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

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

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]}

- $\begin{array}{rl}{j}_{n}(x)& =\sqrt{\frac{\pi}{2x}}{J}_{n+\frac{1}{2}}(x),\\ {y}_{n}(x)& =\sqrt{\frac{\pi}{}}\end{array}$
When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form

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

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]}- $\begin{array}{rl}{j}_{n}(x)& =\sqrt{\frac{\pi}{2x}}{J}_{n+\frac{1}{2}}(x),\\ {y}_{n}(x)\end{array}$
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]}- ${\begin{aligned}j_{n}(x)&=(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\sin x}{x}},\\y_{n}(x)&=-(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\cos x}{x}}.\end{aligned}}$[27]
The first spherical Bessel function

*j*_{0}(*x*) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:^{[28]}- $\begin{array}{rl}{j}_{0}(x)& =\frac{\mathrm{sin}x}{x}.\\ {j}_{1}(x)& =\frac{\mathrm{sin}x}{{x}^{2}}-\frac{\mathrm{cos}x}{x},\\ {j}_{2}(x)& =\left(\frac{3}{{x}^{2}}-1\right)\frac{\mathrm{sin}x}{x}-\frac{3\mathrm{cos}x}{{x}^{2}},\\ {j}_{3}(x)& =(and[29]\end{array}$
- $\begin{array}{rl}{y}_{0}(x)& =-{j}_{-1}(x)=-\frac{\mathrm{cos}x}{x},\\ {y}_{1}(x)& ={j}_{-2}(x)=-\frac{\mathrm{cos}x}{{x}^{2}}-\frac{\mathrm{sin}x}{x},\\ {y}_{2}(x)& =-{j}_{-3}(x)=\left(-\frac{3}{{x}^{2}}+1\right)\frac{\mathrm{cos}x}{x}-\frac{3\mathrm{sin}x}{{x}^{2}},\\ {y}_{3}(x)& ={j}_{-4}(x)=\left(-\frac{15}{{x}^{3}}+\frac{6}{x}\right)\frac{\mathrm{cos}x}{x}-(\frac{15}{{x}^{2}}\end{array}$
The spherical Bessel functions have the generating functions

^{[30]}- ${\begin{aligned}{\frac {1}{z}}\cos \left({\sqrt {z^{2}-2zt}}\right)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}j_{n-1}(z),\\{\frac {1}{z}}\sin \left({\sqrt {z^{2}-2zt}}\right)&=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}y_{n-1}(z).\end{aligned}}$

#### 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]}- $\begin{array}{rl}{\left(\frac{1}{z}\frac{d}{dz}\right)}^{m}\left({z}^{n+1}{f}_{n}(z)\right)& ={z}^{n-m+1}{f}_{n-m}(z),\\ {}^{}\end{array}$
In the following, f

_{n}is any of j_{n}, y_{n},*h*^{(1)}_{n},*h*^{(2)}_{n}for*n*= 0, ±1, ±2, ...^{[31]}- ${\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}}$${\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:

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

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

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*j*_{0}(*x*) = sin*x*/*x*and*y*_{0}(*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:

- $\begin{array}{rl}{S}_{n}(x)& =x{j}_{n}(x)=\sqrt{\frac{\pi x}{2}}{J}_{n+\frac{}{}}\end{array}$
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

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

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

When α is a negative integer, we have

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

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

- $Y}_{\alpha}(z)\sim \{\begin{array}{l}{\displaystyle \frac{2}{\pi}}\left(\mathrm{ln}\left({\displaystyle \frac{z}{2}}\right)+\gamma \right)For\; example,\; this\; kind\; of\; differential\; equation\; appears\; inquantum\; mechanicswhile\; solving\; the\; radial\; component\; of\; theSchr\xf6dinger\text{'}s\; equationwith\; hypothetical\; cylindrical\; infinite\; potential\; barrier.[32]This\; differential\; equation,\; and\; the\; Riccati\u2013Bessel\; solutions,\; also\; arises\; in\; the\; problem\; of\; scattering\; of\; electromagnetic\; waves\; by\; a\; sphere,\; known\; asMie\; scatteringafter\; the\; first\; published\; solution\; by\; Mie\; (1908).\; See\; e.g.,\; Du\; (2004)[33]for\; recent\; developments\; and\; references.\end{array$
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]}- $J_{\alpha }(z)\sim {\frac {(-1)^{\alpha }}{(-\alpha )!}}\left({\frac {2}{z}}\right)^{\alpha }.$

- $}$

- $\begin{array}{rl}{y}_{0}(x)& =-{j}_{-1}(x)=-\frac{\mathrm{cos}x}{x},\\ {y}_{1}(x)& ={j}_{-2}(x)=-\frac{\mathrm{cos}x}{{x}^{2}}-\frac{\mathrm{sin}x}{x},\\ {y}_{2}(x)& =-{j}_{-3}(x)=\left(-\frac{3}{{x}^{2}}+1\right)\frac{\mathrm{cos}x}{x}-\frac{3\mathrm{sin}x}{{x}^{2}},\\ {y}_{3}(x)& ={j}_{-4}(x)=\left(-\frac{15}{{x}^{3}}+\frac{6}{x}\right)\frac{\mathrm{cos}x}{x}-(\frac{15}{{x}^{2}}\end{array}$

- $\begin{array}{rl}{j}_{0}(x)& =\frac{\mathrm{sin}x}{x}.\\ {j}_{1}(x)& =\frac{\mathrm{sin}x}{{x}^{2}}-\frac{\mathrm{cos}x}{x},\\ {j}_{2}(x)& =\left(\frac{3}{{x}^{2}}-1\right)\frac{\mathrm{sin}x}{x}-\frac{3\mathrm{cos}x}{{x}^{2}},\\ {j}_{3}(x)& =(and[29]\end{array}$

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

- $\begin{array}{rl}{j}_{n}(x)& =\sqrt{\frac{\pi}{2x}}{J}_{n+\frac{1}{2}}(x),\\ {y}_{n}(x)\end{array}$