Coordinate surfaces of parabolic cylindrical coordinates. Parabolic cylinder functions occur when separation of variables is used on Laplace's equation in these coordinates
Plot of the parabolic cylinder function D ν (z ) with ν = 5 in the complex plane from −2 − 2i to 2 + 2i
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 ).
There are independent even and odd solutions of the form (A ). These are given by (following the notation of 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] 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\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 ) (a notation dating back to Whittaker (1902))[ 3] that are themselves sometimes called parabolic cylinder functions:[ 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 ) was introduced by Whittaker and Watson as a solution of eq.~(1 ) with
a
~
=
−
1
4
,
b
~
=
0
,
c
~
=
a
+
1
2
{\textstyle {\tilde {a}}=-{\frac {1}{4}},{\tilde {b}}=0,{\tilde {c}}=a+{\frac {1}{2}}}
bounded at
+
∞
{\displaystyle +\infty }
.[ 4] It can be expressed in terms of confluent hypergeometric functions as
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]
Parabolic Cylinder U(a,z) function[ edit ]
Integral representation [ edit ]
Integrals along the real line,[ 6]
U
(
a
,
z
)
=
e
−
1
4
z
2
Γ
(
a
+
1
2
)
∫
0
∞
e
−
z
t
t
a
−
1
2
e
−
1
2
t
2
d
t
,
ℜ
a
>
−
1
2
,
{\displaystyle U(a,z)={\frac {e^{-{\frac {1}{4}}z^{2}}}{\Gamma \left(a+{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-zt}t^{a-{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re a>-{\frac {1}{2}}\;,}
U
(
a
,
z
)
=
2
π
e
1
4
z
2
∫
0
∞
cos
(
z
t
+
π
2
a
+
π
4
)
t
−
a
−
1
2
e
−
1
2
t
2
d
t
,
ℜ
a
<
1
2
.
{\displaystyle U(a,z)={\sqrt {\frac {2}{\pi }}}e^{{\frac {1}{4}}z^{2}}\int _{0}^{\infty }\cos \left(zt+{\frac {\pi }{2}}a+{\frac {\pi }{4}}\right)t^{-a-{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re a<{\frac {1}{2}}\;.}
The fact that these integrals are solutions to equation (A ) can be easily checked by direct substitution.
Differentiating the integrals with respect to
z
{\displaystyle z}
gives two expressions for
U
′
(
a
,
z
)
{\displaystyle U'(a,z)}
,
U
′
(
a
,
z
)
=
−
z
2
U
(
a
,
z
)
−
e
−
1
4
z
2
Γ
(
a
+
1
2
)
∫
0
∞
e
−
z
t
t
a
+
1
2
e
−
1
2
t
2
d
t
=
−
z
2
U
(
a
,
z
)
−
(
a
+
1
2
)
U
(
a
+
1
,
z
)
,
{\displaystyle U'(a,z)=-{\frac {z}{2}}U(a,z)-{\frac {e^{-{\frac {1}{4}}z^{2}}}{\Gamma \left(a+{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-zt}t^{a+{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt=-{\frac {z}{2}}U(a,z)-\left(a+{\frac {1}{2}}\right)U(a+1,z)\;,}
U
′
(
a
,
z
)
=
z
2
U
(
a
,
z
)
−
2
π
e
1
4
z
2
∫
0
∞
sin
(
z
t
+
π
2
a
+
π
4
)
t
−
a
+
1
2
e
−
1
2
t
2
d
t
=
z
2
U
(
a
,
z
)
−
U
(
a
−
1
,
z
)
.
{\displaystyle U'(a,z)={\frac {z}{2}}U(a,z)-{\sqrt {\frac {2}{\pi }}}e^{{\frac {1}{4}}z^{2}}\int _{0}^{\infty }\sin \left(zt+{\frac {\pi }{2}}a+{\frac {\pi }{4}}\right)t^{-a+{\frac {1}{2}}}e^{-{\frac {1}{2}}t^{2}}dt={\frac {z}{2}}U(a,z)-U(a-1,z)\;.}
Adding the two gives another expression for the derivative,
2
U
′
(
a
,
z
)
=
−
(
a
+
1
2
)
U
(
a
+
1
,
z
)
−
U
(
a
−
1
,
z
)
.
{\displaystyle 2U'(a,z)=-\left(a+{\frac {1}{2}}\right)U(a+1,z)-U(a-1,z)\;.}
Recurrence relation [ edit ]
Subtracting the first two expressions for the derivative gives the recurrence relation,
z
U
(
a
,
z
)
=
U
(
a
−
1
,
z
)
−
(
a
+
1
2
)
U
(
a
+
1
,
z
)
.
{\displaystyle zU(a,z)=U(a-1,z)-\left(a+{\frac {1}{2}}\right)U(a+1,z)\;.}
Asymptotic expansion [ edit ]
Expanding
e
−
1
2
t
2
=
1
−
1
2
t
2
+
1
8
t
4
−
…
{\displaystyle e^{-{\frac {1}{2}}t^{2}}=1-{\frac {1}{2}}t^{2}+{\frac {1}{8}}t^{4}-\dots \;}
in the integrand of the integral representation
gives the asymptotic expansion of
U
(
a
,
z
)
{\displaystyle U(a,z)}
,
U
(
a
,
z
)
=
e
−
1
4
z
2
z
−
a
−
1
2
(
1
−
(
a
+
1
2
)
(
a
+
3
2
)
2
1
z
2
+
(
a
+
1
2
)
(
a
+
3
2
)
(
a
+
5
2
)
(
a
+
7
2
)
8
1
z
4
−
…
)
.
{\displaystyle U(a,z)=e^{-{\frac {1}{4}}z^{2}}z^{-a-{\frac {1}{2}}}\left(1-{\frac {(a+{\frac {1}{2}})(a+{\frac {3}{2}})}{2}}{\frac {1}{z^{2}}}+{\frac {(a+{\frac {1}{2}})(a+{\frac {3}{2}})(a+{\frac {5}{2}})(a+{\frac {7}{2}})}{8}}{\frac {1}{z^{4}}}-\dots \right).}
Expanding the integral representation in powers of
z
{\displaystyle z}
gives
U
(
a
,
z
)
=
π
2
−
a
2
−
1
4
Γ
(
a
2
+
3
4
)
−
π
2
−
a
2
+
1
4
Γ
(
a
2
+
1
4
)
z
+
π
2
−
a
2
−
5
4
Γ
(
a
2
+
3
4
)
z
2
−
…
.
{\displaystyle U(a,z)={\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}-{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {3}{4}}\right)}}-{\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}+{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {1}{4}}\right)}}z+{\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}-{\frac {5}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {3}{4}}\right)}}z^{2}-\dots \;.}
From the power series one immediately gets
U
(
a
,
0
)
=
π
2
−
a
2
−
1
4
Γ
(
a
2
+
3
4
)
,
{\displaystyle U(a,0)={\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}-{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {3}{4}}\right)}}\;,}
U
′
(
a
,
0
)
=
−
π
2
−
a
2
+
1
4
Γ
(
a
2
+
1
4
)
.
{\displaystyle U'(a,0)=-{\frac {{\sqrt {\pi }}\,2^{-{\frac {a}{2}}+{\frac {1}{4}}}}{\Gamma \left({\frac {a}{2}}+{\frac {1}{4}}\right)}}\;.}
Parabolic cylinder Dν (z) function[ edit ]
Parabolic cylinder function
D
ν
(
z
)
{\displaystyle D_{\nu }(z)}
is the solution to the Weber differential equation,
u
″
+
(
ν
+
1
2
−
1
4
z
2
)
u
=
0
,
{\displaystyle u''+\left(\nu +{\frac {1}{2}}-{\frac {1}{4}}z^{2}\right)u=0\,,}
that is regular at
ℜ
z
→
+
∞
{\displaystyle \Re z\to +\infty }
with the asymptotics
D
ν
(
z
)
→
e
−
1
4
z
2
z
ν
.
{\displaystyle D_{\nu }(z)\to e^{-{\frac {1}{4}}z^{2}}z^{\nu }\,.}
It is thus given as
D
ν
(
z
)
=
U
(
−
ν
−
1
/
2
,
z
)
{\displaystyle D_{\nu }(z)=U(-\nu -1/2,z)}
and its properties then directly follow from those of the
U
{\displaystyle U}
-function.
Integral representation [ edit ]
D
ν
(
z
)
=
e
−
1
4
z
2
Γ
(
−
ν
)
∫
0
∞
e
−
z
t
t
−
ν
−
1
e
−
1
2
t
2
d
t
,
ℜ
ν
<
0
,
ℜ
z
>
0
,
{\displaystyle D_{\nu }(z)={\frac {e^{-{\frac {1}{4}}z^{2}}}{\Gamma (-\nu )}}\int _{0}^{\infty }e^{-zt}t^{-\nu -1}e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re \nu <0\,,\;\Re z>0\;,}
D
ν
(
z
)
=
2
π
e
1
4
z
2
∫
0
∞
cos
(
z
t
−
ν
π
2
)
t
ν
e
−
1
2
t
2
d
t
,
ℜ
ν
>
−
1
.
{\displaystyle D_{\nu }(z)={\sqrt {\frac {2}{\pi }}}e^{{\frac {1}{4}}z^{2}}\int _{0}^{\infty }\cos \left(zt-\nu {\frac {\pi }{2}}\right)t^{\nu }e^{-{\frac {1}{2}}t^{2}}dt\,,\;\Re \nu >-1\;.}
Asymptotic expansion [ edit ]
D
ν
(
z
)
=
e
−
1
4
z
2
z
ν
(
1
−
ν
(
ν
−
1
)
2
1
z
2
+
ν
(
ν
−
1
)
(
ν
−
2
)
(
ν
−
3
)
8
1
z
4
−
…
)
,
ℜ
z
→
+
∞
.
{\displaystyle D_{\nu }(z)=e^{-{\frac {1}{4}}z^{2}}z^{\nu }\left(1-{\frac {\nu (\nu -1)}{2}}{\frac {1}{z^{2}}}+{\frac {\nu (\nu -1)(\nu -2)(\nu -3)}{8}}{\frac {1}{z^{4}}}-\dots \right)\,,\;\Re z\to +\infty .}
If
ν
{\displaystyle \nu }
is a non-negative integer this series terminates and turns into a polynomial, namely the Hermite polynomial ,
D
n
(
z
)
=
e
−
1
4
z
2
2
−
n
/
2
H
n
(
z
2
)
,
n
=
0
,
1
,
2
,
…
.
{\displaystyle D_{n}(z)=e^{-{\frac {1}{4}}z^{2}}\;2^{-n/2}H_{n}\left({\frac {z}{\sqrt {2}}}\right)\,,n=0,1,2,\dots \;.}
Connection with quantum harmonic oscillator [ edit ]
Parabolic cylinder
D
ν
(
z
)
{\displaystyle D_{\nu }(z)}
function appears naturally in the Schrödinger equation for the one-dimensional quantum harmonic oscillator (a quantum particle in the oscillator potential),
[
−
ℏ
2
2
m
∂
2
∂
x
2
+
1
2
m
ω
2
x
2
]
ψ
(
x
)
=
E
ψ
(
x
)
,
{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\right]\psi (x)=E\psi (x)\;,}
where
ℏ
{\displaystyle \hbar }
is the reduced Planck constant,
m
{\displaystyle m}
is the mass of the particle,
x
{\displaystyle x}
is the coordinate of the particle,
ω
{\displaystyle \omega }
is the frequency of the oscillator,
E
{\displaystyle E}
is the energy,
and
ψ
(
x
)
{\displaystyle \psi (x)}
is the particle's wave-function. Indeed introducing the new quantities
z
=
x
b
o
,
ν
=
E
ℏ
ω
−
1
2
,
b
o
=
ℏ
2
m
ω
,
{\displaystyle z={\frac {x}{b_{o}}}\,,\;\nu ={\frac {E}{\hbar \omega }}-{\frac {1}{2}}\,,\;b_{o}={\sqrt {\frac {\hbar }{2m\omega }}}\,,}
turns the above equation into the Weber's equation for the function
u
(
z
)
=
ψ
(
z
b
o
)
{\displaystyle u(z)=\psi (zb_{o})}
,
u
″
+
(
ν
+
1
2
−
1
4
z
2
)
u
=
0
.
{\displaystyle u''+\left(\nu +{\frac {1}{2}}-{\frac {1}{4}}z^{2}\right)u=0\,.}
^ 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 . Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 686. 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.
^ NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/ , Release 1.2.2 of 2024-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.