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 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 ). 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]
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 . 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.