数学论文_英文翻译
-
毕业设计
(
论文
)
附录
（翻译）
课
题
名
称
一些周期性的二阶线性微分方程解的方法
学
生
姓
名
万
益
目
录
1
．毕业设计（论文）附录（翻译）
英文
2
．毕业设计（论文）附录（翻
译）中文
2014
年
5
月
25
日
Some Properties of Solutions of
Periodic Second Order
Linear
Differential Equations
1.
Introduction and main results
In
this
paper,
we
shall
assume
that
the
reader
is
familiar
with
the
fundamental
results
and
the
stardard notations of the Nevanlinna's
value distribution theory of meromorphic functions
[12, 14,
(
f
)
and
(
f
)
to denote
respectively the order
16]. In
addition, we will use the notation
(
f
)
，
of
growth,
the
lower
order
of
growth
and
the
exponent
of
convergence
of
the
zeros
of
a
meromorphic function
f
，
e
(
f
)
（
[see 8]
）
，
the
e-type order of f(z), is defined to be
e
(
f
)
lim
log
T
(
r
,
f
)
r
r
Simila
rly,
e
(
f
)
，
the e-type
exponent of convergence of the zeros of
meromorphic function
f
,
is defined to be
log
< br>
N
(
r
,
1
/
f
)
e
(
f
)
lim
r
r
We say that
f
(
z
)
has regular order of
growth if a meromorphic function
f
(
z
)
satisfies
(
f
)
lim
log
T
(
r
,
f
)
r
log
r
We
consider the second order linear differential
equation
f
Af
0
Where
A
(
z
)
B
(
e
z
)
is
a
periodic
entire
function
with
period
< br>2
i
/
.
The
complex
oscillation theory of (1.1) was first
investigated by Bank and Laine [6]. Studies
concerning (1.1)
have
een
carried
on
and
various
oscillation
theorems
have
been
obtained
[2{11,
13,
17{19].
When
A
(
z
)
is
rational in
e
，
Bank and Laine
[6] proved the following theorem
Theorem
A
Let
A
(
z
)
B
(
e
z
)
be
a
periodic
entire
function
with
period
2
i
/
and
rational
in
e
z
z
.If
B
(
)
has
poles
of
odd
order
at
both
and
0
,
then
for
every
solution
f
(
z
)(
0
)
of (1.1),
(
f
)
Bank
[5]
generalized
this
result:
The
above
conclusion
still
holds
if
we
just
suppose
that
both
and
0
are
poles
of
B
< br>(
)
,
and
at
least
one
is
of
odd
order.
In
addition,
the
stronger conclusion
log
N
(
r
,
1
/
f
)
o
(
r
)
(1.2)
holds. When
A
(
z
)
is
transcendental in
e
, Gao [10]
proved the following theorem
Theorem
B
Let
B
(
)
g
(
1
/
)<
/p>
j
b
j
1
j
,where
g
(
t
)
is a
transcendental entire function
p
z
z
with
(
g
)
< br>
1
,
p
is
an
odd
positive
integer
and
b
p
0
，
Let
A
(
z
)
B
(
< br>e
)
.Then
any
non-trivia
solution
f
of
(1.1)
must
have
(
f
)
.
In
fact,
the
stronger
conclusion
(1.2)
holds.
An example was given
in [10] showing that Theorem
B does not
hold when
(
g
)
is any
positive
integer. If the order
(
g
)
1
, but is not a positive integer,
what can we say? Chiang
and Gao [8]
obtained the following theorems
z
Theorem
C
Let
A
(
z
)
<
/p>
B
(
e
)
,where
B
(
)
g
1
(
1
/
)
g
2
(
)
,
< br>g
1
and
g
< br>2
are
entire
fu
nctions
g
2
transcen
dental and
(
g
2
)
not equal to a
positive integer or infinity,
and
g
1
arbitrary.
(i)
(
g
2
)
1
.
(a)
If
f
is
a
non-trivial
solution
of
(1.1)
with
e
(
f
)
<
/p>
(
g
2
)
;
then
f
(<
/p>
z
)
and
f<
/p>
(
z
2
i
)
are
linearly
dependent.
(b)
If
f
1
and
f
2
are
any
two
linearly
independent
solutions of (1.1), then
e<
/p>
(
f
)
(
g
2
)
.
Suppose
(
g
2
)
1
(a)
If
f
is
a
non-trivial
solution
of
(1.1)
with
e
(
f
)
1<
/p>
,
f
(
z
)
and
f
(
z
2
i
)
are
linearly
dependent.
< br>If
f
1
and
f
2
are
any
two
linearly independent
solutions of (1.1),then
e
(
f
1
f
2
)
1
.
Theorem D
Let
g
(
)
be a transcendental entire function and its order
be not a positive integer or
(ii)
Suppose
infinity.
Let
A
(
z
)
B
(
e
z
)
;
where
B
(
)
g
(
1
< br>/
)
p
j
b
and
p
is
an
odd
positive <
/p>
j
j
1
integer. Then
(
f
)
or each non-trivial solution f to (1.1). In
fact, the stronger conclusion
(1.2)
holds.
Examples
were
also
given
in
[8]
showing
that
Theorem
D
is
no
longer
valid <
/p>
when
(
g
)
is
infinity.
The
main
purpose
of
this
paper
is
to
improve
above
results
in
the
case
when
B
(
)
is
transcendental. Specially, we
find a condition under which Theorem D still holds
in the case when
(
g
)
is a positive integer or
infinity. We will prove the following results in
Section 3.
Theorem
1
Let
A
(
z
)
<
/p>
B
(
e
)
,where
B
(
)
g
1
(
1
/
)
g
2
(
)
,
< br>g
1
and
g
< br>2
are
entire
functions
with
g
2
transcendental
and
z
(
g
2
)
not
equal
to
a
positive
integer
or
infinity,
and<
/p>
g
1
arbitrary.
If
Some
properties
of
solutions
of
periodic
second
order
linear
differential
equations
f
(
z
)
< br>
and
f
(
z
2
i
)
are two linearly
independent solutions of (1.1), then
<
/p>
e
(
f
)
Or
e
(
f
)
1
(
g
2
< br>)
1
2
We remark that the
conclusion of Theorem 1 remains valid if we assume
(
g
1
)
is
not
equal
to
a
positive
integer
or
infinity,
and<
/p>
g
2
arbitrary
and
still
assume<
/p>
B
(
)
g
1
(
1
/
)
g
2
(
)
,In the case
when
g
1
is
transcendental with its lower order not
equal
to
an
integer
or
infinity
and
g
2
is
arbitrary,
we
need
only
to
consider
B
*
(
)
B
(
1
< br>/
)
g
1
(
)
g
2
(
1
/
)
in
0
,
1
/
< br>.
Corollary 1
Let
< br>A
(
z
)
B
(
e
)
,where
B
(
)
g
1
(
1
/
<
/p>
)
g
2
(
)
,
g
1
and
g
2
are
entire functions
with
g
2
transcendental and
(a)
(b)
z
(
g
2
)
no more than 1/2, and
g
1
arbitrary.
If
f
is
a
non-trivial
solution
of
(1.1)
with
e
(
f
)
,then
f
(
z
)
and
f
(
z
2
i
)
ar
e linearly dependent.
If
f
1
and
f
2
are
any
two
linearly
independent
solutions
of
(1.1),
then
e
(
f
1
f
2
)
.
Theorem 2
L
et
g
(
)
be a transcendental entire function and
its lower order be no more than 1/2.
z<
/p>
Let
A
(
z<
/p>
)
B
(
e
)
,where
B
(
)
g
(
1
/
)
p
b
j
< br>
1
j
j
and
p
is
an
odd
positive
integer,
then
(
f
)
for
each
non-trivial
solution
f
to
(1.1).
In
fact,
the
stronger
conclusion
(1.2)
holds.
We remark that the above
conclusion remains valid if
B
(
)
g
(
)
b
j
j
j
1
p
< br>We note that Theorem 2 generalizes Theorem D wh en
(
g
)
is a positive integer or infinity but <
/p>
(
g
)
1
/
2
. Combining Theorem D with Theorem 2, we have
z
Corollary
2
Let
g
(
)
be
a
transcendental
entire
function.
Let
A
(
z
)
B
(
e<
/p>
)
where
B
(
)
g
(
1
/
)
j
< br>
1
b
j
j
and
p
is
an
odd
positive
integer.
Suppose
that
either
(i)
or
(ii)
below holds:
(i)
(
g
)
is not a
positive integer or infinity;
(ii)
(
g
)
1
/
2
< br>;
then
(
f
)
for
each
non-trivial
solution
f
to
(1.1).
In
fact,
the
stronger
conclusion
(1.2)
holds.
2.
Lemmas for the proofs of Theorems
Lemma 1
([7]) Suppose that
k
2
and t
hat
A
0
,.....
A
k
2
< br>are entire functions of period
2
i
,and that
f is a non-trivial solution of
< br>p
y
(
k
)
A
j
(
z
)
y
(
j
)
(
z
)
0
i
0
< br>k
2
Suppose
further
that
f
satisfies
log
N
(
r
,
< br>1
/
f
)
o
(
r
)
;
that
A
0
is
non-constant
and
rational
z
in
e
，
and
that
if
k
3<
/p>
，
then
A
1
,.....
A
k
2
are
constants.
Then
there
exists
an
integer
q
with
1
q
k
such
that
f
(
z
)
and
f
(<
/p>
z
q
2
i
)
are
linearly
dependent.
The
same
conclusion
z
holds
if
A
0
is
transcendental
in
e
，
and f satisfi
es
log
N
(
r
,
1
/
f
)
o
(
r
)
，
and
if
k
3
，
then
through
a <
/p>
set
L
1
r<
/p>
k
2
.
have
T<
/p>
(
r
,
A
j
)
o
(
T
(
r
,
A
j
))
< br>for
j
1
< br>,.....
as
z
of
infinite
measure,
1
we
and be
Lemma 2
([10]) Let
A
(
z
)
B
(
e
z
)
be a periodic entire function with p
eriod
2
i
transcendent
al in
e
,
B
< br>(
)
is
transcendental and analytic on
0
< br>
.If
B
(
)
has a pole of
odd order
at
or
0<
/p>
(including those which can be changed
into this case by varying the
period of
A
(
z
)
and
Eq
.
(1.1) has a solution
f
(
z
)
0
< br>which satisfies
log
N
(
r
,
1
/
f
)
o
(
r
)
,
then
f
(
z
)
and
f
(
z
)
ar
e linearly independent.
3.
Proofs of main results
The
proof of main results are based on [8] and [15].
Proof
of
Theorem
1
Let
us
assume
e
(
f
)
.Since
f
(
z
)
and
f
(
z
2
i<
/p>
)
are
linearly
independent,
Lemma
1
implies
that
f
(
z
)
and
f
(
z
4
i
)
must
be
linearly
dependent.
< br>Let
E
(
z
< br>)
f
(
z
)
f
(
z
2
i
)
,Then
E
(
z
)
satisfies the
differential equation
E
(
z
)
2
E
(
< br>z
)
c
2
,
(2.1)
4
A
(
z
)
(
)
<
/p>
2
E
(
z
)
E
(
z
)
E
(
z
)
2
Where
c
0
is
the
Wronskian
of<
/p>
f
1
and
f<
/p>
2
(see
[12,
p.
5]
or
[1,
p.
354]),
and
E
(
z
2
i
)
c
1
E
(
z
)
or some non-zero constant
c
1
.Clearly,
E
/
E
and
E
/
E
are
both
periodic
functions
with
period
2
i
,while
A
(
z
)
is
periodic
by
definition.
2
Hence
(2.1)
shows
that
E
(
z
)
is
also
periodic
with
period
2
i
.Thus
we
can
find
an
analytic
function
(
)
in
0
yields
< br>
,so
that
E
(
z
)
2
(
e
z
)
Substituting
this
expression
into
(2.1)
c
2
3
<
/p>
4
B
(
)
2
(
)
2
2
(2.2)
4
Since both
B
(
)
and
(
)
are analytic in
C
*
:
1
,the
Valiron theory [21, p. 15]
gives their
representations as
n
B
(
)
R
(
)
b
(
<
/p>
)
，
(
)
n
1
R
1
(
)
(
)
，
（
2.3
）
n
1
are some
integers,
R
(
)
and
R
1
(
)
are
functions that are analytic and non-vanishing
where
n
，
on
C
*
{
}
，
b
(
)
and
(
)
are
entire functions. Following the same arguments as
used in [8],
we have
T
(
,
< br>)
N
(
,
1
/
)
T
(
,
b
)
S
(
,
)
，
< br>
（
2.4
）
where
S
(
,
)
o
(
T
(
,
))
.Furthermore, the following properties
hold [8]
e
(
f
)
e
(
E
)
e
(
E
2
)
ma
x{
eR
(
E
2
),
e
L
(
E
2
)}
,
eR
(
E
2
)
1
(
)
(
)
,
Where
eR
(
E
2
)
(resp,
eL
(
E
2
)
) is defined to be
< br>log
N
R
< br>(
r
,
1
/
E
2
)
l
og
N
R
(
r
,
1
/
E
2
)
lim
(resp,
lim
),
r
r
r
r
< br>Some properties of solutions of periodic second order linear differential equations
)
(resp.
N
< br>L
(
r
,
1
/
E
2
)
denotes a counting function that only
counts the zeros
2
of
E
(
z
)
in
the
right-half
plane
(resp.
in
the
left-half
plane),
1
(
)
is
the
exponent
of
convergence of the zeros of
in
C
*
,
which is defined to be
log
<
/p>
N
(
,
1
/
)
1
(
)
lim
log
Recall the condition
e
(
f
)
,we obtain
(
)
< br>
.
where
N
R
(
r
,
1
/
E
< br>Now substituting (2.3) into (2.2) yields
2
n
R
< br>
3
n
R
c
2
4
R
(
)
b
(
)
n
1
< br>
(
1
1
)
2
(
1
1
)
2
R
1
4
< br>R
1
R
1
(
)
(
)
R
1
n
1
R
1
< br>n
1
R
1
R
1
2
n
1
(
n
1
1
)
< br>(
2
2
2
)
（
2.5
）
2
R
1
R
1
R
1
n
Proof of Corollary 1
We can easily
deduce Corollary 1 (a) from Theorem 1 .
Proof
of
Corollary
1
(b).
Suppose
f
1
< br>and
f
2
are
linearly
independent
and
e
(
f
1
f
2
)
,then
e
(
f
< br>1
)
,and
Corollary
1
(a)
that
Let
f
j
(
z
< br>)
and
e
(
f
2
)
.We deduce from the
conclusion of
f
j
(
z
2
i
)
are
linearly
dependent,
j
=
1;
2.
E
(
z
)
f
1
(
z
)
f
2
(
z
)
.Then
we
can
find
a
non-zero
constant
c
2
such
that
E
(
z
2
i
)
c
2
E<
/p>
(
z
)
.Rep
eating the same arguments as used in Theorem 1 by
using the fact
2
that
E
(
z
)
is also periodic, we obtain
< br>e
(
E
)
1
(
g
2
)
1
2
,a
contradiction since
(
g
2
)
< br>1
/
2
.Hence
e
(
f
1
f
2
)
< br>
.
Proof
of
Theorem
2
Suppose
there
exists
a
non-
trivial
solution
f
of
(1.1)
that
satisfies
log
N
(
r
,
1
/
f
)
o
(
r
)
.
We
deduce
e
(
f
)
0
,
so
f
(
z
)
and
f
(
z
2
i
)
are
linearly
dependent by
Corollary 1 (a). However, Lemma 2 implies that
f
(
z
)
and
f
(
z
2
i
)
are linearly
independent.
This
is
a
contradiction.
Hence
log
N
(
r
,
1
/
f
)
o
(
r
)
holds
for
each
non-trivial
solution f of (1.1). This completes the
proof of Theorem 2.
Acknowledgments
The
authors
would
like
to
thank
the
referees
for
helpful
suggestions
to
improve this paper.
References
[1] ARSCOTT F M.
Periodic Di®erential Equations [M]. The Macmillan
Co., New York, 1964.
[2]
BAESCH
A.
On
the
explicit
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