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.}
This equation is found, for example, when the technique of separation of variables is used on differential equations which are 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 (Weber 1869 ) harv error: no target: CITEREFWeber1869 (help ) :
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}
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.}
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
There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and Stegun (1965)):
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)}
confluent hypergeometric function .
Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity:
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 ) approaches zero for large values of |z| and |arg(z )| < π/2, while V (a , z ) diverges for large values of positive real z .
lim
|
z
|
→
∞
U
(
a
,
z
)
/
e
−
z
2
/
4
z
−
a
−
1
/
2
=
1
(
for
|
arg
(
z
)
|
<
π
/
2
)
{\displaystyle \lim _{|z|\rightarrow \infty }U(a,z)/e^{-z^{2}/4}z^{-a-1/2}=1\,\,\,\,({\text{for}}\,|\arg(z)|<\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|\rightarrow \infty }V(a,z)/{\sqrt {\frac {2}{\pi }}}e^{z^{2}/4}z^{a-1/2}=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 ) (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)):
U
(
a
,
x
)
=
D
−
a
−
1
2
(
x
)
,
{\displaystyle U(a,x)=D_{-a-{\tfrac {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 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)].}
References
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 .Rozov, N.Kh. (2001) [1994], "Weber equation" , Encyclopedia of Mathematics EMS Press Temme, N. M. (2010), "Parabolic cylinder function" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 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. , 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. 
![{\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]}](/media/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]}](/media/api/rest_v1/media/math/render/svg/d71265ff3675d05ec58aa4da0391d9130c22d27b)
![{\displaystyle 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)].}](/media/api/rest_v1/media/math/render/svg/986fdc7a57024b172e3f9304db50fbf569973e95)
