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Helmholtz Differential Equation--Parabolic Cylindrical Coordinates

In parabolic cylindrical coordinates, the scale factors are h_u=h_v=sqrt(u^2+v^2), h_z=1 and the separation functions are f_1(u)=f_2(v)=f_3(z)=1, giving Stäckel determinant of s=u^2+v^2. the Helmholtz differential equation is

1/(u^2+v^2)((partial^2f)/(partialu^2)+(partial^2f)/(partialv^2))+(partial^2f)/(partialz^2)+k^2f=0.
(1)

attempt separation of variables by writing

f(u,v,z)=u(u)v(v)z(z),
(2)

then the Helmholtz differential equation becomes

1/(u^2+v^2)(VZ(d^2U)/(du^2)+UZ(d^2V)/(dv^2))+UV(d^2Z)/(dz^2)+k^2UVZ=0.
(3)

Divide by UVZ,

1/(u^2+v^2)(1/U(d^2U)/(du^2)+1/V(d^2V)/(dv^2))+1/Z(d^2Z)/(dz^2)+k^2=0.
(4)

Separating the Z part,

1/Z(d^2Z)/(dz^2)=-(k^2+m^2)
(5)
1/(u^2+v^2)(1/U(d^2U)/(du^2)+1/V(d^2V)/(dv^2))-k^2=0
(6)
1/U(d^2U)/(du^2)+1/V(d^2V)/(dv^2)-k^2(u^2+v^2)=0,
(7)

so

(d^2Z)/(dz^2)=-(k^2+m^2)Z,
(8)

which has solution

Z(z)=Acos(sqrt(k^2+m^2)z)+Bsin(sqrt(k^2+m^2)z),
(9)

and

(1/U(d^2U)/(du^2)-k^2u^2)+(1/V(d^2V)/(dv^2)-k^2v^2)=0.
(10)

This can be separated

1/U(d^2U)/(du^2)-k^2u^2=c
(11)
1/V(d^2V)/(dv^2)-k^2v^2=-c,
(12)

so

(d^2U)/(du^2)-(c+k^2u^2)U=0
(13)
(d^2V)/(dv^2)+(c-k^2v^2)V=0.
(14)

These are the Weber differential equations, and the solutions are known as Parabolic Cylinder Functions.

See also

Helmholtz Differential Equation, Parabolic Cylinder Function, Parabolic Cylindrical Coordinates, Weber Differential Equations

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References

Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 36, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 515 and 658, 1953.

Cite this as:

Weisstein, Eric W. "Helmholtz Differential Equation--Parabolic Cylindrical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationParabolicCylindricalCoordinates.html

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