Volume 2010, Article ID 898274, 11 pagesdoi:10.1155/2010/898274 Research Article Approximation of Analytic Functions by Kummer Functions Soon-Mo Jung Mathematics Section, College of Scie
Trang 1Volume 2010, Article ID 898274, 11 pages
doi:10.1155/2010/898274
Research Article
Approximation of Analytic Functions by
Kummer Functions
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea
Correspondence should be addressed to Soon-Mo Jung,smjung@hongik.ac.kr
Received 3 February 2010; Revised 27 March 2010; Accepted 31 March 2010
Academic Editor: Alberto Cabada
Copyrightq 2010 Soon-Mo Jung This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We solve the inhomogeneous Kummer differential equation of the form xy β − xy− αy
∞
m0 a m x m and apply this result to the proof of a local Hyers-Ulam stability of the Kummer
differential equation in a special class of analytic functions
1 Introduction
Assume thatX and Y are a topological vector space and a normed space, respectively, and
thatI is an open subset of X If for any function f : I → Y satisfying the differential inequality
a n xy n x a n−1 xy n−1 x · · · a1xyx a0xyx hx ≤ ε 1.1 for allx ∈ I and for some ε ≥ 0, there exists a solution f0:I → Y of the differential equation
a n xy n x a n−1 xy n−1 x · · · a1xyx a0xyx hx 0 1.2
such thatfx−f0x ≤ Kε for any x ∈ I, where Kε depends on ε only, then we say that
the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam stability if the domainI is not the whole space X We may apply this terminology for other
differential equations For more detailed definition of the Hyers-Ulam stability, refer to 1 6 Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations see 7,8 Here, we will introduce a result of Alsina and Ger
see 9 If a differentiable function f : I → R is a solution of the differential inequality
Trang 2|yx − yx| ≤ ε, where I is an open subinterval of R, then there exists a solution f0 :I → R
of the differential equation yx yx such that |fx − f0x| ≤ 3ε for any x ∈ I.
This result of Alsina and Ger has been generalized by Takahasi et al They proved
in 10 that the Hyers-Ulam stability holds true for the Banach space valued differential equationyx λyx see also 11
Using the conventional power series method, the author12 investigated the general solution of the inhomogeneous Legendre differential equation of the form
1− x2
yx − 2xyx pp 1yx ∞
m0
a m x m 1.3
under some specific conditions, wherep is a real number and the convergence radius of the
power series is positive Moreover, he applied this result to prove that every analytic function can be approximated in a neighborhood of 0 by the Legendre function with an error bound expressed byCx2/1 − x2 see 13–16
general solution of the inhomogeneous Kummerdifferential equation
xyx β − xyx − αyx ∞
m0
a m x m , 1.4
whereα and β are constants and the coefficients a mof the power series are given such that the radius of convergence isρ > 0, whose value is in general permitted to be infinite Moreover,
using the idea from12,13,15, we will prove the Hyers-Ulam stability of the Kummer’s equation in a class of special analytic functionssee the class CKinSection 3
In this paper,N0 and Z denote the set of all nonnegative integers and the set of all integers, respectively For each real number
α, that is, the least integer not less than α.
2 General Solution of 1.4
The Kummerdifferential equation
xyx β − xyx − αyx 0, 2.1
which is also called the confluent hypergeometric differential equation, appears frequently in practical problems and applications The Kummer’s equation2.1 has a regular singularity
atx 0 and an irregular singularity at ∞ A power series solution of 2.1 is given by
Mα, β, x∞
m0
α m
m!βm x m , 2.2
whereα mis the factorial function defined byα0 1 and α m αα1α2 · · · αm−1
for allm ∈ N The above power series solution is called the Kummer function or the confluent
Trang 3hypergeometric function We know that if neitherα nor β is a nonpositive integer, then the
power series forMα, β, x converges for all values of x.
Let us define
Uα, β, x π
sinβπ
Mα, β, x
Γ1 α − βΓβ − x1−β
M1 α − β, 2 − β, x ΓαΓ2− β . 2.3
We know that if β / 1 then Mα, β, x and Uα, β, x are independent solutions of the
Kummer’s equation 2.1 When β > 1, Uα, β, x is not defined at x 0 because of the
factorx1−βin the above definition ofUα, β, x.
By considering this fact, we define
I ρ
⎧
⎨
⎩
−ρ, ρ, forβ < 1,
−ρ, 0∪0, ρ, forβ > 1, 2.4
for any 0 < ρ ≤ ∞ It should be remarked that if β /∈ Z and both α and 1 α − β are not
nonpositive integers, thenMα, β, x and Uα, β, x converge for all x ∈ I∞see 17, Section
13.1.3.
Theorem 2.1 Let α and β be real constants such that β /∈ Z and neither α nor 1α−β is a nonpositive
integer Assume that the radius of convergence of the power series∞
m0 a m x m is ρ > 0 and that there exists a real number μ ≥ 0 with
m − 1!βm a m
α m1
≤ μ
m−1
i0
i!βi a i
α i1
for all sufficiently large integers m Let us define ρ0 min{ρ, 1/μ} and 1/0 ∞ Then, every
solution y : I ρ0 → C of the inhomogeneous Kummer’s equation 1.4 can be represented by
yx y h x ∞
m1
m−1
i0
i!α mβi a i
m!α i1βm x m , 2.6 where y h x is a solution of the Kummer’s equation 2.1.
Proof Assume that a function y : I ρ0 → C is given by 2.6 We first prove that the function
y p x, defined by yx − y h x, satisfies the inhomogeneous Kummer’s equation 1.4 Since
y
p x ∞
m1
m−1
i0
i!α mβi a i
m − 1!α i1βm x m−1
∞
m0
m
i0
i!α m1βi a i
m!α i1βm1 x m ,
y
p x ∞
m1
m
i0
i!α m1βi a i
m − 1!α i1βm1 x m−1 ,
2.7
Trang 4we have
xy
p x β − xy
p x − αy p x a0∞
m1
m
i0
i!α m1βim βa i m!α i1βm1 x m
−∞
m1
m−1
i0
i!α mβi m αa i
m!α i1βm x m
a0∞
m1
a m x m ,
2.8
which proves thaty p x is a particular solution of the inhomogeneous Kummer’s equation
1.4
We now apply the ratio test to the power series expression ofy p x as follows:
y p x ∞
m1
m−1
i0
i!α mβi a i
m!α i1βm x m
∞
m1
c m x m 2.9
Then, it follows from2.5 that
lim
m → ∞
c m1
c m
≤ limm → ∞α m β m
⎡
⎣ 1
m 1
m
m 1
m − 1!βm a m
α m1
m−1
i0
i!βi a i
α i1
−1⎤
⎦
≤ μ.
2.10
Therefore, the power series expression of y p x converges for all x ∈ I1 /μ Moreover, the convergence region of the power series for y p x is the same as those of power series for
y
p x and y
p x In this paper, the convergence region will denote the maximum open set
where the relevant power series converges Hence, the power series expression forxy
p x
β − xy
p x − αy p x has the same convergence region as that of y p x This implies that
y p x is well defined on I ρ0 and so does for yx in 2.6 because y h x converges for all
x ∈ I∞under our hypotheses forα and β see aboveTheorem 2.1
Since every solution to 1.4 can be expressed as a sum of a solution y h x of the
homogeneous equation and a particular solutiony p x of the inhomogeneous equation, every
solution of1.4 is certainly in the form of 2.6
Remark 2.2 We fix α 1 and β 10/3, and we define
a0 10
3 , a m 1 4m2− 6m − 3
3m2m 1 2.11
Trang 5for everym ∈ N Then, since lim m → ∞ a m /a m−1 1, there exists a real number μ > 1 such that
m − 1!βm a m
α m1
10· 13 · 16 · · · 3m 4
m3 m−1 a m−1·3m 7
3m ·
a m
a m−1·
m
m 1
m − 1!
βm−1 a m−1
α m ·
3m 7
3m ·
a m
a m−1·m 1 m
≤ μ m − 1!
βm−1 a m−1
α m
≤ μ
m−1
i0
i!βi a i
α i1
2.12
for all sufficiently large integers m Hence, the sequence {am} satisfies condition 2.5 for all sufficiently large integers m
3 Hyers-Ulam Stability of 2.1
In this section, letα and β be real constants and assume that ρ is a constant with 0 < ρ ≤ ∞.
For a givenK ≥ 0, let us denote C Kthe set of all functionsy : I ρ → C with the properties a andb:
a yx is represented by a power series∞m0 b m x mwhose radius of convergence is at leastρ;
b it holds true that∞m0 |a m x m | ≤ K|∞m0 a m x m | for all x ∈ I ρ, where a m m
βm 1b m1 − m αb mfor eachm ∈ N0
It should be remarked that the power series∞
m0 a m x minb has the same radius of convergence as that of∞
m0 b m x mgiven ina
In the following theorem, we will prove a local Hyers-Ulam stability of the Kummer’s equation under some additional conditions More precisely, if an analytic function satisfies some conditions given in the following theorem, then it can be approximated by a
“combination” of Kummer functions such asMα, β, x and M1 α − β, 2 − β, x see the
first part ofSection 2
Theorem 3.1 Let α and β be real constants such that β /∈ Z and neither α nor 1α−β is a nonpositive
integer Suppose a function y : I ρ → C is representable by a power series∞m0 b m x m whose radius
of convergence is at least ρ > 0 Assume that there exist nonnegative constants μ / 0 and ν satisfying the condition
m − 1!βm a m
α m1
≤ μ
m−1
i0
i!βi a i
α i1
≤ ν
m 1!βm a m
α m1
3.1
Trang 6for all m ∈ N0, where a m m βm 1b m1 − m αb m Indeed, it is sufficient for the first inequality in3.1 to hold true for all sufficiently large integers m Let us define ρ0 min{ρ, 1/μ} If
y ∈ C K and it satisfies the differential inequality
xyx β − xyx − αyx ≤ ε 3.2
for all x ∈ I ρ0and for some ε ≥ 0, then there exists a solution y h:I∞ → C of the Kummer’s equation
2.1 such that
yx − y h x ≤
⎧
⎪
⎪
⎪
⎪
ν
μ·
2α − 1
α Kε for α > 1, ν
μ
m
0 −1
m0
m α m 1 −m 1 α m 2 m0 1
m0 α Kε for α ≤ 1,
3.3
for any x ∈ I ρ0, where m0
Proof By the definition of a m, we have
xyx β − xyx − αyx
∞
m0
m βm 1b m1 − m αb m
x m
∞
m0
a m x m
3.4
for allx ∈ I ρ So by3.2 we have
∞
m0
a m x m
for anyx ∈ I ρ0 Sincey ∈ C K, this inequality together withb yields
∞
m0
|a m x m | ≤ K
∞
m0
a m x m
for eachx ∈ I ρ0
By Abel’s formulasee 18, Theorem 6.30, we have
n
m0
|a m x m|
m 1 m α
n
i0
a i x i
n 1 α n 2
n
m0
m
i0
a i x i
m 1 m α
−m 1 α m 2
3.7
Trang 7for anyx ∈ I ρ0andn ∈ N With m0
if α > 1, then m α m 1 < m 1 α m 2 form ≥ 0;
if α ≤ 1, then m α m 1 ≥ m 1 α m 2 form ≥ m0.
3.8
Due to3.4, it follows fromTheorem 2.1and2.6 that there exists a solution y h x of
the Kummer’s equation2.1 such that
yx y h x ∞
m0
m−1
i0
i!α mβi a i
for allx ∈ I ρ0 By using3.1, 3.6, 3.7, and 3.8, we can estimate
yx − y h x ≤∞
m0
a m x m m 1
m α
α m1
m 1!βm a m
m−1
i0
i!βi a i
α i1
≤ ν
μ n → ∞lim
n
m0
|a m x m|
m α m 1
≤
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ν
μ n → ∞lim
Kε
n 1 α n 2 n
m0
Kε
m 2
m 1 α −
m 1
m α for α > 1,
ν
μ n → ∞lim
Kε
n 1 α n 2 m0−1
m0
Kε
m α m 1 −m 1 α m 2
n
mm0
Kε
m 1
m α−
m 2
⎧
⎪
⎪
⎪
⎪
ν
μ·
2α − 1
α Kε for α > 1, ν
μ
m
0 −1
m0
m 1 m α −m 1 α m 2 m0 1
m0 α Kε for α ≤ 1
3.10 for allx ∈ I ρ0
We now assume a stronger condition, in comparison with3.1, to approximate the given functionyx by a solution y h x of the Kummer’s equation on a larger punctured
interval
Corollary 3.2 Let α and β be real constants such that β /∈ Z and neither α nor 1α−β is a nonpositive
integer Suppose a function y : I∞ → C is representable by a power series ∞
m0 b m x m which
Trang 8converges for all x ∈ I∞ For every m ∈ N0, let us define a m m βm 1b m1 − m αb m Moreover, assume that
lim
m → ∞
m − 1!βm a m
α m1 0, 0<
∞
i0
i!βi a i
α i1
< ∞ 3.11
and there exists a nonnegative constant ν satisfying
m−1
i0
i!βi a i
α i1
≤ ν
m 1!βm a m
α m1
for all m ∈ N0 If y ∈ C K and it satisfies the differential inequality 3.2 for all x ∈ I∞ and for some
ε ≥ 0, then there exists a solution y n:I∞ → C of the Kummer’s equation 2.1 such that
yx − y n x ≤
⎧
⎪
⎪
⎪
⎪
ν · 2α − 1
α Kε for α > 1, ν
m
0 −1
m0
m α m 1 −m 1 α m 2 m0 1
m0 α Kε for α ≤ 1
3.13
for any x ∈ I n , where m0
Proof In view of3.11 and 3.12, we can choose a sufficiently large integer n with
m − 1!βm a m
α m1
≤
1
n
m−1
i0
i!βi a i
α i1
≤ ν n
m 1!βm a m
α m1
, 3.14
where the first inequality holds true for all sufficiently large m, and the second one holds true for allm ∈ N0
If we defineρ0 n, thenTheorem 3.1implies that there exists a solutiony n :I∞ → C
of the Kummer’s equation such that the inequality given for|yx − y n x| holds true for any
x ∈ I n
4 An Example
We fixα 1, β 10/3, ε > 0, and 0 < ρ < 1 And we define
b0 0, b m ε
s ·
1
Trang 9for allm ∈ N, where we set s 5/32 − ρ/1 − ρ We further define
yx ∞
m0
for anyx ∈ I ρ
Then, we seta m m βm 1b m1 − m αb m, that is,
a0 10
3 ·ε
s , a m
14m2− 6m − 3
3m2m 1
ε
s ≤
5
3·ε
for everym ∈ N Obviously, all a ms are positive, and the sequence{a m} is strictly monotone decreasing, from the 4th term on, to ε/s More precisely, a0 > a1 < a2 < a3 < a4 > a5 >
a6> · · ·
Since
a0 10
3 ·ε s > 1
6 ·ε s41
36·ε s a1 a3 , 4.4
we get
∞
m0
a m x m
a0 a1 x a
2x2 a3 x3a4x4 a5 x5
a6x6 a7 x7
· · ·
≥a0 a1 x a
2x2 a3 x3
≥ a0 − a1 − a3
73
36·ε s
4.5
for eachx ∈ I ρand
∞
m0
|a m x m| ≤∞
m0
a m ρ m≤
10
3 ∞
m1
5
3ρ m
ε
s ε 4.6
for allx ∈ I ρ Hence, we obtain
∞
m0
|a m x m | ≤ K
∞
m0
a m x m
for anyx ∈ I ρ, whereK 60/73 · 2 − ρ/1 − ρ, implying that y ∈ C K
Trang 10We will now show that{a m} satisfies condition 3.1 For any m ∈ N, we have
m−1
i0
i!βi a i
α i1
a0m−1
i1
10· 13 · 16 · · · 3i 7
i 13 i a i
≤
10
3 m−1
i1
10· 13 · 16 · · · 3i 7
i 13 i ·5
3
ε
s ,
m 1!βm a m
α m1
≥
10· 13 · 16 · · · 3m 7
3m ·1
6·ε s ,
4.8
since limm → ∞ a m ε/s.
It follows from4.8 that
m−1
i0
i!βi a i
α i1
≤10
1
3 m−1
i1
10· 13 · 16 · · · 3i 7
i 13 i ·1
6
ε s
10
1
3 10· 13 · · · 3m 7
3m
m−1
i1
3m−i
3i 10 · · · 3m 7·
1
i 1·
1 6
ε s
≤ 10
1
3 10· 13 · 16 · · · 3m 7
3m
m−1
i1
1
i 12 ·1
6
ε s
≤ 1010· 13 · 16 · · · 3m 7
3m
1
101
6ζ2 − 1
ε
s
5π2− 12
3 ·10· 13 · 16 · · · 3m 7
3m ·1
6·ε s
≤ 5π2− 12 3
m 1!βm a m
α m1
.
4.9
We know that the inequality4.9 is also true for m 0.
On the other hand, in view ofRemark 2.2, there exists a constant μ > 1 such that
inequality 2.12 holds true for all sufficiently large integers m By 2.12 and 4.9, we conclude that{a m} satisfies condition 3.1 with ν 5π2− 12μ/3.
Finally, it follows from4.6 that
xyx β − xyx − αyx ∞
m0
a m x m
≤
∞
m0
|a m x m | ≤ ε 4.10
for allx ∈ I ρ withρ0 min{ρ, 1/μ}.