抛物柱面函数

抛物柱面函数是满足下列微分方程的特殊函数:

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

在利用分离变数法处理在抛物柱面坐标在的拉普拉斯方程时，自然出现上列方程

通过解二次代数方程和变数代换可以将上列方程表示为两种标准形式:

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

及

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

如果

${\displaystyle f(a,z)\,}$是一个解，则

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

上列方程有奇数解和偶数解两类

偶数解

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

奇数解

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

where ${\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)}$ is the 合流超几何函数.

• Rozov, N.Kh., Weber equation, Hazewinkel, Michiel (编), 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. (编), 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.