AngouriMath
SolveByFoldingAnExponentialOfALogarithm(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
An exponential of a multiple of a logarithm is a power of the argument:
e^(k ln(q)) is q^k , since e^(k ln q) is the definition of the
principal power for every complexq other than zero. That is the spelling the
parser gives every inverse hyperbolic function --acoth(a x) is
1/2 ln((a x + 1)/(a x - 1)) -- so e^acoth(a x) x^3 arrives as
e^(1/2 ln((a x + 1)/(a x - 1))) x^3 , which no exponential rule reads, and is
x^3 sqrt((a x + 1)/(a x - 1)) , a root of a quotient of linears, which the
radical substitution answers. Rubi's 7.4.2. The simplifier folds the same shape
since #1430; the integrand is not simplified before the rules see it, so the fold
is a rule here, asked as a question of its own so the closed rules meet it at the top.
https://github.com/asc-community/AngouriMath/issues/718
principal power for every complex
parser gives every inverse hyperbolic function --
radical substitution answers. Rubi's 7.4.2. The simplifier folds the same shape
since #1430; the integrand is not simplified before the rules see it, so the fold
is a rule here, asked as a question of its own so the closed rules meet it at the top.
https://github.com/asc-community/AngouriMath/issues/718
Remarks
The exponential the fold leaves free of the variable, e^a from
a + b ln(c x^n) , is asked as a symbol k and written back into the answer,
conditions and all:d/dx F(x, k) = R(x, k) for every k , so for
k = e^a too. As an exponential it was arithmetic the rules never finished --
the division bye^(2 a) writes e^(-2 a) , and the two do not cancel --
andtanh(a + 2 ln(x))^2/x^3 (Rubi's 6.3.2) came back a piecewise of 7,796
characters, six arms on the signs of4 e^(2 a) and 4 e^(-2 a) , with an
e^(-56 a) in its derivative's condition that a decimal evaluation takes for zero
ata = 1.07 ; in k it is 181. The two spellings are asked and the shorter
answer kept, since ink the factorisation reads the coefficient and splits
k^2 x^4 - 1 into arms on the sign of k : coth(a + 2 ln(x))/x^2 is
3,577 characters ink and 413 as the exponential. Only where the folded
integrand is a rational function of x -- beside a symbolic power of x the name cost
four times the allocation for an answer as long -- and every such exponential is a
whole power of one of them, so that the relations between them survive the naming.
https://github.com/asc-community/AngouriMath/issues/233
conditions and all:
the division by
and
characters, six arms on the signs of
at
answer kept, since in
3,577 characters in
integrand is a rational function of x -- beside a symbolic power of x the name cost
four times the allocation for an answer as long -- and every such exponential is a
whole power of one of them, so that the relations between them survive the naming.
https://github.com/asc-community/AngouriMath/issues/233
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