- Research Article
- Open Access
Approximation of Analytic Functions by Kummer Functions
© Soon-Mo Jung. 2010
- Received: 3 February 2010
- Accepted: 31 March 2010
- Published: 6 April 2010
We solve the inhomogeneous Kummer differential equation of the form 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.
- Power Series
- Real Constant
- Topological Vector Space
- Large Integer
- Legendre Function
such that for any , where depends on only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). 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 ). If a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any .
under some specific conditions, where 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 by the Legendre function with an error bound expressed by (see [13–16]).
where and are constants and the coefficients of the power series are given such that the radius of convergence is , whose value is in general permitted to be infinite. Moreover, using the idea from [12, 13, 15], we will prove the Hyers-Ulam stability of the Kummer's equation in a class of special analytic functions (see the class in Section 3).
In this paper, and denote the set of all nonnegative integers and the set of all integers, respectively. For each real number , we use the notation to denote the ceiling of , that is, the least integer not less than .
where is the factorial function defined by and for all . The above power series solution is called the Kummer function or the confluent hypergeometric function. We know that if neither nor is a nonpositive integer, then the power series for converges for all values of .
We know that if then and are independent solutions of the Kummer's equation (2.1). When is not defined at because of the factor in the above definition of .
for any . It should be remarked that if and both and are not nonpositive integers, then and converge for all (see [17, Section ]).
where is a solution of the Kummer's equation (2.1).
which proves that is a particular solution of the inhomogeneous Kummer's equation (1.4).
Therefore, the power series expression of converges for all . Moreover, the convergence region of the power series for is the same as those of power series for and . In this paper, the convergence region will denote the maximum open set where the relevant power series converges. Hence, the power series expression for has the same convergence region as that of . This implies that is well defined on and so does for in (2.6) because converges for all under our hypotheses for and (see above Theorem 2.1).
Since every solution to (1.4) can be expressed as a sum of a solution of the homogeneous equation and a particular solution of the inhomogeneous equation, every solution of (1.4) is certainly in the form of (2.6).
for all sufficiently large integers . Hence, the sequence satisfies condition (2.5) for all sufficiently large integers .
In this section, let and be real constants and assume that is a constant with . For a given , let us denote the set of all functions with the properties (a) and (b):
(a) is represented by a power series whose radius of convergence is at least ;
(b)it holds true that for all , where for each .
It should be remarked that the power series in (b) has the same radius of convergence as that of given in (a).
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 as and (see the first part of Section 2).
for any , where .
for each .
for all .
We now assume a stronger condition, in comparison with (3.1), to approximate the given function by a solution of the Kummer's equation on a larger (punctured) interval.
for any , where and is a sufficiently large integer.
where the first inequality holds true for all sufficiently large and the second one holds true for all .
If we define , then Theorem 3.1 implies that there exists a solution of the Kummer's equation such that the inequality given for holds true for any .
for any .
for every . Obviously, all s are positive, and the sequence is strictly monotone decreasing, from the 4th term on, to . More precisely, .
for any , where , implying that .
We know that the inequality (4.9) is also true for .
On the other hand, in view of Remark 2.2, there exists a constant such that inequality (2.12) holds true for all sufficiently large integers . By (2.12) and (4.9), we conclude that satisfies condition (3.1) with .
for all with .
for all .
The author would like to express his cordial thanks to the referee for his/her useful comments. This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (No. 2009-0071206).
- Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar
- Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
- Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar
- Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
- Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978, 72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
- Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.Google Scholar
- Obłoza M: Hyers stability of the linear differential equation. Rocznik Naukowo-Dydaktyczny. Prace Matematyczne 1993, (13):259–270.MathSciNetMATHGoogle Scholar
- Obłoza M: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Naukowo-Dydaktyczny. Prace Matematyczne 1997, (14):141–146.MathSciNetMATHGoogle Scholar
- Alsina C, Ger R: On some inequalities and stability results related to the exponential function. Journal of Inequalities and Applications 1998, 2(4):373–380.MathSciNetMATHGoogle Scholar
- Takahasi S-E, Miura T, Miyajima S: On the Hyers-Ulam stability of the Banach space-valued differential equation . Bulletin of the Korean Mathematical Society 2002, 39(2):309–315. 10.4134/BKMS.2002.39.2.309MathSciNetView ArticleMATHGoogle Scholar
- Miura T, Jung S-M, Takahasi S-E: Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations . Journal of the Korean Mathematical Society 2004, 41(6):995–1005. 10.4134/JKMS.2004.41.6.995MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Legendre's differential equation and its Hyers-Ulam stability. Abstract and Applied Analysis 2007, 2007:-14.Google Scholar
- Jung S-M: Approximation of analytic functions by Airy functions. Integral Transforms and Special Functions 2008, 19(12):885–891. 10.1080/10652460802321287MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: An approximation property of exponential functions. Acta Mathematica Hungarica 2009, 124(1–2):155–163. 10.1007/s10474-009-8167-1MathSciNetView ArticleMATHGoogle Scholar
- Jung S-M: Approximation of analytic functions by Hermite functions. Bulletin des Sciences Mathématiques 2009, 133(7):756–764.View ArticleMathSciNetMATHGoogle Scholar
- Kim B, Jung S-M: Bessel's differential equation and its Hyers-Ulam stability. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
- Abramowitz M, Stegun IA: Handbook of Mathematical Functions. Dover, New York, NY, USA; 1972.MATHGoogle Scholar
- Wade WR: An Introduction to Analysis. 2nd edition. Prentice Hall, Upper Saddle River, NJ, USA; 2000.MATHGoogle Scholar
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