In mathematics, the Stieltjes transformation Sρ(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
S ρ ( z ) = ∫ I ρ ( t ) d t t − z , z ∈ C ∖ I . {\displaystyle S_{\rho }(z)=\int _{I}{\frac {\rho (t)\,dt}{t-z}},\qquad z\in \mathbb {C} \setminus I.}
Inverse formula 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
ρ ( x ) = lim ε → 0 + S ρ ( x + i ε ) − S ρ ( x − i ε ) 2 i π . {\displaystyle \rho (x)=\lim _{\varepsilon \to 0^{+}}{\frac {S_{\rho }(x+i\varepsilon )-S_{\rho }(x-i\varepsilon )}{2i\pi }}.}
Derivation of formula Recall from basic calculus that
∫ − ∞ ∞ 1 x 2 + 1 d x = lim x → ∞ arctan x − lim x → − ∞ arctan x = π 2 − ( − π 2 ) = π . {\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}dx=\lim _{x\to \infty }\arctan x-\lim _{x\to -\infty }\arctan x={\tfrac {\pi }{2}}-(-{\tfrac {\pi }{2}})=\pi {\text{.}}}
Hence f ( x ) = 1 π ( x 2 + 1 ) − 1 {\displaystyle f(x)={\tfrac {1}{\pi }}(x^{2}+1)^{-1}} is the probability density function of a distribution—a Cauchy distribution. Via the change of variables x = ( t − t 0 ) / ε {\displaystyle x=(t-t_{0})/\varepsilon } we get the full family of Cauchy distributions:
1 = ∫ − ∞ ∞ 1 / π x 2 + 1 d x = ∫ − ∞ ∞ 1 / π ( t − t 0 ε ) 2 + 1 d x d t d t = ∫ − ∞ ∞ ε / π ( t − t 0 ) 2 + ε 2 d t {\displaystyle 1=\int _{-\infty }^{\infty }{\frac {1/\pi }{x^{2}+1}}dx=\int _{-\infty }^{\infty }{\frac {1/\pi }{({\frac {t-t_{0}}{\varepsilon }})^{2}+1}}{\frac {dx}{dt}}dt=\int _{-\infty }^{\infty }{\frac {\varepsilon /\pi }{(t-t_{0})^{2}+\varepsilon ^{2}}}dt}
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