Parabolic cylinder functions revisited using the Laplace transform
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arXiv
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| Format: | Preprint |
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2024
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| author | Costin, Rodica D. Mavrogiannis, Georgios |
| author_facet | Costin, Rodica D. Mavrogiannis, Georgios |
| contents | In this paper we gather and extend classical results for parabolic cylinder functions, namely solutions of the Weber differential equations, using a systematic approach by Borel-Laplace methods.
We revisit the definition and construction of the standard solutions $U,V$ of the Weber differential equation \begin{equation*} w''(z)-\left(\frac{z^2}{4}+a\right)w(z)=0 \end{equation*} and provide representations by Laplace integrals extended to include all values of the complex parameter $a$; we find an integral integral representation for the function $V$; none was previously available.
For the Weber equation in the form \begin{equation*}
u''(x)+\left(\frac{x^2}{4}-a\right)u(x)=0, \end{equation*} we define a new fundamental system $E_\pm$ which is analytic in $a\in\mathbb{C}$, based on asymptotic behavior; they appropriately extend and modify the classical solutions $E,E^*$ of the real Weber equation to the complex domain.
The techniques used are general and we include details and motivations for the approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parabolic cylinder functions revisited using the Laplace transform Costin, Rodica D. Mavrogiannis, Georgios Classical Analysis and ODEs 33C15, 34A25, 34A30, 44A10 In this paper we gather and extend classical results for parabolic cylinder functions, namely solutions of the Weber differential equations, using a systematic approach by Borel-Laplace methods. We revisit the definition and construction of the standard solutions $U,V$ of the Weber differential equation \begin{equation*} w''(z)-\left(\frac{z^2}{4}+a\right)w(z)=0 \end{equation*} and provide representations by Laplace integrals extended to include all values of the complex parameter $a$; we find an integral integral representation for the function $V$; none was previously available. For the Weber equation in the form \begin{equation*} u''(x)+\left(\frac{x^2}{4}-a\right)u(x)=0, \end{equation*} we define a new fundamental system $E_\pm$ which is analytic in $a\in\mathbb{C}$, based on asymptotic behavior; they appropriately extend and modify the classical solutions $E,E^*$ of the real Weber equation to the complex domain. The techniques used are general and we include details and motivations for the approach. |
| title | Parabolic cylinder functions revisited using the Laplace transform |
| topic | Classical Analysis and ODEs 33C15, 34A25, 34A30, 44A10 |
| url | https://arxiv.org/abs/2407.20403 |