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mathematical analysis > functional analysis > operator > integral transform > Stieltjes transform

mathematical analysis > functional analysis > transformation > integral transform > Stieltjes transform

Preferred term

Stieltjes transform

Definition(s)

In mathematics, the Stieltjes transformation $S ρ (z)$ of a measure of density $ρ$ on a real interval $I$ is the function of the complex variable $z$ defined outside $I$ by the formula
${\\displaystyle S_{\ ho }(z)=\\int _{I}{\ rac {\ ho (t)\\,dt}{z-t}},\\qquad z\\in \\mathbb {C} \\setminus I.}$
Under certain conditions we can reconstitute the density function $ρ$ starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density $ρ$ is continuous throughout $I$ , one will have inside this interval
${\\displaystyle \ ho (x)=\\lim _{\\varepsilon \ o 0^{+}}{\ rac {S_{\ ho }(x-i\\varepsilon )-S_{\ ho }(x+i\\varepsilon )}{2i\\pi }}.}$

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Stieltjes_transformation)

Broader concept(s)

integral transform

Synonym(s)

Stieltjes transformation

In other languages

transformée de Stieltjes

French
transformation de Stieltjes

URI

http://data.loterre.fr/ark:/67375/PSR-CL3SVR3H-P

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