AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug ~~Donate~~ Contacts

SolveAProductOfExponentialsOfQuadratics(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A product of exponentials of polynomials of degree at most two, whole powers of sines
and cosines of such polynomials, whole powers of sums of those exponentials -- which is
how a hyperbolic function is written -- and a polynomial, where some exponent is left a
quadratic. A sine or cosine is written as exponentials, sin(q) = (e^(i q) - e^(-i q))/(2i),
and the product multiplied out is a sum of terms k P(x) e^(Q(x)), with Q of
degree at most two: the Gaussian's moments where Q is a quadratic, and elementary
where it is not. Rubi's 4.7.6, 6.1.5 and 6.2.5, f^(a + b x + c x^2) sin(d + e x + f x^2)^n and f^(a + b x + c x^2) cosh(d + e x + f x^2)^n, 6.1.4's x^2 sinh(a + b x + c x^2)^2,
and 2.3's f^(a + b x + c x^2) g^(d + e x + f x^2).
https://github.com/asc-community/AngouriMath/issues/1501

Remarks

A sine or cosine is taken only beside an exponential. Alone, or beside a polynomial,
the sine of a quadratic is a Fresnel integral, and x sin(x^2) is -cos(x^2)/2,
which the substitution answers with no imaginary unit in it. A sum of exponentials is
taken beside another exponential, or in a power: x^2 sinh(q) alone is a sum the
integrator splits, and each of its terms is the Gaussian's.
A term's exponent is kept as whole multiples of the exponents the integrand writes, not
as their sum: e^(i q) times e^(-i q) is then the multiple 0, decidably,
where the sum i q + (-i) q is not read as zero, and a Gaussian of it would
divide by a square's coefficient that is.

AngouriMath

Table of contents

AngouriMath

SolveAProductOfExponentialsOfQuadratics​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

SolveAProductOfExponentialsOfQuadratics(AngouriMath.Entity,AngouriMath.Entity.Variable)