B. Special functions

When solving boundary value problems by separation of variables, the partial differential equation splits into ordinary differential equations in each coordinate. Depending on the coordinate system and the geometry of the domain, these equations lead to different families of orthogonal functions.

The Bessel functions of the first kind Jm(r)J_m(r). They arise when separating variables in polar and cylindrical coordinates: the radial part of the Helmholtz equation reduces to the Bessel equation

r2⋅d2y(r)dr2\displaystyle r^2 \cdot \frac{\displaystyle d^2 y(r)}{\displaystyle dr^2} +r⋅dy(r)dr\displaystyle + r \cdot \frac{\displaystyle d y(r)}{\displaystyle dr} +(r2−m2)⋅y(r)\displaystyle + (r^2 - m^2) \cdot y(r) =0.\displaystyle = 0.

The graphs of the first orders are shown in figure (B.1).

Bessel functions of the first kind of orders 0, 1, 2, 3
Fig. B.1. The Bessel functions of the first kind Jm(r)J_m(r) of orders mm =0,1,2,3= 0, 1, 2, 3.

The Neumann functions Ym(r)Y_m(r) (Bessel functions of the second kind). They arise as the second linearly independent solution of the same Bessel equation. Unlike JmJ_m, they grow unboundedly in absolute value near the axis: Ym(r)Y_m(r) →−∞\to -\infty for rr →0\to 0. Therefore, in problems where the domain includes the axis rr =0= 0, they are discarded by the requirement that the solution be bounded, but in annular domains (not containing the axis) they enter the solution on a par with JmJ_m. The graphs of the first orders are shown in figure (B.2).

Neumann functions (Bessel functions of the second kind) of orders 0, 1, 2, 3
Fig. B.2. The Neumann functions Ym(r)Y_m(r) of orders mm =0,1,2,3= 0, 1, 2, 3.

The spherical Bessel functions jk(r)j_k(r). They appear in the radial part of the Helmholtz equation in spherical coordinates

r2⋅d2y(r)dr2\displaystyle r^2 \cdot \frac{\displaystyle d^2 y(r)}{\displaystyle dr^2} +2⋅r⋅dy(r)dr\displaystyle + 2 \cdot r \cdot \frac{\displaystyle d y(r)}{\displaystyle dr} +(r2−k⋅(k+1))⋅y(r)\displaystyle + \left( r^2 - k \cdot (k+1) \right) \cdot y(r) =0.\displaystyle = 0.

The substitution R(r)R(r) =1r⋅R^(r)= \dfrac{1}{\sqrt{r}} \cdot \widehat{R}(r) reduces it to the Bessel equation of half-integer order, so the solutions are expressed in terms of ordinary Bessel functions

jk(r)\displaystyle j_k(r) =π2⋅r⋅Jk+1/2(r).\displaystyle = \sqrt{\dfrac{\pi}{2 \cdot r}} \cdot J_{k + 1/2}(r).

As in the cylindrical case, jkj_k oscillate with decreasing amplitude; their graphs are shown in figure (B.3).

Spherical Bessel functions of orders 0, 1, 2, 3
Fig. B.3. The spherical Bessel functions jk(r)j_k(r) of orders kk =0,1,2,3= 0, 1, 2, 3.

The Legendre polynomials Pk(z)P_k(z). They arise from the angular (polar) part of the Helmholtz equation in spherical coordinates. After the substitution zz =cos⁡(θ)= \cos(\theta) and for mm =0= 0 the equation takes the form of the Legendre equation

(1−z2)⋅d2y(z)dz2\displaystyle (1 - z^2) \cdot \frac{\displaystyle d^2 y(z)}{\displaystyle dz^2} −2⋅z⋅dy(z)dz\displaystyle - 2 \cdot z \cdot \frac{\displaystyle d y(z)}{\displaystyle dz} +k⋅(k+1)⋅y(z)\displaystyle + k \cdot (k+1) \cdot y(z) =0,\displaystyle = 0,

whose solutions bounded on the interval zz ∈[−1,1]\in [-1, 1] are the Legendre polynomials of degree kk. They form an orthogonal basis on [−1,1][-1, 1]; the first few are shown in figure (B.4).

Legendre polynomials of degrees 0, 1, 2, 3
Fig. B.4. The Legendre polynomials Pk(z)P_k(z) of degrees kk =0,1,2,3= 0, 1, 2, 3.

The associated Legendre functions Pk(m)(z)P_k^{(m)}(z). If in the polar part mm ≠0\neq 0, the Legendre equation generalizes to the associated equation

(1−z2)⋅d2y(z)dz2\displaystyle (1 - z^2) \cdot \frac{\displaystyle d^2 y(z)}{\displaystyle dz^2} −2⋅z⋅dy(z)dz\displaystyle - 2 \cdot z \cdot \frac{\displaystyle d y(z)}{\displaystyle dz} +[k⋅(k+1)−m21−z2]⋅y(z)\displaystyle + \left[ k \cdot (k+1) - \dfrac{m^2}{1 - z^2} \right] \cdot y(z) =0,\displaystyle = 0,

whose solutions are the associated Legendre functions Pk(m)(z)P_k^{(m)}(z), expressed in terms of the Legendre polynomials

Pk(m)(z)\displaystyle P_k^{(m)}(z) =(1−z2)m/2⋅dmdzmPk(z).\displaystyle = (1 - z^2)^{m/2} \cdot \frac{\displaystyle d^m}{\displaystyle dz^m} P_k(z).

For mm =0= 0 they turn into the Legendre polynomials Pk(z)P_k(z), and together with the azimuthal factor cos⁡(m⋅ϕ)\cos(m \cdot \phi) or sin⁡(m⋅ϕ)\sin(m \cdot \phi) form the angular part of the eigenfunction.