THE SOLUTION IN THE LARGE OF A CERTAIN SECOND ORDER
ORDINARY LINEAR DIFFERENTIAL EQUATION OF RANK n
SENIOR THESIS
by
SUSAN C. MAYER
ADVISOR:
T. K. PUTTA SWAMY
BALL STATE UNIVERSITY
MUNCIE, INDIANA
WINTER, 1968-69
'_I' ..,_
TABLE OF CONTENTS
Page
Chapter
INTRODUCTION
1
II.
SOLUTION OF THE DIFFERENTIAL EQUATION AT z = 0 •••
3
III.
ASYMPTOTIC DEVELOPMENT OF SOLUTIONS AT z = 0 •••••
6
.......
10
V.
DETERMINATION OF STOKE'S MULTIPLIERS •••••••••••••
13
VI.
CRITIQUE. • . . . . .. . . . . • .. . . .. . . . . • .. . • .. .. .. .. . • . . • . .. . • . .. • •
14
VII..
BIBLlOORA.P.H:Y....... • . . . . • . . .. • • .. . • .. . .. .. • • .. .. .. .. .. . .. .. .. .. ..
15
I.
IV.
ASYMPTOTIC STRUCTURE OF SOLUTIONS AT z =
eo
INTRODUCTION
1.
In order to introduce the investigations of the present paper,
let us take for consideration the ordinary homogeneous linear differential
equation of second order,
2
d y
z
+ k
dz2
dv
=-
dz
2 2n-l y= 0
+ mz
(1.0)
Let the variable
z be regarded as complex, as likewise the constants
k
n be any positive integer.
and
m.
equation
.Let
(1.0)
will have in the language of Fuch's theory [lJ, a
z = 0, and an irregular singular point
finite regular singular point at
at
z = ~ , so that in all
complex
Then, the differential
(1.0)
has two singular points in the extended
z-plane.
z = 0
The indicial equation corresponding to
f(h)
=h(h
+ k - 1)
=0
(1.1)
The two roots of this equation are h = 0
(1.0)
h = 1 - k
and
complex their difference is not an integer.
assures that
is found to be,
and
h2
will have two fundamental solutions of the form,
being the two roots of (1.1).
(1.2)
We know that each of these
solutions will converge within a circle drawn about the point
and extending to the point
divergent.
k
z =
~.
At z =
In particular, the behavior of
is
Then, the established theory
i = 1,2
hl
Since
~
z = 0
, however, each becomes
Y , i = 1,2
i
in the
2
neighborhood of
z =
~
is not available, at least not without serious
z = ..
difficulties so far as Fuch1s theory itself is concerned, since
is an irregular singular point.
However, from the established theory of
linear differential equations it is known that in such a case, there will
exist two linearly independent solutions
neighborhood of the point
means of normal series.
z =
~
Yi'
i = 1,2
which in the
may be expressed asymptotically by
It is known that there exist a set of connection
coefficients, namely, Stokels multipliers
c
ij
such that,
2
I
Yi(z) =
j=l
Until
SO~l
c .. YJ.(z)
lJ
i = 1,2
(1.3)
method of computing these coefficients is devised, the
solution of the problem in the large cannot be considered as solved.
It
is to the solution of this problem in the large that the present thesis
is addressed.
In order to do this, we shall look upon the
~question
determinicog the asymptotic developments of two functions
as that of
Yi'
i = 1,2
when these are regarded as defined merely by means of their Maclaurin
development (1.2), that is without reference to the fact that they are
the solutions of
(1.0).
In this manner we shall arrive independently
at the fact that each is expressed when
combination of the functions
i
the precise nature of this dependence.
]z]
= 1,2
is large as a linear
and we shall arrive also at
3
=1.
SOLUTION OF THE DIFFERENTIAL EQUATION AT z = 0
Introducing the operator
2
£...z2
z
+ k
dv
U = z
d
the differential equation,
dz '
2 2n-l
+ m z
y = 0 becomes
~
(1.0)
dz
f(U)y + z2ng(U)Y = 0
where
feU) = u(u + k - 1)
corresponding to
=0
z
and
(2.0)
2
g(U) = m
The indicial equation
is found to be,
f(h) = h(h + k - 1) = 0
The roots of this equation are
hl
(2.1)
=0
and
~
=1
- k.
Since their
difference is not a multiple of the skip number 2n[2, page 418-423]
the differential equation (1.0) has two solutions of the form,
Yi
=z
h.
1-
co
I
c
r.
z
2rn
i
= 1,2
(2.2)
1-
r=o
Substituting (2.2) in (1.0) and equating the coefficient of z2n(r+l)+\
throughouG, we get the two term recurrence relation,
c +1 . [2n(r+l) + h. ] [2n(r+l) + h. - 1] +
r
,1.
1.
1.
2
k c +1 . [2n(r+l) + h. ] + m c
r
,1.
or
1.
2
-m c
[2n(r+l)
.= 0
r,l
.
r,1-
+ h. - 1 +k J
1-
4
2
-m
[2n(r+l) + h. 1[2n(r+l) + h. - 1 +k 1
So,
~
~
and
lim
cr+l,i
r~~
c-
= O.
Hence (2.2) converges for all finite
z
by
.
r,~
the ratio test.
Also, iteration of (2.3) yields,
=
j=l
where
Taking
i = 1
c
and
rl
i = 1,2
and
2n
hi = hl = 0, we get,
=
r
(2n)2r
Similarly taking
k - 1
a =
is arbitrary,
j=l
i = 2,
(r+l)fj
5=1
hi = h2 = 1 - k = -2na,
(2n)2r
(2.5)
(j+a)
we get
(2.6)
1
r
(r+l)
Tf
(j - a)
j=l
Here col
and
c o2
1
r (1 c
c
are arbitrary.
Choosing
col =
and
we get,
a)
rl
=
r2
=
~_m2~r
(2n)2r
r
(r+l)
r
(r + a + 1)
( _m2 )r
(2n)2r
r
(2.7)
(2. S)
(r+l) r(r - a + 1)
I)
Substitutjng this in (2.2) we get,
rr
( im zn)2r
2n
(r - cr + 1)
r=o
which converge for all finite
Let
r (r + 1)
z.
im n
S = -2n z
(2.10)
co
Then,
Yl =
I
r=o
Y2 =
~ s-2cr
'"
(s)2r
f' (r + cr + 1)
~s~2r
r (r + 1)
I
r=o f'(r-cr+l) r (r + 1)
which converge for all finite
,
where
S in the extended complex
~
= (~n)-2cr
lIn
S-plane.
6
ASYMPTarIC DEVELOPMENT OF SOL1ITIONS AT z = 0
III.
Nrn, we are in a position to apply W. B. Ford's VIIth Theorem
[3, p. 27, J.
Let
f(S)
be the function of the complex variable
S
defined by the series
f(S) =
I
(3.0)
r=o
in which
k l , k2
are constants (real or complex) and in which her)
may be regarded as a function
h(s)
of the complex variable
s = x + iy
and as such satisfies the two following conditions:'
h(s)
(a)
of
s
is a single valued, analytic function
res +~) res + k 2 )
throughout the finite
(b)
h(s)
s-space,
is such that when considered for values of
large modulus lying in the right half plane
s
Re(s) = x > xo ' where
is some assignable number, it may be expressed in the form,
c
h(s) = c
in which
c
n
Then
2
T(s-+';:k=-2')T(s-+~k-2~+.".1')
o
are constants and
lim
s
f ( S)
of
+ •••
~(s,n) = 0
n = 1,2,3 ...
~CO
has the following asymptotic development
xo
7
fen
f(s)
in which
Cj
e- 2S H2-P
N
2J1t
e
N
2s
'"
I
1=
co!
[
2J1t
R=
<;
Cj
p
0
(2S/
c
is the constant given by
where it is understood that, i f we take
(-s)-p = e-p[ln p + i(1f' - rr)
s-p = e -p rin p + i
3rr
2
o
p =
= c
, where
0
~
+ k2 -
~
J
If']
x0
is that obtained when arg ~ =
co
f(S)
N
-
I
and
o ~ If'
<;
2rr
(3.4)
o ~ If'
<;
2rr
p.5)
may be regarded as an
arbitrarily large negative number, the asymptotic development of
¥,
(3.3)
, we take
S =peilf'
Moreover, if in condition (b) the quantity
when arg S =
(3.2)
rr
- - < arg g < !!.
2
2
;
are determinate constants of which
as in (3.1), where
arg S <
0
""
-p
S
¥
;
(-2S)X
f(g)
¥in the following relation:
he-i)
+
./= 1
)~ o(2~f~
(
+
In all cases the interpretations
-s ) -pe -2s
2/lf
(3.4)
~
.1=0
and
So by the above mentioned theorem
have the following asymptotic behavior for
Yl
~
(-2s) m
(3.5)
and
(3.6)
are to be used.
Y2
(g) large:
as given by (2.10)
8
Yl
e- 2s {_sfPl
N
25
Yl
N
e
2s
~
cj
(-2S):A'
rr
2"
3rr
< arg S <
2
'=0
'"
(sfPl
L.
2Jit
£'=0
cR
(2S)K
l'
- "2
< arg S <
2"rr
(3.8)
(3.9)
where
c
P =a+:h and
=1
120
Similarly
rr
2 < arg S <
3rr
rr
rr
2
- 2" < arg S < 2
(3.10)
(3.11)
_ rr
arg S - 2" (3.12
where
P
,~
=_a+:h
2
and
d0
=1
9
So,
~ ~~-2s)~
'"
Yl - b 1 e
2s
s-a-l/2
-
~<args< 32T1
(3.13)
~<
(3.11)
arg S <
~
1=0
+ ble2Ss-a-l/2
'" (~fsre arg S =1£2
I
(3.15)
x. =0
_ (-1) -
where
~
-
0: -
1/2
2JTi
_ 1
--
,b
1
2Jri
Similarly
(3.16)
T1
T1
- - < arg s< 2
2
(3.17)
arg S =1£
2
where
=..!... ( -1) a-l/2
a
2
~ = (~n)
l1l1
-2a
2Jri
and
(3.18)
10
IV.
ASYMPI'arrC STRUCTURE OF SOLUTIONS AT z =
The point
z =
00
as already noted, is an irregular singular
,
point of the differential equation (1.0).
equation of rank
ro
Since (1.0) is
a differential
n, it would be advantageous to make the following
transformation of the independent variable
z.
Then the differential equation (1.0) becomes
n-l + k
n
n
~ - hs
ds
(h.O)
y = 0
We are now in a position to apply Birkoff's, theorem.
G. D. Birkoff
[h, pp. h63-h68] has completely discussed the existence, form, range of
the asymptotic solutions of the homogeneous linear differential equation
n-l
d nfl
dz
where the coefficient
a (z)
+ ... + a (z)y = 0
n
are developable
r
(Izl > R
= sufficiently
large) either in convergent series, namely
ar(z) = zrk[a
a
ro
+
rl +
z
_a r2+
... ] r =1
"2
2
z
k being zero or a positive integer (the integer
rank of
(h.o) at z
as follows:
= 00).
k+l
... n
(h.l)
is called the
Birkoff's essential results may be summarised
11
Let, for the equation (4.0), the roots
Pl' P , • "P n
2
of the
character'Lstic equation, i.e. the algebraic equation
+ a
be distimt from one another.
yr (r
solutions
no
= 0
Then there exists a set of fundamental
=
= 1,2, ••• n) of (4.0) in the neighborhood of z =
developable asymptotically in forms
r = 1,2, • •• n
where
is a polynomial of degree k + 1 in z, the coefficient
fr(Z)
of whose highest power of
constantEi with
ar
is
fir
k+l
P*
r
where
and
a .
rJ
are
.
=1
0
z
Applying the above mentioned Birkoff's theorem, the differential
equation (4.0) has the characteristic equation
p'1. - 4 = 0
P =
I
2 •
Thus equation ·(4.0) will have two linearly independent solutions
Yl ' Y2' for which
I
where
are normal divergent series having the forms
51
and
s2
sl = e
s2
S
I
is large
-2S SA, [1 + U + U
S
~2
= e 2s S ),~ [1 + U + L 2 .
S
~2
+ ••• ]
+ •••
1
12
where .\ 1
~2
and
are constants whose values may be determined by
formal substitutions of these series forms into (4.0)
found to be
A
1
1
- - a
2
\=-~
-a
and are thus
13
V.
DETERMINA.TION OF STOKE t S MULTIPLIERS
Therefore, we have if
of (1.0) at
z = 0
IsI
is large, the solution Y , j = 1,2
j
may be developed as follows:
2"IT
< arg
[im
n
2n z
1 < 2"
3IT
arg
[im nl
2n z
<
< arg
nl <
-im
2n z
3IT
" - < arg
nl <
-im
2n z
IT
" 2"IT <
IT
2
,
where
Similarly
IT
2"
IT
2
and
~ = ( ~n) "2ct
:un
2
2
14
VI.
-
CRITIQUE
A successful attempt has been made in this paper to solve the
.
differential equation (1.0) in the large, when 1 - k
number.
is a complex
A close examination of (1.0) reveals the noteworthy fact that
the results obtained in this paper hold good, even when
number, provided that
integer.
k
is not equal to
mn + 1
where
k
is a real
m is an
No attempt has been made in this paper to solve (1.0) in the
large under this hypothesis.
Yi' i = 1,2
about
In this case, one of the solutions
z will be logarthmic in character and as a con-
sequence, the evaluation of Stoke's multipliers and also the determination
of the behavior of the solutions in the various sectors awaits a far
more penetrating analysis to solve the problem in the large.
attempt to complete this
One can
part of the analysis as well as to explore
further the scope and limitations of the method that has been used to
solve the problem in the large in this paper completely.
Besides these, there is one other direction of further possible
research.
That is, generalization in the direction of a differential
equation of higher order than the second is evidently a possible field
of study.
15
VII.
BIBLIOGRAPHY
1.
Birko::f, G. D. I! Singular Points of Ordinary Linear Differential
Equations. I! Trans. American ~ Society. Vol. 10, 1909.
pp. 463-468.
2.
_ _--' and Trjitzinsky, W. J. I! Analytic Theory of Singular
Differential Equations.!! Acta Math. Vol. 60. 1933.
pp. 1-83.
-- --
3.
Brand, Lewis. Differential and Difference Equations.
and Som, Inc. New York, 1966. pp. 413-438.
4.
Ford, W. B. Studies ~ Divergent Series and Summability and
ASymptotic Developments of Functions Defined ~ Maclaurin
Series. Chelsea Publishing Company, New York.
5.
Ince, E. L. OrdinaEY Differential Equations.
Inc. New York, 1956.
6.
Wright, E. M. It The Asymptotic Exponent of the Generalized
Hypogeometric Function. u Proceed~s of London Math Soc.
Series 2. Vol. 46, 1940. pp. 38§::08.
John Wiley
Dover Publications,
7. - - -• U The Asymptotic Expansion of Integral Functions and of
the Coefficients in Their Taylor Series.1! Trans. American
Math Society.
Vol. 64, 1948.
pp. 409-438.