skip to main content

Mathematics (thesaurus)

Content language

Home / Explore / Browse / Vocabularies / Mathematics (thesaurus)

mathematical physics > special function > spherical Bessel function

mathematical analysis > functional analysis > special function > spherical Bessel function

Preferred term

spherical Bessel function

Definition(s)

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form
${\\displaystyle x^{2}{\ rac {d^{2}y}{dx^{2}}}+2x{\ rac {dy}{dx}}+\\left(x^{2}-n(n+1)\ ight)y=0.}$
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
${\\displaystyle {\egin{aligned}j_{n}(x)&={\\sqrt {\ rac {\\pi }{2x}}}J_{n+{\ rac {1}{2}}}(x),\\\\y_{n}(x)&={\\sqrt {\ rac {\\pi }{2x}}}Y_{n+{\ rac {1}{2}}}(x)=(-1)^{n+1}{\\sqrt {\ rac {\\pi }{2x}}}J_{-n-{\ rac {1}{2}}}(x).\\end{aligned}}}$
$y n$ is also denoted $n n$ or $η n$ ; some authors call these functions the spherical Neumann functions.
From the relations to the ordinary Bessel functions it is directly seen that:
${\\displaystyle {\egin{aligned}j_{n}(x)&=(-1)^{n}y_{-n-1}(x)\\\\y_{n}(x)&=(-1)^{n+1}j_{-n-1}(x)\\end{aligned}}}$

Broader concept(s)

special function

In other languages

fonction de Bessel sphérique

French

URI

http://data.loterre.fr/ark:/67375/PSR-PC9Q4CNL-L

Download this concept:

RDF/XML TURTLE JSON-LD Created 7/27/23, last modified 7/27/23