Complete systems of recursive integrals and Taylor series for solutions of Sturm – Liouville equations. (English) Zbl 1243.34011
Summary: Given a regular nonvanishing complex valued solution \(y_{0}\) of the equation
\(y''+q(x)y=0\), \(x \in (a,b)\),
assume that it is n times differentiable at a point \(x_{0}\in \)[a,b]. We present explicit formulas for calculating the first \(n\) derivatives at \(x_{0}\) for any solution of the equation
\(u''+q(x)u=\lambda u\).
That is, a map transforming the Taylor expansion of \(y_{0}\) into the Taylor expansion of \(u\) is constructed. The result is obtained by means of the representation of the solutions of the Sturm-Liouville equation.
\(y''+q(x)y=0\), \(x \in (a,b)\),
assume that it is n times differentiable at a point \(x_{0}\in \)[a,b]. We present explicit formulas for calculating the first \(n\) derivatives at \(x_{0}\) for any solution of the equation
\(u''+q(x)u=\lambda u\).
That is, a map transforming the Taylor expansion of \(y_{0}\) into the Taylor expansion of \(u\) is constructed. The result is obtained by means of the representation of the solutions of the Sturm-Liouville equation.
MSC:
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 34B24 | Sturm-Liouville theory |
| 42A65 | Completeness of sets of functions in one variable harmonic analysis |
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 34A30 | Linear ordinary differential equations and systems |
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