AN AB$TRACT OF
T} TISI3
OF
for
(Name)
(Degree)
(Majr)
Date Thesis presented---T i t le ------
__
Abstract Approved: -----------------(Major Professor)
ABSTRACT
The first part of the paper is devoted to the obtainIng of convergent power series solutions and the determination of certain properties of these solutions. In particular, properties are discussed relative to the number of zeros of. a solution and the distances between successive zeros.
The relationships of orthogonality of two different
solutions are shown.
Since it is difficult to vrite down a
general term for the power series, the equation is transforied into one which gives rise to a series with simpler
coefficients. In addition, the relation of the Weber equation to (1) is shovm.
Beginning with Chapter VII the discussion centers on
solutions of (1) asymptotic ink. The theorem of Trjitzins,çy :is apilied to show the asimptotic character of the
formal solutions. In Chapter VIII the solutions asymptotic in the parameter are obtained by applying the method and
existence theorem given by Birkhoff. In both cases, solutians are obtained which are best suited for calculation.
The last Chapter deals with the behavior of the coefficients of a solution due to Stokes' phenomenon.
In the
case of solutions aspiptobic in 'i,explicit formulas are
given for the calculation of the coefficients.
A STUDY OF ThE DIFFERENTIAL EQUATION
d'w
.-+(rfl-1-z
z.
)w
Q
dz
by
C.
GORDON MORRIS
A.
THESIS
submitted to the
OREGON STATE COLLEGE
in partial fulfillment of
the requirements for the
degree of
MASTER OF ARTS
June 1938
APPROVED:
s
Professor of Mathematics in Charge of Major
Head of Department of Mathematics
Chairman or School Graduate Uommittee
Chairman of State College Graduate Council
Acknowledgment
The writer wishes to express his thanks to Dr. W. E.
MiThe who suggested the topic and directed the work on power series solutions, and to Dr. Henry Scheff
who directed
the work on asymptotic solutions and gave valuable assist-
ance in making that part of the thesis possible.
TABLE 0F CONThNTS
Chapter
I.
II.
Introduction
i
Existence of a Solution of
(i)'.(w-)Jo
III.
The Solution of (i) by Means of a
Power Series
IV.
Properties of Solutions of (1) for
real ;2=%i
V.
Solutions of Transformed Equation
VI.
The Relation between Our Equation
and the Weber Equation
VII.
ii
14
15
The Solutions of (1) Asymptotic in
the Parameter
IX.
6
The Solutions of (1) Asymptotic in
the Complex Variable
VIII.
4
in
Asymptotic Connections
26
33
i'
A STUDY OF TEE DIFFERENTIAL EQJJATION
o
c:1
I.
This paper is
s
IbTiODUC ILION
stu.dr of tine
differential equation
LÇ+
:7
(1)
The first part of the paper is devoted to the ob-
taining of convergent power series solutions and the de-
termination of certain properties of these solutions.
In particular, properties are discussed relative to the
solution and the distsnces between
number of zeros of
a
successive zeros.
The relationships of orthogonality
of two different solutions are shown.
Since it is diffi-
cult to write down a general term for the power series,
the equation is transformed into one which gives rise
a
series with simpler coefficients.
In addition,
to
the re-
lation of the Weber equation to (1) is shown.
Beginning with Chapter VII the discussion centers on
solutions of (1) asymptotic in
is given to obtaining real
utation.
the
and particular stress
solutions for practical comp-
The theorem of Trjitzinsky is applied to show
asmptotic character of
the formal solutions.
In Chapter VIII the solutions asymptotic in the
ameter
m
are obtained by applying the method
ence theorem given by Birkhoff.
and.
par-
exist-
Here again solutions are
obtained which are best suited for calculation.
2
In Chapter IX the behavior of the coefficients of a
solution due to Stokes' phenomena is discussed.
case of solutions asymptotic in hi
are
,
In the
explicit formulas
iven for calculation of the coefficients.
3
EXISTFICE OF A SOLUTION OF
II.
(1)
+
(+ f)
i
=
Consider the general homogeneous linear differential
equation of the second order
W
(l-a)
ft
p,"
in which the coefficients p,
out the finite plane.
plex variable
f
P2''
and
Eere '.)
are regular through-
is a function of the com-
It is shown by a general existence
.
ti+
theorem* that (l-a) admits one and only one analytic sol-
ution of form
w
i'0
-t-$,
.....
which
with
its first derivative
at
o
This solution is valid over the entire finite
.
takes
on assigned values
plane.
Equation
(1)
satisfies the conditions necessary in
the type form (l-a).
solution of
Therefore we may assert (1) has
the forni
w
C0i- C,.
C-
a
.....
We shall next dterriine thi coefficients of this series.
*Pierpont, James, Functions of a Complex Variable, p.
and Co., 1914
459, Gum
III.
TETE
SOLUTIOI
OF (1) BY
iEAIS OF A POWER SERIES.
Assume the solution
w=C0tC,tC,,2
Then
*
3çaz
2
If we substitute
--
C-
'ÇA....
-1)C1
the foregoing expressions in (1) we have
(C6+2C#
.....
+(i)(ii)C
Next,
.....
the coefficients are equated to zero, ard
this
gives
cz
cow,
z!
C3: -C,b
C.¿-t2
3!
-mC
The last formula gives
each
the two preceding ones where
ry constants.
C.
C0
coefficient in terms of
and
C,
are two arbitra-
Several of these coefficients have been
/-2\
computed and are given below.
C:c.-;i)
cg
C'
/m2-5\
o
i-)
=(T7 Je
3
-
C
2
7
__/
7,'
I
z
-4LLw8
L
-
I
8
52
-
s-
7i
C
-)
8','
I
-4-ib
- /oo
±
i
t
CO3
5-3
/o
±2f) C
--(-
I,
I/i
-
-
'r'
,
F
Ç
2
û
o
2
-
C '3 -
±
o
ô 4'
fr1
-
277
C,
o
¡3,!
-'
When
z
z
t,
these values for the coefficients are substibu-
ted in the assumed power series solution, we have as the
most general solution for finite values of
k)z
A.J
o
where
CJ
-fi
___
w
- /O
(LA1
______
J
s-
-.
%1
-+
¿4L4w)
i
/
4vi
'o
4
2
7
+
f
3
s
/1,
'I
)
O
)
/0/
-
-t-
+..---.
4
IV.
PROPERTIES OF SOLUTIOIS OF (1) FOR REAL Z)C
A solution of (1) has an infinite number of
1.
zeros.
Proof:
shall consider the equation
We
0
(2)
which has
a
be given any value.
at this value of
Now we multiply (1) by
(2)
from
(i.).
where
may
Since (1) has a solution which has a
let us choose
zero,
(-&),
9% Csin
solution
j
ard (2)
r
byy
and subtract
The result gives us
9y'-.yj7"4.
[(,*i'J-rJyf0
so that the expression in brackets is positive.
Choose
Furthermore, pick the solution
9
such that
'(
& )y(a_.
When the last equation is integrated between the limits
and
r
,
there results the expression
f((/?lZ)/7Jyjdx
vy'-yy:r#
Assume
9
vanishes again first after
&
b
= o.
;
then this
equation will be
_yyJf1f(i21yj
The signs of all
tive.
The
tifle
o.
terms in the equation will be posi-
situation is not possible.
first, we have
6
2
yy ]#f[(øv)-n Jyjd
2
If
y
vanishes
V7t
Here the signs are opposite, and the relation could be
Since (2) has an infinite number of zeros,
satisfied.
the process might be continued an infinite number of
thus showing (1) has an infinite nuiiber of zeros.
times,
2.
The squares of the amplitudes between successive
zeros of a solution of (1) are aiwsys decreasing.
Proof:
Lety
and form the equation
be a solution of (1)
w =
Then
yv'+,1
,
2.
-=
i
yy
=2j
±
/
____
[y"+(w+)y}
F
;;+
=
J -Y--i
-
/
2
_-Y
'
:?:_$
y
2.
(w4')
When
2'
Lety,
a decreasing function.
The slopes at
slopes at
'J:ti
Ji
I
W
,
j
z
i2.
'J
is
¿=1,2,3, ----- , represent
of the intervals, and let
the amplitudes
Since
is always negative, and
is positive
represent
will be zero, and we have
2.
i
w2.
etc.
decreas
>
e
tc
and the proof is complete.
3. The
interrals between successive zeros of
tion of (1) are continually decreasing.
Proof:
a
solu-
Assume
a
solution of equation (i) has successive
and
zeros at
In (i) make the substitution
this
TTon
new
ec'atinn will
become
J2
!J,a
which, since J
Ç.
(1-b)
Equation (l-b) has
,
may be viritten
+
a
which will have a zero
solution y
at a-when the corresponding solution
at
We seek to show that
J
Let us multiply (1) b
The result gives us
tract,
yN- jg"-f
j
(1)
has a zero
will vanish st
before y
y
of
and (l-b)
,9
V-('
-'i)
bvj
j
yy
and sub-
-o.
When we integrate this equation between the limits
3
and
obtain the eouation
we
+j_
As before in this process,
at
the signs of the terms do not make
Ify does vanish before
sible.
satisfied.
which
does not vanish before
if
Thus the interval
a..
,
is
$
the equation ros-
the equation could be
shorter than
indicates that the next interval between theczeros
of a solution
j
is
shorter than the one before it.
The
process may be continued throughout all the intervals to
show that they are continua].1r decreasing.
4.
If corresponding to two different values of
there are two solutionsj andy of (1) both equal to
, then
zero at and
ori('f
JOpBUO
uQT4flb
GU
1K,
z
(2)
n)+
'iC
)&-f
T
n
()
ri
T(
ciçnux
aw
UGLr.AJL
?,Ç(
pu
q
(ç)
(i,)
=
pi
iq
uoGJdxG
BI[;
UÇ
M
,
4-
tt41
;oi;qns oxo4
'
ALGA9
sLaTI pti»
Tí
49q
'f
0
ÇÇ
3JpU
M01 '(JT
OJOf ULütX
UJJJÇp UO4flOS
uoTnb
[UO
O.:»lD
%
UG
(_)
zi o(i-'±ti')+
(o-T)
1
%'
)
UOfliOS
(
JO
7,Ç
uOÇfl1Q$
q
Q
(°-I)
(
7v)
O
.
(-E)
TJiIL
0
7 4
)
j-t:
c1Ç
'
ii;-
0
{(:D4)»»°f
pTuo3
X
D
SOJGZ
-[4T
cï
LTd13T? o
)
J(
$..
(
:
(
s-ç
:i
From this it is seen that
When
¿
1,2,3, -----
From
these resu] ts
Y
.
C
tK.,
sketch
we
at
)Q
o
when
and
t3
t
-
We
may no
write
a
()y
z
L2
(5)
yL;
Z
(6)
now
multiply (5)
from (5).
twe en
-
IÍ
the limi ts
.
'
we.
-
O
I
ncl
(6)
by
y
and subtract
integrate the resulting equation be-
and
O
o
Z
y
b
=
¿
(i-'.)y7
-f
6'J
Let us
-
(K
-yjyx :10i0
we ob
tain
z
Z
)1j/F
I
IiJzdt::O,
The first term of the last equation wil] be zero at these
limits,
and
have,
we
X
2
Z
t(&i
T
whe re
y
K)
y(,K)
J
2
X,
11
V. SOLUTIONS
OF
TRANSFORID EUATION
It is important for some purposes to obtain conver-
which
gent solutions of (1) for
the general term can ex-
Since it is difficult to write a
plicitly be written.
general term for the power series solutions obtained in
the previous section it is desirable to transform the e-
quation into one which will give rise to simpler series
coefficients.
Let
a
w=
e
y
/
z
I
e
=
22.
e
'
czK42 +)v0.
L4/}
With this transformation (1) becomes
(7)
v
v
4
L
Now to remove the term
For
?JE
the equation (7) now becomes
-(-t-w)y
I,
V
We
t*c
series solution
now assume a power
v=
I:
/
ç-
I
ô
as before and have
12
V
'I
()
Z
Nhen these values are substituted in
2
Jz
J-
J>Z
7-2
ç-
+
..r,
(iv1)L
r
'
and equating t.o zero
By collecting the coefficients of
we have
there results
the recurrence formula
=
_p.n-i
(z-3)
2
-
Since
a.,
are arbitrary constants we form now
and
two power series solutions which converge for all
the
// -
a1
/
-F
Cw+z)
(iv+.5j)
+
z.
(vvt1'h -3]Z
(w+ z)Cvtcj)
)2v1
J
2M'.
(-iY' (iwt3i)(wrt.
(2r+1)
The most general solution is now written
\,a0v,
Two
a(V
(t-t
independent solutioìs of (1)
2
14»
(8)
3
u
e.
may now be
)
)
well known theorem on homogeneous linear differen-
By a
tial equations there exist solutions of (1)
Wz
written
(
:
),
real for real
such that
),
(
)
and
13
w, (o)
h./,
'(o)
o
-_,'
a
/
Upon differentiation of equations (8) we find that
Thus it is evident that
and
"I
,
and,
and
Li
in spite of the complex apDearance,
real values for real values of
(1)
are identical with W,
is
au&
they give
The general solution of
VI.
THE RELATION BETWEEN OUR EQUATION AND
THE WEBER EQUATION
In (1) make the substitution
=
V
L
We have now the equation
U(±L
2.
where
-
-
)'/=
1
This is the well known 1eber e-
.
quation whose solutions are the Weber-Hermite parabolic
cylinder functions.*
parameter (except
It is obvious that real values for
rno
)
and independent variable ïn our
equation correspond to imaginary values in the Weber equation.
*WEittaker and Watson, iviodern Analysis, Cambridge
16.5
University Press, 1933, p.347
15
VII.
THE SOLUTIONS OF (1) ASYMPTOTIC IN
C
A.
IE
OMPLEX VARIABLE
Introduction
to have solutions
It is often desirable
of equations
such as (1) which are practical to use for large numerical values of the independent variable.
In order to em-
phasize that the Independent variable is now complex we
denote it by the letter
for (i)
.
It is shown by Fabry* that
there exists a full set** of formal solutions.
These solutions have been proved to be asymptotic under
certain restrictions.***
C
form
where
Q.
(
-
)
is
a
Q()
4
()
e
3
The formal solutions are of the
¿42
polynomial of form
¿
s1(.)=/:;I_'+
and4
Is
Lf....,
a constant.
In general the series
do not converge.
'J
(
)
do not terminate and
Trjitzinsky**** has shown that these
formal solutions are asymptotic expansions of true solutions and has determned their validity.
*MSE. Fabry, These, University of Paris, 1685
**Ijere "full set" means two formal solutions such that
the Wronkian formed from their formal derivatives does not
vanish.
***'V.J. Trjitzinsky, Acta Mathematics, vol.62: 1-2
1933, pp.167-226
****W.J. Trjitzinsky, lac. cit..
16
We
shall proceed by assuming the above solutions
determining their unknown coefficients.
ence of true
solutions asymptotic
and
Then the existformal solutions
to the
and the regions in which the asymptotic developments are
valid will be considered.
The formal
B.
ers
series solutions of (1)
in descending pow-
of.
Assume a
solution
-t
5(a)
(9)
-
where/s
,
r- ,v
,
e
P()
vL)
(zj*
e
.....
-------------------
a..,
are constants
We shall find first the particular sol-
to be determined.
.,,
utions for which a
,4
-
ra
C
ç
¿
'
J-J.
Dïfferentiating, we get
Z
e
iZr
4Ò".*
4,,7
+e
7
('-;)Z
.7
i,-
z-j
Ç
L(
+
.7
-
7
z-7-'
L
e
t -j)a.
7'
-3E-II,
C
(v- v)e
2-r
/3;+)'
¿
.
+
.
)
oi-
e
¿
L 2(3
(4.
.7
i)
4-J
a
4.-J-I
-J)z
J
m
+e
t>± +r*
Z- L^-J) U-/-/J <%*" J
T- e
7= o
The above values may be substituted in (l) and we have
(10)
Equating the coefficients of
Q
£
to zero we
find
K, =
yV + Z
.♦mi
__/,
K, = a- -/
-«-These are corresponding notations, that is, when
/% 9 +1 , yv= + 2ii -J~
etc..
To determine the coefficients, ,
ci
-,
,a-3,
the most
'-'
efficient procedure is to derive a recurrence formula.
P(s)
This is done below.
(10)
the coefficient of e
If
is equated to zero,
Ì3a
(,-*,)
.2.
in
we have
a,ì' (iv-de)
+
i
o.
Solving this equation fora-,we have
.zi (2,V_i#/)+Pn
which gives an expression for the value of each coefficient in terms of a preceding one.
When the values of 4' and ¡9 are substituted in (9)
the equation may further be reduced to
(fl)
a1
2it
±
Since we have shown
with
ô
it
follows that
all
odd subscripts will be zero, since for these by (1l)
will always appear as a factor in the recurrence formula.
The
coefficients
ç, a.1
----,
may now be
calculated.
A few of these are given below:
2
'i
-3Z
ä
±3S41i
(12)
.
.............................
19
I'
±ni[t(2_4)iJ[±(z1-8)2J..
-
4i
the values of these coefficients are substitu-
\1vhen
ted back in (9)
there results the following expression
for S
Z
r
Q,
z
where the value of
For
J,
is given by the
formu1
(12)
(
±
Z
and (13), and for J
/2
J
the upper sign of the symbol
,
to be used in both (12)
lower sign.
(/#-+.t+ ..-«),
)
for
is
the
,
Since the above gives two distinct formal
solutions we may write them as
.*
.
14
P.iL
S,:
(
j
¿z
..... )
14)
2
=
-
and
If
-!L,
e
:
w
are real it is desirable to change the
form of these expressions as follows:
arge
Now
±jL
e
IT1
=
+JI
.
Ko±f
)
,
±jk1
=e1e
)
For
±2
real
.....
I
r
.j
i
r'
è
)
=e
taraL
e
±j(+1oaII)
? pi mii)
tz(--+T
.
Ce
.L
I-
II
20
It is possible to separate the real and imaginary parts of
the last expression by writing
i.LJ.
e
Ctieos)t(--4-.
1oIî]4
¿
2
The equations (14) may now be written for real
v
and. Z
as
sIol)] [v]
S
¡'
r
L-ai
(f +Tk11)] [V()]
Leo
where the constant factor introduced by the multivaluedness of log
has been dropped.
By a linear combination of
formal solutions free from
the leading term of
s
SS
-%f
"
t
we may obtain
Thus if we denote by
2
(set
,
and
1E
/
)
's-r
(15)
-s
-
ii
z
When all the terms of V.
(
-)
are considered these corn-
binations will still be free from
e-,-
the complete formal
r
?
L-r-
.
We then call Ç, and
solutions defined analogously to
(15)
Let
¿
jocJal
21
We then get the following:
r_____________
-I
C
II
Sirt
(os p+L'
-4sinp
C_,::
For most practical purposes when
and
w
are real these
last formulas are better suited for computation since they
The most general formal sol-
involve only real quantities.
ubjon for real
and vn cen hence be written
-
()
C,
C-,
(z)
-+
C GL.
Existence and validity of asymptotic solutions.
C.
Consider the formal solutions just derived
e
s
;z-'(
z
=
S
2.
where
Q:
-
(1* ..... )
wi_j_
and
2
2.
-f
j2L
......
-
,
:i:
In order to apply the
2.
'
theory of Trjitzinsky it is nec-
essary to consider regions R.¿ * bounded by Q-curves which
are defined as follows:
Q-curve is one on which
c'
where
and
Q
Li)
-
[Q(:)J
¿,.7
20
/
¡
2,
represents the real part of
Trjitzinsky,
bc.
cit., p.171
2
(*-).
22
Now
GLJ
cr2J
4
Let
#1(:
then
- y
Zi
-
jz
Finally
I
Qi
oe
{ìJ-z'xy
We must now consider along what curves the equation
O is satisfied.
and
y
o
These are obviously the axes
These are hence our Q-curves, and they divide
.
K.
-plane into four reFions
the
gram.
'Q
as shown in the dia-
They are closed along the Q-curves but open at
X
Trjitzinsky states further** that we must take into
account the possibility of a repion R. where one of the
differences
Q
)
(
-
Q7(
)
has
part and is such th:t for some
(16)
t
0
[Qiy (t))
,-
along both boundsries of
as
*The
.
e
.
heie is,
°
K.
of course, not to be confused with the
of equation (1).
**Trjitzinsky,
non negative real
a
bc.
cit., p.180
23
We note that in every one of
the four regions
bounded by our Q-curves there will be at least one difference
Q1 -Qy wbic
has
a
In both
R,
positive.
In
for the entire plane we have the following:
is negative
and R.3
both
R
and
R
and[Qz,]
positive
is
,
In fact,
non negative rea1 part.
is
aridCQj
is
negative.
Specifically then, we must consider the possibility
of relation (16) for
in
in
-
and
R.3
R.and R4,
and henceforth in referring back to the relation (16) it
will be understood that in any region
mentioned above.
ing it only for the
In the regions
,
and in the regions
R.,
,
R3 the
e
'
R4it
is
iI
(16-a)
we are consider-
left member of (16) is
/3
lI
(16-b)
e
Evidently these expressions approach zero along both
bourdaries of their respective regions,
1C0 ory=
it is
When this happens along both boundaries of
necessary to subdivide
into two subregions R.
each with one boundary in common with R
boundary
dividinR
fromR
o
,
and another
and interior to
j
such that
along it all the left members of (16) increase indefinitely for every (
> O.
24
Take the 45
lines for such boundaries interior to
the following, configuration:
We have
A1
»4.
/
\
X
''
/
s'
s'
"Aq
4,"
OA,
Along the new boundaries
consider the appro-
From (16-a) and (16-b) we
priate left members of (16).
see these are in every case
.,4
e
i
2IyI
Along the new bounderies this becomes
-4
I
e
.
Now along any of these lines
IM
e
-,
is negligible compared
initely since in the limit
Iz,
to
increases indef-
I
Since the conditions are satisfied along these
.
divided into subre-
boundaries we now have the regions
tZ
shown in the diagram.
¿
gions
as
1,2,3,4,
I
,A.
\ZS
\
//
Rs'
fi,'
/
\
/
/?3e
/
113
R,,
I'
J
1,2,
25
Frani the
Trjitzinsky
fundamental existence theorem given by
we may
now state the following:
In any region
l,2,5,4,
i =
of the equation (1)
tions.tJ, ,JJ2.
%%f
ii
¡7
2)
¿J2
for all values of
-
in
R1.
J
here
''1.S
y
a
indicates
obtained by formally differentiating the series
*Trjitzinsky, lac. cit., p.208
true solu-
exist such that
¿f2.
.
1,2,
7
the series
26
VIII.
THE SOLUTIONS OF (1) ASYMPTOTIC IN THE
P AR AME TER
A.
Introduction
In a paper by Birkhoff* the asymptotic character of
tl-ie
solutions of
2
___
I
of
is discussed for large values
Z..
,,
(
parameter,/
)
f) C___
(
+
..
o
('
-1
The coefficients
.
are assumed to be analytic in the complex
o
at,/7.
and to have derivatives of all ord-
in some interva1
ers in the real variable
The equation (1) may be put into this type form by
letting
It then appears as
#z{/
.
vYt
(i')
o-
For (it) the characteristic
the paper becomes
This equstion has the roots
.
-i
and
¿
which will be denot-
It is convenient to postpone the
ision as to which Is
in
1=0
Q
ed by Q, and
)
equation** defined
0<,
dec-
and which is oÇ until later.***
Formal solutions of the form
(r7)
-7
-
e
;
';Lkir)P
j
i=z1
*Transactions American Mathematical Society, vol.9,
1908,
T2l9-232
**Trans.
.
Math. Soc., vol.9,
4
***Considered in part
C,
viii
1908, p.220,
equation
0
27
are then shown to exist.
B.
The f ormsi
solutions.
To determine the
is a solution of
(4,.(
ï
(17) we assume that Ó2
of
)
Then
(1').
cÇ:
JQ
i1O
f4?
I,
= e
7,iu,;
-J #1
(/*
Jç,
e'141° _J*
/o
1
2
:0
When these values are substituted back in (li) we obtain
the equation
(18)
T
u1)
e
Ç-
L u1)f
2Ç
-3tl
L
+
7=0
We now collect the coefficients of
for each
e
('Q(.(
Z
J =
,I.
in (18)
1<.
t__110
('.)
1_1i0
')
o
By integration
This constant
C
may be taken as
C
1
since it is arbitrary.
The next equation becomes
F
f,
L-ha
+
where it is understood that the
-
1.
's
=
are functions ofi.
r
Since
u.¿o =
u.
O
there results
i
-
Z
Zf
or
2o(
We take the constant of integration involved in the de-
termination of u
,
used any functions
,
(,/
of,,-
u
that
so
as
),
(0)
Ö.
If we
constants of in-
tegration we could obtain other asymptotic series which
Likewise the process may be repeat-
are less convenient.
ed for other coefficients,
a
but it is simpler to develop
We have
recurrence formula.
f
Z
From this
--
,.
2
jr (u-Ft.
+
u
-I
or
ìt
020&.
iv-
)&:t,
-X
Íu+
'
Ujg
i
L
By use of this formula a number of the coefficients have
been obtained explicitly and are given below.
/
3
(19)
'-4.
2o(
3
I
uiz
lj
3
-2)
?C-#
/5
/z
29
4
(t-)
u1.,)
j
g
#2
____
r ___
i
c
L
s
II
7
3
"/7ZIo
/156
3
-p--
2.9i
When these coefficients are substituted back in (17)
z
we have
ó
71J
e
L-- '21 ..... J.
¿?,,( 5.
I
«z
'
The derivative series
are of importance,
and they
may be expressed upon differentiation of 6. as follows:
f
\
= e
6
(
2
(47,)z(1+;
.-
--
2±
a
'a
.
It is possible to make linear combinations
of the
formal solutions just obtained sueb that the new form will
'% .
In the original equation (1)
we must consider two oases.
When .-, is real and posiLive
be real for real
//
and
will be real, and we have
w
w2
: [cos,,
'i6
SiflÍQf.]
¿
_
t:cosìz
t
¿
sinfl]
['Jo uiJ (4)/::'l
I
[
tA
The following combinations 2ive the desired result.
H:
sin
cos#o,j.
1
30
2'.*/5
-
¿
Z3,'3
(20)
;_Ws::
H
-t-
-
- _j;:P
COS/.oI
jfl1Q
COS/07C+.]
It is seen that in the solutions
the signs al-
thus
ternate in pairs,
H,
(20-a)
Hz =
CO5f
or
The coefficient of
the term
(19)
(20)
replaced by unity.
oÇ.
-
be inserted from (20-a).
The
(
±
the same as
)
signs can then
Since these coefficients are ob-
the recurrence formula we may
tamed from
is
except for the possible factor of (-1), of
,
with
-1,,..r
write
as many
terms of (20) as needed.
is real and negative ç
When
form
(pl
j
,
real for real
may be taken in the
and the original formal solutions will be
vi
and
In this case we have
Zf+3*3[2l4Lt4
/Z 3)
's
=
e
C
/.5_
tel
((j+z4
I
-J.i;' {2ì /- Iz_1....).
1-
C. Existence and validity of asymptotic solutions.
confined to the real axis we shall define the
For
region S
2
plane as one in which the indices
of the1
i
and
may be so arranged that
fora
in
3
fAA]
Here
.
represents the real part of
the complex number..t
gion S
and
-i
Let
¿
Q(
The corresponding re-
.
will then be one such that
[(i
Since1'
is a complex parameter let
f'where1r
ande
are real.
(+
ì71y
We have
or
o.
Thus the
5 corresponding
above values of
to the
o(
is
the lower halt' of the complex ¡' plane.
Now if we let
0<
i
and o
-
=
?
and proceed similar-
ly to determine the corresponding region
this is
the upper half of
we find that
.5
thee piane.
It is now possible to define the region
of
the
piane as that in which
'b
i
<
ir
arge
(f;',..
,)
--
,
-O
-
,
1 ,_ 2
,.
.
32
Xbe
Let
confined to
'
cluding the point
by
b
_b=
finite portion of the realaXis
a
as interior point.
O
Define this
It is now possible to state the
.
Theorern*- For ,
ist true solutions,
on
(
,
-j
2
f.
r
in
and.
)
there ex-
1,2, of the differential e-
quation (1') such that
o-r.'
=
/
i'yi
F
(4;((,
(
where
=
and
in
theE
and
s
e
)
on the interval (-J,,
bc.
cit.,
p.
-T
L
e
are bounded functions for
*Birkhoff,
in-
j,
).
2e5, 22b
1age
values of
L)
ASYMPTOTIC CONNECTIONS
IX.
A.
Relations of coefficïents for solutions asymptotic ini.
of form
The formal solutions
e
=
are not in general sinple valued due to the factor
Anir
true solution
is single valued since it is a linear
J
combination of the single valued functions
and
isJ,
vsJ
dis-
cussed in Chapter II.
In order to consider the formal solutions,
--
is
(
),
-
study them on the Riemann surface on
as single valued we
which arg
,S.
This will consist of an in-
single valued.
finite number of sheets with branch point at the origin.
like
The regions of validity are extended so that a diagram
Riethat of page 24 is now visualized on each sheet of the
mann surface.
R.
There will then be infinitely many regions
numbered as follows
.
R.1
-
Íz
boundary conditions.
pair of true
(21)
a
arg
.t:T
O,
:t
l,t2,
-ri-
arg..
particular solution
f
by means
of
This is then a linear combination of
solutions,1,
w.
¿J
C'
.
W
¿J.
From the fundamental theorem of
write
=
t':;
(i-t)-
First let us fix
any
1<
(-')2
,
+
and1W
zT
CZ
.
3.
2.
jitzinsky we may also
W4SZ.lCIS,+jICLS
34
in
¿7
1
,7=
1,2,
O,
z
i1,r2,±
3,.....
Now the value of the formal solution 5. in eheet
i
of thé Riemann surface will not be the same as the value
in sheet 2.
Since
W
is single valued,
C
evident that the coefficients,
where in sheet
,
it is therefore
must change some-
and this fact is known as the Stokes'
1,
phenomenon.
To study the changes in the
*
note what happens as
arg
(k.-
is taken across the boundaries OA,.:
&- O,± i,i 2,.
,
)
coefficients we shall
...,
and
theX andY
axes.
At this point it is convenient to define the following nomenclature:
We shall say the formal solution
"dominant" over the formal solution
S,.
,
c
is
in the region
if
z
CQJ
in interior of R.
We may without any resulting confusion apply the word.
to
the corresponding exponential factors in the
From the relationships
sequel.
sin2-
c{Ql
where
8
arg
we
see
that,
in the regions
3, is dominant if
¿
is even,
dominant if
¿
is odd.
Vve
-c[Q,},
now proceed to study the behavior of the coeffi-
cients,C
,
as a curve
DA
is crossed:
On
(-)T
II
Z
K.
(&i )-wi
ii1e
'
2
¿KI«l)
Thus
(-i)
.
Q
Kodd
In the limit
C
e
Qa.
is dominant over
e
even integer,
inteey,
.
-
o
Likewise on OAK,
.
e
is dominant over
Since the regions
e
a2.
are closed along their bound-
aries, we may assert both the relations
along
OA,,
w
,C,
w
,.
C,
S, -i-S
C
S1
C S
S
the common boundary of
,
and R..
,
.
From
the definition of an asymptotic expansion it follows that
,,C,
()J
Cc
e'
(23)
,C,
where all the
-
)-
Dividing both sides of (23) by
it as
o°
onOP,
(Fc2.
alongC)/,
O as
e
we have finally
=
-
,.
and taking the lim-
$6
Thus the coefficient of the dominant solution S1does not
change in R,.,
This proof may be extended to show that the
.
coefficIent of the dominant solution does not change on any
curve
s
e))
If the coefficient of the dominant solution is equal
to zero
in.
R. equation (23) becomes
C
)( )-'
-
where the
Dividing both sides
on
O
as
-I
bye
,C,
(
-along
O.
and taking the limit
as*-o-
have
we
IIC'1
/2,
which shows that the coefficient of'the sub-dominant solution does not change on 01A,.
ed,
This result may be extend-
and we may say that in any region
where the co-
efficient of the dominant solution is equal to zero, the
coefficient of the sub-dominant solution does not change
on the curve
We shall now determine whether or not the coeffIcIents
C,_,charige
on the boundaries of R..
ij
.
Consider the diagram
37
Here it will be necessary to discuss only the case when
OYsince
is taken across
is similar.
the behavior on the other axes
We write
+
z'
-
c,
s,
iz
Ca
5i
c
s
+
n
In other notation we have:
(24)
+ C e
tj=
I-,
(25)
a2.
4 f
w
where all
s
-
O as
-b
on the line arg
-
On the boundaryO"( we may equate the two solutions
(24)
and (25) above since it is common to both
also on this boundary
a
LQ.,]
'_EQ2]
by the definition of
When we do this the result gives us the equa-
Q-curve.
tion
,v,t
(2e)
fLCI
e
'
(J
¿t
-
-
e
z,
-\))
Next, let us substitute the values of
Here
_4-A_
I
4Lj21
-
(L
=
t
:
is complex we n*y write
Since
-V--4
I
=
Now
-I,"
y
andR
4ZPn
$;=
-1-y
-f-
z
rn,
(lorIi+z
.
,
and 4
-r1ogJ_ m%
.-
OnÖ
arg
¿(mioci
e
-e
e w1og
_wy1o(l
¿
K
(27)
sincee
-mY.-im
wy arg)
_
arg.
different
Is a constant, K
from zero.
"
When the value of
tuted in (26),
P
¿(Pv%log I2i
az)[,:LC,
K e
->v
The limit of equation (28) as
um
substi-
r
t-( Ke
f/-t-IZ)\,!()J
Ç,*,.,
,c,(if.t2A,(]+
(29)
is
that equation becomes
-i1og i'
(28)
as given in (27)
C))=a.
ive
-v9.ogII
i
-
e
where
'.2.
13
Z,
..
-
/2
Now we desire to show that
I in
which
If rfl<.
O.
k..
ThereforeA
effect
or
A
oA
O and hence
0,
-'-
G
A=ß
O.
Consider Case
(29) becomes In effect
B = o.
O.
K..O"+
O,
i
If vv,> O, we have in
13=o,
and substituting this back in (29) we have also
o.
In Case II,
11m
rrt>.
O.
Equation (29) becomes
ã(
loggaI-s.?)
A8J°.
39
Now let us assume
A=
hen
O,
B
O.
¿(w loIzJ+.Z)
lin
I
$
If
O,
we have
-
but this equation is absurd since the left side represents
a
vector whose argument is increasing indefinitely.
Thus it is shown
thatA
O and
3=
o in each case
and this gives
C
c
r:
on the Q-curve
any Q-curve,
y
This same process may be aprlied to
.
ives the result that the coefficients
and it
do not change when
-
is taken across these boundaries.
Let us now write the asymptotic solution simply
k'
-'
We have just shown that as
--
CL
is
taken around the origin
S2
neither coefficient can change on a Q-curve.
Furthermore,
the coefficient of the dominant solution will not change
on any of the
boundariesÖA.
The coefficient of the sub-
dominant solution may change on these boundaries unless
the coefficient of
the dominant solution is equal to zero,
in which case it will not change.
-In
close
order to throw more light on the situation we en-
the Q-curves of our Riemann surface in small sectors
or D-regions which may be defined as follows:
(arg-?kh>O,
Sheet
i
will then appear as in the figure where the D-
regions are the shaded areas.
40
Y
Aa
,A1
t
¡
y'
\
'43
,
/
/
/
1<
Let that part of the region R1. exclusive of the
.
Now it can
tJ
has the a-
regions be càlled the restricted region
be shown that in any restricted region
jQ,
D-.
symptotic development
where
is
the solution dominant
in,andC
o.
Likewise if the coefficient of the dominant solution is
equal to zero
where
43
is
we will have
in
the
sub-dominant solution.
It will here be noted that the location of curves
where the sub-dominant coefficient changes is to a certain
extent arbitrary.
extending to owould
Any curve
evi-
dently do equally well.
On the Q-curve the two formal solutions become of equal importance and hence in a D-region it is necessary
that we retain both terms,
w
wiaereC,
SI
is
,-
Q1
S'C
S2
in D
the same as in the adjacent region in which
is dominant and
C±5
the same as in the adjacent
41
region in which
2 is dominant.
If both coefficients C,
1.
Ç should
and
equal zero in
we have the trivial case
:
w
o.
Sunmiarizing this analysis we have found the following
If we write
situation:
simply
AJ-'.'
where '/
the
is
CIS4+ÇSL
any solution fixed by boundary conditions,
only possible change of the constants,C
is
then
that C-
can
change on the lines which bisect the second and fourth
C
quadrants, and
first and third.*
can change on the lines which bisect the
A constant can change on these lines
If by some pro-
only when the other constant is non-zero.
cess like contour integration or the saddle point method
certain minimum information could be found,
this together
with our anslysis of the change of the coefficients would
unravel the complete story of the asymptotic form of the
solution
''J
over the whole
tian just mentioned is
the
t}
.
OA
,
of
a know1edge
O, t 1,
±
2
,
the leading term of
solution
asymptotic expansion of the
the lines
The minimum informa-
-plane.
.
.
.
.
,
W on each of
and furthermore
is is known except for a constant factor.
B. Connections for solutions asymptotic in the parameter rn.
*The reader will recall that the location of these
lines was somewhat arbitrary when we made the subdivisions
to satisfy the requirements of Trjitzinof the regions R.
in the
sky's existence theorem. Any curves extending to
respective quadrants would do equally well, providing on
arg-1I'h-> O, that is, the subdividing
the curve
curve has a different limiting direction from the bounding Q-curve.
42
From the general existence theorem of page
3,Z
one
would suppose that the coefficients of a general solution
C,
would depend upon
1y.
and that these coefficients might
,
1'
change due to Stokes' phenomenon whent
different region 5'r
is taken into a
It is desired to find the conriec-
tion of these coefficients in the different regions, and
if n terms are needed in the expansion we shall now show
how to calculate the coefficients
tions
'øI«
( .%
,j
), K
(,t'
C
for the solu-
)
1,2, principal* at the origin so
that the asymptotic form of these solutions is known ex-
plicitly to n terms.
We indicate these solutions in the
form
C,(& e
T
(',y') =
(30)
U,r()
,QT
(3-r.
I-fl-,
CKZ
e0(2'1L tA(()
j
îz(,1
J,
where the
1sare bounded functions forp sufficiently
large in S
and
on (-ib , b
1G
).
The coefficients will
be found in the form
(#0)
-r
+
Ç+
-
-
-
-
+
C
/
orr times this expression, where
the
explicitly and
forp
(,"
This form, however,
)
is sufficient since
w
*The solutions
such that
(o)
:
is bounded
C
and
ti
t
ar'e
determined
sufficiently large,
the
may be
principal at the origin are
o
w11(o)ao
43
e
absorbed in the
of (30).
Let n be one of the numbers 0,1,2,.....
However,
.
after lt Is chosen it remains fixed In all further calcu-
lations.
Now by
t..i2
'
(
''
,
=
)
e
we
shall denote the n-el terms
7
From the general existence theorem of
write the following:
-e
(31)
E
Here the
(
-b , $
thelr
).
-
o(. 1'«
¡
e'E
/
=
pae 3
we may now
(içe)
('")
&-
are bounded functions for1-
in SD
and
on
We may now write the principal solutions and
derivatives as follows:
C
(32)
I
I
TC,fryI'?C(e)Ç,
where
(33)
)J/,J
i(o)=J,
By making
J
oca
(e)
T
r
I
tA
J
(0)
=
(
-1-I
1
i
J
If
in the following man-
use of (32) we write (33)
ner, remembering that
( 34)
-o
for all n, and
+
P
C I( Z (
(e)u ,)+_L
+ 1C
Jr
E2
¿l,2:
o
'
p
E (o
1
44
Equation (34) gives us
C
the
system which may be solved for
a
providing the characteristic determinant
's
not equal zero.
does
In tbls case
:
,
-
LA1
oÇ-
(o
E6)
-;;-;.
*
,o)
Here
L4
for,'
sufficiently large, we may write the solutions for
(o,,'
)
Ç,and
rÁ
and
,
o
O.
Thus
O,
-
and
These become the following:
i(u2'(oP)+
c'1
(35)
I-
1C1Ço)
.
OL 'C'
¿:::
At this point it
will be noted from the series for
and Ó. of Chapter VIII that
(O,
,o
)
is
is of the form
.i
=
and
)
dAV
of the forni
¿
Thus the
(36)
C
's of
will take the forni
($5)
+,
C('')..-.
(
or
-1-
(
-
-t-.. . +
rl +
times such an expression, where the
for large values of,"
.
E- ts
are bounded
Now (36) by mere "long division"
may be written in the form
- co+f,
-7;
E
?'
)
45
or
the
/
where E
times such an expression,
CtS
are known explicitly.
is bounded,
and
This is the procedure for
calculation we set out to establish.
Fronì
the equation defining a region
on page 3/
we
may now take 7'O,2,4,..... as indicating the regions in
i
which
-j
and
o(--jando<
einiteS1
in a
Likewise
?.
odd integer when
(32)
are calculated
.
Now if the forms
fore
.
wand
,
of
say S0
,
the resulting expressions
are exactly the same for all regions
since we have the
following results:
('y')]
î
C
£u(x,t')\z
.+,
C
'+I ClU
'
1U'(1(-'),
D
C.
1u:c;,
Cgg
Tfi(A1(1,Í7)],
where the brackets denote that the remainder terms are omitted.
Hence it is no longer necessary to affix the sub-
in
script
and
-
the asymptotic formulas.
in all
Sr
We shall
take.i
since it is shown there is no
Stokes' phenomenon.
The four còefficients
of (35) have been calculated
46
for
n4,
and these are given below
('
\JII =1
(ti
-
+
1-
/
;i
Q
L
2
-4-
-a-
77)
s
-_L 7-..L -j----
C a,
4
2
(_4.
L
f
2.
where the
*
taj-),
,°
are bounded functions f
s
&
fi.' I,
or
on (-1
,
The principal solutions (32) may now be expressed
for
n=
follows:
4 as
r,,".i..
w,
L 3 J
+(I
I
If.-:'
6
z
)
+
z
t
j--
L
I
z
/
+
()
Pi.%
e
t&I2
;-
4L
¡
&j2I'
i
'rJ
fo
s-
2
I
L:t
/2-
-
3J4''1
a
_!_;7] -fa. J 7/
Ê3(içe)
,(?1
1-i.
L22 (
22P3}
iYL__
i2
o
F
f
() i
(_
J
BIBLI OGBAPIJY
Birkhoff, G.D.,
iety, volume
Transactions of American Mathematical Soc9, 1908
Fabry, E.E'., These,
University of Paris, 1885
Pierpont, James, Funebions of
Co., 1914
a
Complex Variable. Ginn and
Trjitzinsky, W.J.., Acta Mathematica, volume 62:1-2,
1933
Wbittaker and Vatson, Modern Analysis, Cambridge University Press,
1933