AngouriMath
SolveAnExponentialOfAQuadraticOverTheLinearThroughItsVertex(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
An exponential of a quadratic over a whole power of the linear through its vertex:
F^(A x^2 + B x + C)/(p + q x)^n with 2 A p = q B , which is how
f^(a + b x + c x^2)/(b + 2 c x)^n arrives. Under u = p + q x the exponent is
A/q^2 u^2 + C - B^2/(4 A) , with no linear part, and the integrand
F^(A/q^2 u^2 + C - B^2/(4 A))/(q u^n) is a moment about 0 of the Gaussian: the
exponential integral of the square for an oddn , the error function for an even
one. Rubi's 2.3. https://github.com/asc-community/AngouriMath/issues/233
exponential integral of the square for an odd
one. Rubi's 2.3. https://github.com/asc-community/AngouriMath/issues/233
Remarks
Only through the vertex: anywhere else the shift leaves a linear part in the exponent,
and the moments about 0 do not answer it. The exponent inu is written from the
coefficients, not by substitutingx = (u - p)/q and asking the simplifier to see
that the linear part cancels.
and the moments about 0 do not answer it. The exponent in
coefficients, not by substituting
that the linear part cancels.
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