拋物柱面函數

拋物柱面函數是滿足下列微分方程的特殊函數:

Parabolic cylinder function U

Parabolic cylinder function V

${\displaystyle {\frac {d^{2}f}{dz^{2}}}+\left({\tilde {a}}z^{2}+{\tilde {b}}z+{\tilde {c}}\right)f=0.}$

在利用分離變數法處理在拋物柱面坐標在的拉普拉斯方程時，自然出現上列方程

通過解二次代數方程和變數代換可以將上列方程表示為兩種標準形式:

${\displaystyle {\frac {d^{2}f}{dz^{2}}}-\left({\tfrac {1}{4}}z^{2}+a\right)f=0}$    (A)

及

${\displaystyle {\frac {d^{2}f}{dz^{2}}}+\left({\tfrac {1}{4}}z^{2}-a\right)f=0.}$     (B)

如果

${\displaystyle f(a,z)\,}$是一個解，則

${\displaystyle f(a,-z),f(-a,iz){\text{ and }}f(-a,-iz).\,}$ 也是解。

解析解

上列方程有奇數解和偶數解兩類

偶數解

${\displaystyle y_{1}(a;z)=\exp(-z^{2}/4)\;_{1}F_{1}\left({\tfrac {1}{2}}a+{\tfrac {1}{4}};\;{\tfrac {1}{2}}\;;\;{\frac {z^{2}}{2}}\right)\,}$

奇數解

${\displaystyle y_{2}(a;z)=z\exp(-z^{2}/4)\;_{1}F_{1}\left({\tfrac {1}{2}}a+{\tfrac {3}{4}};\;{\tfrac {3}{2}}\;;\;{\frac {z^{2}}{2}}\right)\,\,\,\,\,\,}$

where ${\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)}$ is the 合流超幾何函數.

參考文獻

• Rozov, N.Kh., Weber equation, Hazewinkel, Michiel (編), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Temme, N. M., 抛物柱面函数, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
• Weber, H.F. (1869) "Ueber die Integration der partiellen Differentialgleichung ${\displaystyle \partial ^{2}u/\partial x^{2}+\partial ^{2}u/\partial y^{2}+k^{2}u=0}$". Math. Ann., 1, 1–36
• Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" Proc. London Math. Soc.35, 417–427.