Coordinate surfaces of parabolic cylindrical coordinates. Parabolic cylinder functions occur when separation of variables is used on Laplace's equation in these coordinatesPlot of the parabolic cylinder function D v(z) with v=5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , the parabolic cylinder functions are special functions defined as solutions to the differential equation
d
2
f
d
z
2
+
(
a
~
z
2
+
b
~
z
+
c
~
)
f
=
0.
{\displaystyle {\frac {d^{2}f}{dz^{2}}}+\left({\tilde {a}}z^{2}+{\tilde {b}}z+{\tilde {c}}\right)f=0.}
(1 )
This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates .
The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling z , called H. F. Weber 's equations:[1]
d
2
f
d
z
2
−
(
1
4
z
2
+
a
)
f
=
0
{\displaystyle {\frac {d^{2}f}{dz^{2}}}-\left({\tfrac {1}{4}}z^{2}+a\right)f=0}
(A )
and
d
2
f
d
z
2
+
(
1
4
z
2
−
a
)
f
=
0.
{\displaystyle {\frac {d^{2}f}{dz^{2}}}+\left({\tfrac {1}{4}}z^{2}-a\right)f=0.}
(B )
If
f
(
a
,
z
)
{\displaystyle f(a,z)}
is a solution, then so are
f
(
a
,
−
z
)
,
f
(
−
a
,
i
z
)
and
f
(
−
a
,
−
i
z
)
.
{\displaystyle f(a,-z),f(-a,iz){\text{ and }}f(-a,-iz).}
If
f
(
a
,
z
)
{\displaystyle f(a,z)\,}
is a solution of equation (
A ), then
f
(
−
i
a
,
z
e
(
1
/
4
)
π
i
)
{\displaystyle f(-ia,ze^{(1/4)\pi i})}
is a solution of (
B ), and, by symmetry,
f
(
−
i
a
,
−
z
e
(
1
/
4
)
π
i
)
,
f
(
i
a
,
−
z
e
−
(
1
/
4
)
π
i
)
and
f
(
i
a
,
z
e
−
(
1
/
4
)
π
i
)
{\displaystyle f(-ia,-ze^{(1/4)\pi i}),f(ia,-ze^{-(1/4)\pi i}){\text{ and }}f(ia,ze^{-(1/4)\pi i})}
are also solutions of (
B ).
Solutions [ edit ] There are independent even and odd solutions of the form (A Abramowitz and Stegun (1965)):[2]
y
1
(
a
;
z
)
=
exp
(
−
z
2
/
4
)
1
F
1
(
1
2
a
+
1
4
;
1
2
;
z
2
2
)
(
e
v
e
n
)
{\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)\,\,\,\,\,\,(\mathrm {even} )}
and
y
2
(
a
;
z
)
=
z
exp
(
−
z
2
/
4
)
1
F
1
(
1
2
a
+
3
4
;
3
2
;
z
2
2
)
(
o
d
d
)
{\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)\,\,\,\,\,\,(\mathrm {odd} )}
where
1
F
1
(
a
;
b
;
z
)
=
M
(
a
;
b
;
z
)
{\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)}
is the
confluent hypergeometric function .
Other pairs of independent solutions may be formed from linear combinations of the above solutions.[2]
U
(
a
,
z
)
=
1
2
ξ
π
[
cos
(
ξ
π
)
Γ
(
1
/
2
−
ξ
)
y
1
(
a
,
z
)
−
2
sin
(
ξ
π
)
Γ
(
1
−
ξ
)
y
2
(
a
,
z
)
]
{\displaystyle U(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}}}\left[\cos(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)-{\sqrt {2}}\sin(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}
V
(
a
,
z
)
=
1
2
ξ
π
Γ
[
1
/
2
−
a
]
[
sin
(
ξ
π
)
Γ
(
1
/
2
−
ξ
)
y
1
(
a
,
z
)
+
2
cos
(
ξ
π
)
Γ
(
1
−
ξ
)
y
2
(
a
,
z
)
]
{\displaystyle V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}
where
ξ
=
1
2
a
+
1
4
.
{\displaystyle \xi ={\frac {1}{2}}a+{\frac {1}{4}}.}
The function U (a , z )z and |arg(z )| < π /2 , while V (a , z )z .
lim
z
→
∞
U
(
a
,
z
)
/
(
e
−
z
2
/
4
z
−
a
−
1
/
2
)
=
1
(
for
|
arg
(
z
)
|
<
π
/
2
)
{\displaystyle \lim _{z\to \infty }U(a,z)/\left(e^{-z^{2}/4}z^{-a-1/2}\right)=1\,\,\,\,({\text{for}}\,\left|\arg(z)\right|<\pi /2)}
and
lim
z
→
∞
V
(
a
,
z
)
/
(
2
π
e
z
2
/
4
z
a
−
1
/
2
)
=
1
(
for
arg
(
z
)
=
0
)
.
{\displaystyle \lim _{z\to \infty }V(a,z)/\left({\sqrt {\frac {2}{\pi }}}e^{z^{2}/4}z^{a-1/2}\right)=1\,\,\,\,({\text{for}}\,\arg(z)=0).}
For half-integer values of a , these (that is, U and V ) can be re-expressed in terms of Hermite polynomials ; alternatively, they can also be expressed in terms of Bessel functions .
The functions U and V can also be related to the functions Dp (x )[3] [2]
U
(
a
,
x
)
=
D
−
a
−
1
2
(
x
)
,
V
(
a
,
x
)
=
Γ
(
1
2
+
a
)
π
[
sin
(
π
a
)
D
−
a
−
1
2
(
x
)
+
D
−
a
−
1
2
(
−
x
)
]
.
{\displaystyle {\begin{aligned}U(a,x)&=D_{-a-{\tfrac {1}{2}}}(x),\\V(a,x)&={\frac {\Gamma ({\tfrac {1}{2}}+a)}{\pi }}[\sin(\pi a)D_{-a-{\tfrac {1}{2}}}(x)+D_{-a-{\tfrac {1}{2}}}(-x)].\end{aligned}}}
Function Da (z )1
a
~
=
−
1
4
,
b
~
=
0
,
c
~
=
a
+
1
2
{\textstyle {\tilde {a}}=-{\frac {1}{4}},{\tilde {b}}=0,{\tilde {c}}=a+{\frac {1}{2}}}
+
∞
{\displaystyle +\infty }
[4]
D
a
(
z
)
=
1
π
2
a
/
2
e
−
z
2
4
(
cos
(
π
a
2
)
Γ
(
a
+
1
2
)
1
F
1
(
−
a
2
;
1
2
;
z
2
2
)
+
2
z
sin
(
π
a
2
)
Γ
(
a
2
+
1
)
1
F
1
(
1
2
−
a
2
;
3
2
;
z
2
2
)
)
.
{\displaystyle D_{a}(z)={\frac {1}{\sqrt {\pi }}}{2^{a/2}e^{-{\frac {z^{2}}{4}}}\left(\cos \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a+1}{2}}\right)\,_{1}F_{1}\left(-{\frac {a}{2}};{\frac {1}{2}};{\frac {z^{2}}{2}}\right)+{\sqrt {2}}z\sin \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a}{2}}+1\right)\,_{1}F_{1}\left({\frac {1}{2}}-{\frac {a}{2}};{\frac {3}{2}};{\frac {z^{2}}{2}}\right)\right)}.}
Power series for this function have been obtained by Abadir (1993).[5]
References [ edit ]
^ Weber, H.F. (1869), "Ueber die Integration der partiellen Differentialgleichung
∂
2
u
/
∂
x
2
+
∂
2
u
/
∂
y
2
+
k
2
u
=
0
{\displaystyle \partial ^{2}u/\partial x^{2}+\partial ^{2}u/\partial y^{2}+k^{2}u=0}
Math. Ann. , vol. 1, pp. 1–36 ^ a b c Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. "Chapter 19" . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ISBN 978-0-486-61272-0 LCCN 64-60036 . MR 0167642 . LCCN 65-12253 .
^ Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" Proc. London Math. Soc. , 35, 417–427.
^ Whittaker, E. T. and Watson, G. N. (1990) "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348.
^ Abadir, K. M. (1993) "Expansions for some confluent hypergeometric functions." Journal of Physics A , 26, 4059-4066.
_with_v%3D5_in_the_complex_plane_from_-2-2i_to_2%2B2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg)
![{\displaystyle U(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}}}\left[\cos(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)-{\sqrt {2}}\sin(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f23625c2c6830ba58f8fa8a65769115cf44cbfd8)
![{\displaystyle V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d71265ff3675d05ec58aa4da0391d9130c22d27b)
![{\displaystyle {\begin{aligned}U(a,x)&=D_{-a-{\tfrac {1}{2}}}(x),\\V(a,x)&={\frac {\Gamma ({\tfrac {1}{2}}+a)}{\pi }}[\sin(\pi a)D_{-a-{\tfrac {1}{2}}}(x)+D_{-a-{\tfrac {1}{2}}}(-x)].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9dc0e547116ec9cf1c56d5997eb3d74dc62e4c)
