AngouriMath
SolveAnExponentialBesideAMultipleOfTheDerivativeOfItsExponentialIntegral(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
An exponential of a rational function h times a rational function R that is
the derivative ofe^h N/D and of a multiple of the exponential integral of h :
e^h R = (e^h N/D)' + c e^h h'/h , whose integral is e^h N/D + c Ei(h) .
Hebisch's(2 + 2x + 3x^2 - x^3 + 2x^4) e^(x/(2 + x^2))/(2x + x^3) is
(2 + x^2) e^(x/(2 + x^2)) + Ei(x/(2 + x^2)) .
the derivative of
Hebisch's
Remarks
At a zero r of h of order k where h has no pole, h'/h has
the residuek , and (N/D)' + h' N/D none: a pole of N/D there is one of
order two or more in it. Soc is the residue of R at r over k , and
R - c h'/h is asked of SolveByExponentialAnsatz(AngouriMath.Entity,AngouriMath.Entity.Variable)'s ansatz, which
declinesR whole: by Liouville's theorem e^h R has no elementary
antiderivative wherec is not zero. Each linear factor of the numerator of h ,
as written, is a candidater . The answer is differentiated back at sampled points.
https://github.com/asc-community/AngouriMath/issues/233
the residue
order two or more in it. So
declines
antiderivative where
as written, is a candidate
https://github.com/asc-community/AngouriMath/issues/233
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