
e2
7T/ST
22.0
eT
30.0
-Tr
e1TV
5
38,7
3
T
48.6
4
3
2
e7
eT
T
61.1
7
--Tr
4
1
90
-21'
7
T
iiTT
e
--g
'TA-5
eT
--Tr
4
5
-21T
eT
1
7
e
e27T
0
T
1
r' = 17
r
-27r sin cP
ffiT
ffirT
1
-21A-75
-LiTT
e4
e--rri/T
e
iliTT
e7
IT/TT
e
1
43
We see that
rmax
r'
max
Thus, while
= e
2ff
% 530,
= 1
r'
min
e=
-27r
=
1;
0.0019
.
traverses the arc AB, the image point
T
traverses a spiral-like curve
starting at
tion
rmin =
8 = 0
C'
z
in the negative direc-
with rin and terminating at
As T proceeds along the
m ax = 1.
arc BC the z-point travels along an "outer" Spiral C
-2n
with
r'
in the positive direction, starting at
rmax = e2n
The "inner" spiral
.
0 = 0
and terminating at
rmin = rax
M
= 1
8 =-2n with
keeps (with the
C'
exception of the immediate neighborhood of
close to the origin (for
still only
spiral
C,
rx,
-315°
0
r = 1,
a
-2n)
the value of
0.05), while the values of
starting with
with
r
very
r'
is
for the "outer"
increase rapidly.
This fact is illustrated in Figure 2 (the numbers at
certain points of
C
denote the distance from the origin).
It is, of course, not possible to sketch the "inner"
spiral
C'
because of its relatively very small linear
dimensions and we therefore omit it in Figures 2 and 3.
r'
(Note that
max
rMin
rmax
rm.
e
2n
ix,
530
44
z -plane
-21T < arg z < 0
Figure 2.
Mapping of the upper half-circle of the T plane
into the z-plane.