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Borel's Expansion


Let phi(t)=sum_(n=0)^(infty)A_nt^n be any function for which the integral

 I(x)=int_0^inftye^(-tx)t^pphi(t)dt

converges. Then the expansion

 I(x)=(Gamma(p+1))/(x^(p+1))[A_0+(p+1)(A_1)/x+(p+1)(p+2)(A_2)/(x^2)+...],

where Gamma(z) is the gamma function, is usually an asymptotic series for I(x).


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Cite this as:

Weisstein, Eric W. "Borel's Expansion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BorelsExpansion.html

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