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Weber Differential Equations

Weber differential equations include the differential equation satisfied by

w=z^(-1/2)W_(k,-1/4)(1/2z^2),
(1)

where W is a Whittaker function, which is given by

d/(zdz)[(d(wz^(1/2)))/(zdz)]+(-1/4+(2k)/(z^2)+3/(4z^4))wz^(1/2)=0
(2)
(d^2w)/(dz^2)+(2k-1/4z^2)w=0
(3)

(Moon and Spencer 1961, p. 153; Zwillinger 1997, p. 128). This is usually rewritten

(d^2D_n(z))/(dz^2)+(n+1/2-1/4z^2)D_n(z)=0.
(4)

The solutions are parabolic cylinder functions. For arbitrary complex number n, the general solution is

w(z)=C_1D_n(z)+C_2D_(-n-1)(iz).
(5)

The ordered Wronskian of this pair, using W[f,g]=fg^'-f^'g, is

W[D_n(z),D_(-n-1)(iz)]=-ie^(-ipin/2).
(6)

More generally, for a!=0, the equation w^('')-(az^2+c)w=0 has general solution

w(z)=C_1D_p(sz)+C_2D_(-p-1)(isz),
(7)

where r^2=a, s^2=2r, and p=-(r+c)/(2r). Its two displayed solutions have ordered Wronskian

W[D_p(sz),D_(-p-1)(isz)]=-ise^(-ipip/2).
(8)

The equations

(d^2U)/(du^2)-(c+k^2u^2)U=0
(9)
(d^2V)/(dv^2)+(c-k^2v^2)V=0,
(10)

which arise by separating variables in Laplace's equation in parabolic cylindrical coordinates, are also known as the Weber differential equations. As above, the solutions are known as parabolic cylinder functions.

Zwillinger (1997, p. 127) calls

y^('')+(y^')/x+(1-(nu^2)/(x^2))y=-1/(pix^2)[x+nu+(x-nu)cos(nupi)]
(11)

the Weber differential equation (Gradshteyn and Ryzhik 2000, p. 989). On a simply connected domain not containing 0, its general solution, with consistent branches, is

y(x)=C_1J_nu(x)+C_2Y_nu(x)+E_nu(x),
(12)

where J_nu is a Bessel function of the first kind, Y_nu a Bessel function of the second kind, and E_nu a Weber function.

See also

Anger Differential Equation, Parabolic Cylinder Function

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 989, 2000.Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 127, 1997.

Referenced on Wolfram|Alpha

Weber Differential Equations

Cite this as:

Weisstein, Eric W. "Weber Differential Equations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeberDifferentialEquations.html

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