AngouriMath

Navigation

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

FactorANonnegativeVariableOutOfRadicals(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

The radicals of expr with every power of u that a radical can hold taken out of it, for a u that is not
negative: (u^2 (1 + u))^(1/2) is u sqrt(1 + u), and
(2u/(1 + u))^(1/2) is sqrt(2u)/sqrt(1 + u).

Remarks

For the substitutions whose variable is one by construction -- u = e^(k x),
u = (a x + b)^(1/q) with q even, u = x^(1/q) likewise -- and
for no one else: the simplifier is right not to call sqrt(u^2 (1 + u))u sqrt(1 + u) for a u it knows nothing about, and the substitution
knows exactly this about its own. Two identities, each exact for u >= 0:
(a b)^r = a^r b^r whenever a is a non-negative real, since then
arg(a b) = arg(b); and (P/Q)^r = P^r/Q^r whenever Q is a positive
real, for the same reason -- which a polynomial in u with non-negative
coefficients and a positive constant term is, at every u >= 0.
Without it 1/sqrt(t + t^(3/2)) under u = sqrt(t) is
2u/sqrt(u^2 + u^3), a root of a cubic, and with it 2/sqrt(1 + u);
sqrt(1 + tanh(4x)) under u = e^(8x) is a root of 2u/(1 + u),
which nothing rationalises, and with it sqrt(2) sqrt(u)/sqrt(1 + u).

AngouriMath

Table of contents

AngouriMath

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

FactorANonnegativeVariableOutOfRadicals(AngouriMath.Entity,AngouriMath.Entity.Variable)