When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}+\left(x^{2}-n(n+1)\right)y=0.}$

The two linearly independent solutions to this equation are called the spherical Bessel functions j_n and y_n, and are related to the ordinary Bessel functions J_n and Y_n by^[26]

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+2x{\frac {dy}{dx}}+\left(x^{2}-n(n+1)\right)y=0.}$

The two linearly independent solutions to this equation are called the spherical Bessel functions j_n and y_n, and are related to the ordinary Bessel functions J_n and Y_n by^[26]

_n and y_n, and are related to the ordinary Bessel functions J_n and Y_n by^[26]

y_n is also denoted n_n or η_n; some authors call these functions the spherical Neumann functions.

The spherical Bessel functions can also be written as (Rayleigh's formulas)^[27]

${\displaystyle {\begin{aligned}j_{n}(x)&=(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\sin x}{x}},\\y_{n}(x)&=-(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\cos x}{x}}.\end{aligned}}}$[27]

The first spherical Bessel function j₀(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:^[28]

The spherical Bessel functions have the generating functions^[30]

${\displaystyle {\begin{aligned}{\frac {1}{z}}\cos \left({\sqrt {z^{2}-2zt}}\right)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}j_{n-1}(z),\\{\frac {1}{z}}\sin \left({\sqrt {z^{2}-2zt}}\right)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}y_{n-1}(z).\end{aligned}}}$

Differential relations

In the following, f_n is any of j_n, y_n, h⁽¹⁾
_n, h⁽²⁾
_n for n = 0, ±1, ±2, ...^[31]

In the following, f_n is any of j_n, y_n, h⁽¹⁾
_n, h⁽²⁾
_n for n = 0, ±1, ±2, ...^[31]

${\displaystyle {\begin{aligned}\left({\frac {1}{z}}{\frac {d}{dz}}\right)^{m}\left(z^{n+1}f_{n}(z)\right)&=z^{n-m+1}f_{n-m}(z),\\\left({\frac {1}{z}}{\frac {d}{dz}}\right)^{m}\left(z^{-n}f_{n}(z)\right)&=(-1)^{m}z^{-n-m}f_{n+m}(z).\end{aligned}}}$${\displaystyle {\begin{aligned}h_{n}^{(1)}(x)&=j_{n}(x)+iy_{n}(x),\\h_{n}^{(2)}(x)&=j_{n}(x)-iy_{n}(x).\end{aligned}}}$

In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers n:

${\displaystyle h_{n}^{(1)}(x)=(-i)^{n+1}{\frac {e^{ix}}{x}}\sum _{m=0}^{n}{\frac {i^{m}}{m!\,(2x)^{m}}}{\frac {(n+m)!}{(n-m)!}},}$

and h^{half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers n:}

${\displaystyle h_{n}^{(1)}(x)=(-i)^{n+1}{\frac {e^{ix}}{x}}\sum _{m=0}^{n}{\frac {i^{m}}{m!\,(2x)^{m}}}{\frac {(n+m)!}{(n-m)!}},}$

and h⁽²⁾
_n is the complex-conjugate of this (for real h⁽²⁾
_n is the complex-conjugate of this (for real x). It follows, for example, that j₀(x) = sin x/x and y₀(x) = −cos x/x, and so on.

The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.

Riccati–Bessel functions: S_n, C_n, ξ_n, ζ_n

Riccati–Bessel functions only slightly differ from spherical Bessel functions:

The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.

Riccati–Bessel functions only slightly differ from spherical Bessel functions:

They satisfy the differential equation

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+\left(x^{2}-n(n+1)\right)y=0.}$

For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's equation with hypothetical cylindrical infinite potential barrier.^[32] This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g., Du (2004)^[33] for recent developments and references.

Following Debye (1909), the notation ψ_n, χ_n is sometimes used instead of S_n, C_n.

Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments 0 < z ≪ √α + 1, one obtains, when α is not a negative integer:^[3]

${\displaystyle J_{\alpha }(z)\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha }.}$

When α is a negative integer, we have

${\displaystyle J_{\alpha }(z)\sim {\frac {(-1)^{\alpha }}{(-\alpha )!}}\left({\frac {2}{z}}\right)^{\alpha }.}$

For the Bessel function of the second kind we have three cases:

Following Debye (1909), the notation ψ_n, χ_n is sometimes used instead of S_n, C_n.

Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments 0 < z ≪ √α + 1, one obtains, when α is not a negative integer:^[3]

${\displaystyle J_{\alpha }(z)\sim {\frac {(-1)^{\alpha }}{(-\alpha )!}}\left({\frac {2}{z}}\right)^{\alpha }.}$