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mathematical analysis > functional analysis > operator > integral transform > Abel transform

mathematical analysis > functional analysis > transformation > integral transform > Abel transform

Preferred term

Abel transform

Definition(s)

In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by

${\\displaystyle F(y)=2\\int _{y}^{\\infty }{\ rac {f(r)r}{\\sqrt {r^{2}-y^{2}}}}\\,dr.}$

Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by

${\\displaystyle f(r)=-{\ rac {1}{\\pi }}\\int _{r}^{\\infty }{\ rac {dF}{dy}}\\,{\ rac {dy}{\\sqrt {y^{2}-r^{2}}}}.}$

Broader concept(s)

integral transform

In other languages

transformée d'Abel

French

URI

http://data.loterre.fr/ark:/67375/PSR-TQC1MD4T-P

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