OPi
sufficient
for
those
the logical
the
starting
point
left class
and the
(ii)
Every
rational
number
x
Theorie
real, number x is then
naturally
defined
by
in any
convenient manner;
that symbolised
we may
of any
fresh type
most general
be
connected
or
amplitude,
or
phase,
of
z.
y
the
real
which
z^
is the
given
an
arbitrary
number
N
(no
matter
7?o
such
that
tend
to
a
limit
numbers
such
that
rational
numbers
into
two
classes,
the
Z-class
x
for
which
positive
number
f,
we
can
choose
no
such
i^-class
of them,
sequence
{xn)'
The
sequence
{x^)
terms of the
and
necessary,
i.e.
that
it
is
satisfied
{xn)
satisfies
Cauchy's
condition
positive
number
e,
an
a;i,
a-g,
1'4,
the sum on the left is gi-eater than or equal to
j
G
of
{x^).
(II)
and
{y^)
exist.
Therefore,
by
converge
positive
number
e,
we
as
of the series
1 1 1
the
n
2
a„
moduli
other
series
to
Thus, given
7n
infinity;
therefore,
by
of moduli of
divergent
to increase the
terms of the
Si',i
equal to, the
^,
circumference
over which
there is
no intercommunication,
n > ni, '
be
^ '
moduli, and therefore is less
than
2
e-
Therefore
>S^^ be the
been
shewn
by
Riemann
2'5.
Double
series\.
Let
all positive
Pringsh&iui,
the
which the
n
columns
of
series should
b}^
J
as a
this
double
series
be
cr.
Then
given
e
we
j)
> m,
q>
convergent
double
columns
be t^^
exists,
and
similarly
the
a^Tn,
by
... are
independent
are
due
to
Cauchy,
Analyse
from d'Alembert's
the
whole
^r-plane.
On
convergence of
S
the same circle
zero.
1
Now let
infinite
—
verify
that
V
mTTJ
\m-/
that the
infinite determinant
[-^«>i-J?,i-=-<»...+oo
to form the principal
k
to
form
the
by l+a^,
denoted
is
der reellen
;
+
of
to
of
nP^'
; and
consequently,
reasoning
as
sufficient
condition
(1897),
pp.
101-152.
of
2sin2s-4sin-»2
(de la
where
 ?»v
number.
(Ceskro.)
values
part is positive, when
integer.
M,i^
w,i
a„2
of a
of lines
is to
^x
^b.
If there exists a
1+
limit
of
tinuity
of
g
(x)
\f(x)
Example.
From
f{x)
f{x)
may
tinuously on X
two
equivalent
real function
f{x),
we
ordinary discontinuities
at certain
is
as
was
shewn
joined by a
called
its
boundary;
is
then
called
a
an
open
one-dimensional
region.
region
then
under
consideration.
If
8
should
p.
indifferent
of the
=
non-uniformly
convergent
sum is
shall
shew
that
i_
negative.
The
series
or
positive.
There
is
conversely. Thus
continuous if
condition
series
*S'
corresponding
terms
in
a
convergent
series
fact that,
moduli of its terms
of
convergence
which
work
is
is
S>0,
convergent series independent
/^. (Hardy.)
n
of Dirichlet and Abel for
convergence,
 
is not
of in
points
z
number
uniformly*
depends
on
the
the sets
this
statement
divided into a
Pr§.
15).
t
Examples
theorem (which is
[CHAP.
Ill
suitable either
terminate
not
satisfy
condition
(-4).
theorem* the
interval CD
find
a
positive
number
S^,
a
proof
may
out
a
process
of
finite
hounded
number
k
independent
of
x
[Let
n
be
the
divided.
Let
l/(.^2)-/(^2)|<if,...
The corresponding
continuous function of x when a^x^b. Form a
section
in
which
the
r
f(x)>a
of
\f{x')g{x')-f{x)g{x)\^\f{:>f)\.\g{x')-g{x)\ +
\g{^)\-\f{^')-fi.^)l
\z\-^\Zq\.
function of
0^a;^l;
convergent
when
I
whenever
In
some
text-books
a
more
elaborate
proof,
sums
We
and Paris,
(Camb. Math. Tracts,
2sini
(Cauchy and Cajori.)
Ill
of
the
resulting
the series
limit of
4 11.
be S, s.
continuous complex
'
longest of
them tends
function.
continuous
the
theory
of
Mat.
(3)
vn.
(1902),
Mat.
(3)
vn.
(1902),
pp.
231-359.
See
the theorem
P
dx
does
not
depend
on
p
that
r
\^
value
theorem,
\fa
{x,
a)
-fa
(x,
a)
I
a function
a continuous function
would
have
x), instead
of assuming
y
b}-
lines
parallel
to
the function
f{x,
„(
=
=
an infinite
;
2
/(m).
/
so does integral, and if the
integral
(§
Oeuvrcs
(2),
vii.
p.
269.
X
Journal
is
uniform
when
a
in any
.
^.
of
improper integral.
neither
Watson's
being
the
same
(but
in
opposite
That
is
con-
vergent
series
of
continuous
functions
considered.
Therefore
if
x
except
to
,r
of
values
;
the sum
of the
to e'^' -,
which is
 
j
x
sin
{x^
unifonuly
in
the
range
(a^?/^6)
times, then
ot
values
of
z
sense
in
which
the
The
reader
z
function
of
z^
we
say
subsequently,
however,
the
elementary
functions
already
discussed)
-^
—
-
—
-
on any
derivate.]
The
above
inequality
is
obviously
starting-point.
We
i.
(1900),
p.
14.
t
It
function
fact that ZqAZBzq
depends
only
is
only
if
Cq
and
Ci
were
is taken
the region
is kept
Annales
in the
the
whenever
integi-als are
is
called
the
derivate
of
Similarly
it
can
possessing
a
derivate
equal
to
2^
f
f(z)dz_
27ri
in
all
uniformly
along
a
»=o
inside
C;
on
C,
let
S
inside its circle of con-
vergence.
531.
Analytic
functions
of integration (a,
J
a
function
region >S' if
satisfied
deals
with
uniformity
of
importance
is
afforded
by
finite
number
M.
Therefore,
by
the
path
of
=
Dr
Brook
Taylor
(1715)
in
his
Methodus
Incrementonim.
=
arithmetic
2
2
cv,
for
positive
constant,
and
(1
polynomial
so
of an
ellipse whose
A
the
smaller
given by
Lagrange for
the case
theorems enable us
and to
convergence, and
used
to
define
series with
us
to
define
the
function
at
other
points,
at
of
it,
starting
from
a
find
any
number
of
point
of
which
curve
PBQ
into
advanced
Math.
Soc.
xxxv.
(1903),
pp.
388-416.
II
impossible to pass.
article,
equivalent
is
1
define
a
of
C
which
C in the sense
+
of which
is
valid
defined
form
Theorem,
if
C
denote
the
circle
a
a
interior
point,
of the
It
singularities.
general, an
bounded
has simple
is
a
one-one
for the function valid
these singularities, the
series converges absolutely,
l)~i
outside
this
(_)fc-ia-2A-»
dz
b'k
obvious that/(/)
fvr
Math,
lxxxviii.
(1880),
pp.
277-310,
little difficulty
in seeing
such a function
of
the
function
has
a
Example.
Prove
that if the different values of a^, corresponding to a given
value
of
z,
be
of a
rational value of x
of
would
appear
to
be
of
of
differential
equation
/'(--)=/(^)-0(4
of
all
the
values
of
the
series
 2
0,^^
V
2
e-2(am)^-m-l
m
(i'
0),
converges absolutely for all values of z except purely imaginary values
; and
(Weierstrass, Berliner
z,
of z
region
and
let
de
277
Hence
jo
1^
.
.
,^
—
,
/
J
on
|
that
as ic—
contour
C
consisting
real axis joining
t
Q(z) has no poles above the real axis outside the contour.
W. M.
A. 8
Q
(x)
dx
upper
half
constant)
and
p,^
and
a
ends
of s be
on the real axis.
8,-^0;
diminish
the
area
indenting
the
contour.
r
and x^Q{x)—¥^
consisting
simple poles on the
that
P
j
a; ~^
Q
(x)
dx
poles
(2)
contained
in
the
interior
origin, we have
of 2.
the
tangent
describes
the
curve
C
real
and
different,
then
between
two
only of/'
describes a
number of
(Amigues,
than n
,
(1827), p.
the formula of Darboux,
be
Conjectandi,
p.
97
(published
posthumously,
1713).
X
An
explicit
expression
for
the
,162,
n(^\,
nth
be seen
from the
by
Bernoulli.
by
Maclaurin
(1732),
[Published
1738],
pp.
68-97,
to
and
such
that,
if
f
Example
1.
Prove
that
where
z
=
to
any
point
p
theorem :
of smother function, whereas we
shall
now
discuss
positive and negative powers
tion
we
shall
follow
an inner curve
equation
f
j_r
e'(z)dz
1
rr
known, A^ and
£2, 42,
G^...{2n-2)%
represented
by
equation is
we write
of
G
sequence
of
circles
we take
if we suppose
(except
equal
moduli.
origin.
But,
at
points
on
C,
\~^)
;
there-
fore
zero.
From
(
—.
—
--„ —
points*
Oi,
tts.
«3,
(§
and
pol6s
oi
SERIES 139
Now let
values of
f(z)
of/(s) as
—
at
y—>+
x
as
«=o
that
of
inverse factorials have been studied in
an
important
the point
e^O
Example
+
the
expansion
1
oc
the ovals
'
certain region of the
exjianded in an infinite
Philosophie de la Technie,
equation
«
=
—
the series
an integer,
+
SERIES 149
22. If
points
0,
x.
and
C.
Let
By
lAy
\AJ
the
expression
Sn-
large, the right-hand member
x
asymptotic expansio
where
e
We
shall
confine
our
for all
as
to
whether
a
is
in
to the
the
numbers
ao,
00
of
convergence
of
outside this
the
coefficients
in
the
S
j
a„
unlimited
number
of
as in
on
|,j.
really be a function of n and might be o
(1)
qua
function
Series Divergentes
to
du
recent
developments
of
»
,
.
particular
has been
involved
are
real.
t
modification
and bounded in the range
(
-
minima and a finite
Hobson's
Functions
of
a
Real
inequality
log
a^-
y
that in which
em-
is
proved.
See
by Abel's
+
-)'^(^''-...\
^
values
of
integer.
value wherever
/
(,/-)
strip
in
is not
Fourier
series
from
to
x
throughout
f{t),
in this
the
Dirichlet's
^ao+
S
(on
will
be
used
in
this
seines.
Let
f(x)
be
1765
by
Lagrange,
range ^x^n.
i
(tt
a
mere
expanded
and
Example
5.
series.
93.
number
fluctuation
and
that
they
left
Now
as
/(a;)
should
be
b,n
series and
+
the
function
f
(f)
the two
fix
number e,
we can
obviously independent
and
lower
bounds
Corollary
is independent
Jo
Jo'
and
the
upper
bound
of
a
finite
|j
number
Fourier
magnitude
of
We
shall
now
prove
the
theorem
permissible
to
+
^
by
luith
i{/(^-
that
function in the
in which
f{t)
proof.
Cours d'
point, the
Then,
by
by
depends only
about
uniformity
or
all
real
values
series by means
9 5.
bound of
\f{x)\
by
giving
can
restrictions
been proved
equal to
the sura
at the
follows
by
=
lim
{(In
left.
e,, ;
therefore
for
±
to
w.
to have other expansions of (say) the
form
oq
^i<x<^.^,
F{x)
is a continuous
the range
;
^1
9632.
Proof
of
Rieiiianns
Theorem.
We
being
uniformly
convergent,
is
=
so we can
This
process
gives
point of the
integer m
so that
I
\
to x
-tt
and
+n
(Berger.)
11.
all real
values of
inside the octahedron bounded
circle of
/'°sin(2n
In
we shall be concerned with the
investigation of extensive and important classes
of functions which satisfy
a
domain
8
equation,
and
let
St
be
the
8^
is
a
point
of
^S',
in
works
devoted
to
the
subject,
For a method
For this
for
=
identical.
Writing
(z) is
that
in
and
^2,
-^3,
find
1
and
be one-valued ; i.e. it may not return to the same
value
when
z
of
to
linear
equations
second
order
are
not
constant
point, the indicial equation
+
equation.
Example.
shew that
the series
justified,
J
of
zero the
the
point
analytic
a
\)z}-^^-ahu
^^S- +4 -^(^'-»')»=°'
equation
has
order
we
=
forms
satisfies
(which
exponents.
Example. Shew that
-1
When
a
1911.)
cos {xt)
is an
§
chapter.
of integral equations
has been investigated
K{x,
theory of
origin.
The
greatest
(numerical)
and
is
obviously
bounded,
the
ordinary theory of maxima and minima is applicable, and if
we consider
2
condition
2
10.
The
be generalised in
generalisations.
Bulletin
the
reasoning
given
below
in
method of justifying
(when
5-*-0)
^x^b, it is
X3
)i
*
The
factorials appear because each determinant of s rows and columns occurs
s
in
equation.
u^oo
M^oo
(71+1)5
fore
r<j>(^)D(w,^;\)d^-r{D{w,y;\)-\D(\)K{x,y)]<j>(y)dy
J
a
J
a
Fredholm's
equation
formulae
2)(X)
bounded in the ranges a-^x,
y
-^1),^
if
Z)
y
reciprocal
functions,
function, we
solutions,
obtained
by
giving
y
various
values,
is
number.
+
<^{x)
equation.
first
Fredholm's
that
Ki
(x,
y)
which
for the range
expression
on
to
y;
are
all
real.
W.
permissible), we
numbers are
(*)
71
equation
than
one
orthogonal
function
to
each
characteristic
number.
X
of
Fredhohns
equation.
continuous
solution
u
(x)
Let
(f)(x)=
I
u
(I)
o?|,
negative the
then,
when
x
in
iDlace
of
\f/,
defined
by
the
,^
Changing the
order of
Equations (Cambridge
J
a
is
0(^)=
2
in
it by means of an infinite product of Weierstrass' canonical
form.
Consider
the
product
ze^^
decimals.]
For, resolving
converge, the number of
denominator,
and
; and thus
we have
n
First
Kind.
integrals
proceed as follows : when
a less precise inequality
{z)
article is
limit,
since
result
(0>.r>-i).
is the
define Pj
question
is
due
to
Hankel*;
this
arg
+
Gamma-functions, and thence
;
frequently
written
by t
z is
e-i *')'
rf«
A=(J
z is
posijiive;
we
may
.
-0,
as
2
V
{z).
useful
real
values
e(^~^)^°°*
and
exceed
taken to be
the regions bounded
=
contour
integral
which
starts
from
P,
encircles
the
point
of the integrand is
pass
over
the
branch
points
0,
1
of
the
integrand,
the
 
r((x,
thatt
r
(x)
r
(h)
+
r)
h
I
sin''
joiniug
line. Let
for
-
 ^
and find the values
...(n-t)'
+
functions
in
Theory
of
Prime
Hurwitz, Zeitschrift fiir
to
Euler's
When a
function.
Also,
being
contour {2NTr
to
-
Consequently
Hermite's
and
A
summary
of
his
memoir
is
contained
in
a
>1.
From
the
formula
been
-*-
When cr>
^{a+[t])^~'^ (a
Since arc tan i/(l
Uiy-^)-^-''''
a uniformly
cr
C{s,a)
We
semicircle of (large) radius
<r
[
^^
|
<
lower
signs
be
taken,
the
function defined
satisfy
the
inequality
the
formula
just
obtained,
we
write
seen
279
1905.)
9,
and
then
making
a
is
Kth term
z
convenient
is an
(a,
a',
a),
(yS,
y8',
h),
(y,
y',
c)
we
make
such
changes
in
the
above
solutions,
they
{a,
the original four
hypergeometric
equation
P' *.
relation
counectiug
Mj
with
«i3,
was
discovered
in
1730
r(—
5),
viz.
s
Oils
5)
F
(1
integer.
Now,
by
this
function
with
centre
of
If
C
be
defined
to
be
the
semicircle
if
p-s-oo
is readily seen
as
the
integrand
at
-f-S-y
any pole
however,
both sides of the equation are analytic functions of, say,
a:
r
(a
''^TTi
J
_x
i
b;
r{s-t)ri-s)(~zYds\-
much
difficulty
that
=
appear. With
2
liypergeometric
function
by
means
+
, and consequently
hypergeometric
equation.
its
JC
•where
C
not depend
the differential
equation defined
of that
we
^=
greater
so
\z
series
series of integer powers of
z
argument z,
be
shew
how
to
obtain
(a
-ta-],v'-l5
unaltered.
t
Pa'-l
of
the
above
functions
above twelve contiguous
as
to
be
established
for
the
rest
/3,
/3',
y,
y
by
relation, the
article.
REFERENCES.
C.
F.
j
the former
x
between
and
1,
the
solution
of
the
equation
is
F{y±\\
following
results :
-(l-)8^+y(l-)5j;5^+{r-(«+e+l)-}5;j'-*^-.^/-,=0,
d^F^ d^F^
integer and
see Barnes,
0<
.r
angle,
 />„(cos5)|<
2.
4.
..2m
r
(In
fin
P< 71+1
taken
round
a
contour
j'
{(^^-ir}^._,
the
integral
(
-
n
which
are
not
after
the
right
the
contour
would
is
Lastly, writing n
we
have
+
of
=
Then it is
^
and
so
case
expected
that
a
formula
can
be
obtained
giving
Pn(z)
in
Substituting
of n.
§
|
some
to
Mr
J.
Hodgkinaon.
Ztt
J
conditions are both
{2^
-I-
§
Pn{z)
a
pure
written
by
considering
2
A
j
{z
range of
straight
line,
square
root
after
describing
it.
second kind
part
of
n
it
is
{z'
-l)Q'n
(2)
range
of
z, are
at the
apparent
from
these
of z.
positive vahies of
and
therefore
if
z
the
substitution
wj=0
the
expansion
of
(1
as
the
basis
of
a
general
Pn^Az)Qn{t)-Pn{z)Qn+x{t)
_\
by two ellipses.
regard
to
If
n
is
f(T3;^)(^)
^(-«...+i;i-».;i-J^);
and
Barnes
has
given
a
definition
of
expression has to
cos
<o
Let
a
that ^±/3' he
 
^'
of
(1896),
pp.
443-531.
Cylinder
that when
n is
1-30
(Peterhouse,
1907.)
I
—\n
ffn
can
be
of
(Glaisher,
17.
r=
a
been
performed.
=
that
.
27r TO r(m-a
of I/2, when
/V
(2)
is
defined
+
1896.)
See
also
n
and
6
—
{r(wp
;,io^~)
r(n
shew
that,
when
—h
<x
<h,
in
a
equation which
given
by
the
series
modified
by
Barnes,
-j^
of
symmetry
various functions
of Applied
negative
integer,
then
^-^--il/,,„(^)
an
integer.
Shew that the formula for the eri'or function is true
for
Wi.,,^_i\ in(x),
;
Soldner. Jahnke und
z)
+
analytic at
(—
integrand
between
the
contours,
and
since
Making
that
the
2Trl
J
W]c,
the
serai-
circle
associated
with
of
^ <
-
xxxv.
(1903),
right is not
has
only
been
given
the range
get
i:„(0)=-2isin(«
(1908),
pp.
253-279.
E.
T.
then
+
and Besso
then
|J
e
and
if
or
negative.
(Watson.)
16.
cosine or sine is
and o-(^)
shew that,
to
1,
o- (/')
decreases
i'=l
\ Jijn)
2.
(AdamofF.)
3-35...
ia;2
otherwise
(1764)
[Published
1766],
pp.
243-260,
xxv.
(1769)
[PubUshed
in 1824 by
welcher
appeared
in
researches
which
is
now
occurs in 1732 in Daniel
Bernoulli's
memoir
that the
Bessel
Jn{z)
linear differential equation
order
n.
For
(§
with regard
which
(B).
z
From
(A)
(C),
and
the
critical cases when 2/; is a negative integer ; when n is half of
a negative
z,
all
differential
coeflficients
of
y
vanish,
and
y
numerically smallest
end of the
been
given
by
continuous
Example
(real
or
that the
(1871),
p.
148.
does
so
when
we
we
have
zero and,
integration,
when R{n)>0,
real,
Ju
hypergeometric
functions.
When
I
arg
' '^'
^ *'^'
(r+l)th
term is
the point
Example
1.
Shew
thatt
lim
8,
of
e/o
(2)
for
Asymptotic expansions for
The
asymptotic
The
reader
(iii)
equation
various
writers,
notably
Macdonald.
The
Eiemann, Ann,
function
Bessel
coefficients,
in
the
form
on the
+
=
e-'
{?<±^/(«2
and
so
diflFerent
character
from
that
of
thus
have
f(x)
=
writers:
alternating
currents
in
wires,
have
been
J'o
Emde,
Funktionentafeln
(Leipzig,
1909).
REFERENCES.
(1896), pp.
6,
express
2**
coefficients.
3.
integer),
integers,
(Math.
Trip.
1893.)
where
z
=
1
5.
Shew
that,
if
form
X
(2),
vii.
(1909),
p.
349
(2),
xviii.
(1920),
pp.
163-200.
The
formal
expansion
w
of
n)
by
Lommel
—
Mag.
(6),
xvi.
(1908),
p.
39. Shew that,
p^-'j'
of z
and integrating
c,
C
whose
centres
are
at
the
origin,
establish
the
expansion
/
result of
positive integer,
Ann.
ill.)
satisfies
the
equation
di/
dz^
(iv) The
homogeneous
mathematical investi-
solid
of
electric
waves
the
equation
satisfied
by
each
component
of
the
electric
or
vibrations,
it
is
the
body
and
inversely
when
to
attempt
an
exhaustive
with
special
of
the
equations
of
con-
ditions
at points
in discussing
being exposed to the
atmosphere ; in this problem we are given, effectively, the velocity potential
at
the
former in
and
difficult*.
form of a
to
be
analytic
at
(xo,
yo,
•S'o).
§§
infinite
+
under
of
Laplace's
equation.
This
;
This
V=f{x-\-iy)+g{x-ii/)
is
the
general
solution
of
ox^
oy-
primitive of
an ordinary
into
every
solution
the
forms
see
equation of zero
Q
I
(z
so
P„
(cos
d)
called
(see
S
mtb
into
quadrangles
a homogeneous
polynomial of
equation.
of rectangular coordinates*
say
a,
of
the
Green's
functions.
For
properties
of
V
(r,
6,
cf))
involving
for V {r,
4>')
cos
(m(f))
deduce that 27rz'
density
can be reduced to Laplace's equation by the change of
variable
r=t/,„
1iTJ{ckj.)
coefficients
(see
the
cylinder
is
dx^
§
c.
The
solution
(ff,
(f)),
and
equation of
This
equation
determines
k;
in which
+
OZ,
Deduce that
lying in
axis of
z, where
function
(Tp
at
IS
zero.
of
masses
X +
the initial
C;P„(cos<9),
Bessel functions.
b,
as
where
A;,
Another proof is
motions is
\
motions, shew that
equations
jr
21,
which
lies
beyond
functions
associated
section,
the
boundary
condition
is
shape
problem
just
discussed.
af-2
u
eccentric angle of
the
decay of magnetic force
of Rigid Dynamics which
of
the
Lunar
Perigee,
xvii.
p.
41.
nature
of
G
{z)
forming
=
Mathieu's
and
yet, been discovered.
of
the
general
solution
where
<f>
(6)
Mathieu's equation
formed with
Mathieu
that mh
in the
gives
rise
to
ce^
{z,
q)
(1
^i(z),
/SjC^),
but
(ii)
19 4.
integral of
not
necessarily
we see
and a
equation or Hill's
u
g
in this
where
<f)
real
before,
00
2
On
converges
absolutely.
«=o
Defining
have
to
§
carry
out.]
On
(,i
»
through
a
remarkably
good
approximation
been obtained
by Lindemann*
xiv
irregular singularity
at x
equation for
TFj._^(3)
has one
+
coefficients
c„,
namely
where
(n
the recurrence
necessary to
^.
right can
be -sNTitten*
§
theorem
is
parameters a and
acquainted
with
the
method
will
work.
Write
a
positive
of cosines, i.e.
f/',^
r=l r=l
above.
where A,,
is the
when
^=0.
On
substituting
for
p
in
inevitable, as
q
complex
that A
fj.
and
a
in
us
to
—
functions
yet
been
determined*.
+
=
=
^,
write
u
q
above
the
Mathieu functions.
sign
in
case,
the
as
confluent
form
2
(1915),
pp.
2-15.
the
series
provided
that
function of
 
whose ratio*
is not
f(z)
analytic
no
singularities
other
in
the
theory
of
elliptic
by
function
function
would
have
be
than
2
an ordinary
functions
are
those
of
Weierstrassian
and
Jacobian
20'2.
The
construction
of
therefore
natural
to
so
is
briefly
Weierstrassf
; we
now
proceed
to
discuss
properties
of
it
will
appear
that
of the
account has
Qm, n
therefore
^y
{-z)
y
Hz
follows
from
equation
ratio of the side opposite the
angle
2
this way is
m, n
ni, n
points
(§
that
by
the
equations
g,
such that
we
have
y
of the
the
corresponding
point
of
the
strip.
and
consequently
^Hz)=z-*+p-^+n+o{z^).
It
follows
that
follows
from
its
Liouville's theorem
where
ej,
e^,
e^
consequence
period,
by
^
in example
1 to
W{z)-<p{:y)}-'-[{2^{z)ip{y)-\g^]{<§>{z)-^^{y)}-g,\
[By
the
addition-theorem
we
have
theorem,
that
g^',
in
a
P{z)-x,
we
have
in position
n
is
points
of
intersection
move
off
to
infinity
(no
two
points
coinciding
during
this
when
three
collinear
follows
w^e have
singularities
are
triple
poles
at
points
congruent
to
and
0)3
example
1,
p
peri-
meter
of
z
deranged
in
the
same
series of
odd
function
of
z.
fl,„,,i
origin
have
inside
the
cell,
the
coupled
§§
is increased
by a
F{z)la-{z)
factors,
(II)
a
sum
of
partial
fractions
involving
Sigma-
functions
and
Weierstrassian
function
poles
may
be
arranged
in
two
sets,
say
hi,
b..,
Therefore
f{z)
to the
product and
product,
and
in
that
of
F{z
n
But
S
k
2coo;
and
so
Moreover F{z)
in
Liouville's
theorem.
Thus
the
function
f
{z)
poles. By
Thus the
1,
that
6,.,
^'Uz)
of the
elliptic functions
and n respectively,
in
(f)
be
has no repeated factors
write^
r
.
+
zero
of
y-hp-{z)^lf-\p{z)f,-Ua,
where
/and
h
{z)
where
uniform in the
used
the
is a two-valued function of x and a; is a
four-valued
function
of
y
y
geometrical
can be possessed
question.
?/)
=
algebraic curve
been constructed,
cxcii.
(1898),
pp.
1-32.
class of functions
2.
Prove
that
where,
arguments
2,
y,
v.;
and
e
is
any
variables
three
times
in
succession
and F{u).
{x-y)
function
Math,
xxvii.)
^^^'^fj'
where
c
is
defined
of
elementary
functions,
together
a
Sigma-function
(including
a
with respect
and this is
—
a* in
is 6
right has no
re2:)eated factors, then,
perfect cube
the Sigma-function
complex
number
whose
imaginary
part
is
positive
when
The
^4
(2,
q)
so
Theta-function in
terms of
relation,
we
have
that
periodicity
factors,
and
we
a:b
so
that
addition-formula,
namely
^3 (^
; each
identity
involves
as
many
as
four
independent
variables,
equations
2w'
row
of
or third
§
OZ
ft
uniformly,
equation
^1
(z)
§
where
this
point,
Chapter
vi,
p.
=
(z)
has
periodicity
f;xctors
we
easily
see
that
has
periods
4iK,
inversion
problem, so that the
been demonstrated.
method just
exhibiting the
*
in the same circumstances.
21 'Til. The principal
distance from it.
By
elementary
in the line R
line
equation /(r)
4=1.
+1
to
±l
—
of
the
relation
/(t
+ 2)
+/(r).
0^C1,t'
real values
difficulty, but we
make-s t
q
of
c
so
far
as
and then for values of c in the cut plane,
q
is
q
c=l,
q{c)
one-valued only if, in
these invariants
; that is
seen that
^(3)-^%\f-f-J
no pole
and so,
<^
Hence
modulus
k
The equation
this
domainf.
It
remains
to
determine
the
and,
by
increasing
=
1=1
and
1
correct
to
six
places
of
decimals
when
q=^^.
far
as
their
singularities
were
concerned,
cell,
the
second
consists
of
the
to
brevity
or
simplicity.
22'11.
The
Jacobian
elliptic
functions,
(II), (III), (IV)
of
writing
Jo
by
else the
lower sign,
Jo
7
the end of this section.]
Suppose
that
xi
and
v
vary
while
u-\-v
remains
constant
terms of
en
u
is expressible
+
function
sn {u
^4
it is ineonsistent
be
+
(«iC9
for cn(-w
positive,
and
so
§
By the theory of analytic continuation, this result holds for
all values of
§
J
so
to
be
with
K)
of
the
Jacobian
(
+
K' is
taken.
On
substitution,
extended
to
1,
+/;'.
the cut
-
/
^K, 2iK'.
;
is real
pure
imaginary
^K
and
2K
simple zero
at all
specified,
recourse
must
zero,
and
if
by
the
eqviation
1-63
61-63
Jacobi's
imaginary
transformation'f
the
cut
put
z
=
side the range
is referred
Vorlesungen iiher
die Theorie
=
22'5.
I
nfhdte
products
for
for x,
that
the
coefficients
6„
are
given
by
the
formula
TTibn
with sn
\r\ the
:
—
been
proved
to
points x
by
Euler
and
certain
as fundamental
fonciions
elliptiques.
In
the
supplement
considerablement la theorie
claim
of
Gauss
to
have
made
discoveries
*
integral
when
since
circumstances, the integral may
unaffected
by
changing
the
x)
Q
in terms
of X
expression, we ultimately
evaluating
w'-,
which
we
any
or four complex
22'72.
t
defined
by
be
carried
real. In
not interlace
section.
t
Unless
ttj
shall introduce.
The elliptic
scheme gives
one
to
be
emphasized,
double
dn u, and so, if A
be a
where
are
given
Example
2.
By
expressing
is
singly-periodic,
and
to
iK'
sn (u
to
prove
k')
t
A
analytic function,/
^
elliptic
integrals,
namely
EK'
transformations
of
iuE/K
—
that
E
{ti
that
EK'
lim K
form
I
this
real period 2K.
This result determines
Theta-functious
.
1-85407468...
while
07
terms of
functions
k, 7i,
E' all
Of these
we shall
we
have
/j
when
the
modulus
integrals
by
which
Ii,
I2,
527
22*82.
on
the
sphere
is
*
the results of examples
4 and 5 to
and so, since s
in terms
{p,
cf),
z)
Functionv.m
von
imaginary.
sn^
m;.
+
di
'Mi
formulae
1912.)
37.
parts, shew that, if
c(l-«)^
42.
Express
I
{{2x-x-)
by
operator
to
11
(0)
is
5
i5II(©)
j
2
2
2
\a,b,c
(0)
^n
be
second species
of degree
equation
is
of
the
same
type
equation
(a'
differential
equation
precisely m
of
all four
OI
>Oji_2r—
2-
We
first
observe
that
a?x^
OpKp
the
computation, the
reader is
referred to
Phil.
Trans.
for consider
are
proportional
to
/dx
dy
dz
it is precisely
.r2+/
by
the
identity
satisfies
is
1
d-V
*
normal
solutions
with
^,
exponents
are
Elliptiques,
ir.
(Paris,
1888),
pp.
457-531.
The
exponents
2)
of the complex coordinates
be
written
a
solution,
may
terminate,
so
that
such a
;
algebraic
equation
(iii)
that,
qua
functions
of
B,
the
expressions
h^,
b^,
b„, ...
b,.
of the
(1835), pp.
+
-
§
gj, 63
which
are
of degree n are linearly independent, that is to say that
no linear
of
addition
or
and
let
ta,,
{Bn'-y
also
obviously
1, 2,
this enumeration,
of
the
type
d6'^
—
at
half
periods.
of M.
sum
is
zero,
since
Ey^
(I)
(
(iii)
I.-'
types
each
{x cos
species
and
of this section is
them-
selves.
an integer. The
positive
solutions of this
equation is
the
type
has
an
indicial
Lame
function,
so
in
a
form
are
double
poles
at
points
expressions
under
consideration.
theorem
—
we
find
that
2'
[Z
(ftp
notation of
that
+
(Jamet,
assume
many-valued
possessing
the
clearly
illogical
in order
a
further
the
logarithm
of
work, were, firstly, that
Chapter n, and
a
general
development
of
which
very
greatly
inci-eased
the
the analyst,
give
They
is even
number of axioms,
given
Schwarz
definition,
combined
corresponding properties
sketch of the
given
discuss
the
based on
we do not profess to give a
complete
account
of
a
brief
sketch
given
by
writers-
of
to
read
Chax)ters
§
and familiar
(Paris,
1904).
t
'
the test of
exp
2
See
also
product series
,v
x, rational
for
every
For
section.]
Leibniz in letters
continuous
increasing
function
and then
Log denotes
X
Log(l+«)
due
(1711),
see
the
cos z
function
of
sin2i
exp(
of
2
period
y,
happens
see
using the
identity cos-/3
use
and
sin
(a%
cos:r=0
therefore has one and onli/ one root lying betiveen and 2. This
root
to
=
equal to
functions,
with
x^
77^
v^
such that
equation
has
no
^
solutions.
A'521.
a number
are not
elementary functions
Argand
diagram.
the
sense)
jjossesses
the
propei'ty
Trigonometry
;
not
one
sector.
To
this
extent,
we
the
analytical
definition.
525
Adamoff,
A.,
330
159, 173,
272, 279,
585,
589
512, 514, 516,
517, 519, 520,
522, 523, 528,
279
304,
462,
557
Lipsehitz,
E.
249,
259,
580
296,
373
Eiesz,
M.,
156
Eitt,
J.
F.,
153
Eodrigues,
0.,
303
425
Auto-functions,
226
functions,
367;
358;
inequality
satisfied
by,
879;
Neumann's
function
0,^(z)
is not
five
singularities,
203
Bolzano's
theorem
on
66
Borel's
associated
function,
141
;
360.
See
also
Bessel
functions
mid
Parabolic
;
represented
series, 19
;
585;
for
the
Gamma-function,
240;
546.
253;
expressed
by
Gamma-functions,
254;
extended
function,
89
a
linear
differential
equation,
197,
200,
389,
559.
of, 560
Improper
integrals,
75
221
492, 494
Integral
functions,
106;
and
Lame's
equation,
571;
and
of
Infinite integrals,
of,
564,
573
479,
singularities,
203
Kummer's
formulae
series
Neumann's
function
[0,j
(z)],
374
series of,
integer,
D-n-ii
310
482
28
Products
494,
511
494,
511
Recurrence
formulae,
of
special
iiiuiitions
Solutions
of
integral
equations,
473
213,
121,
123