calc: Unsafe Simplifications
11.3.3 “Unsafe” Simplifications
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Calc is capable of performing some simplifications which may sometimes
be desired but which are not “safe” in all cases. The ‘a e’
(‘calc-simplify-extended’) [‘esimplify’] command applies the algebraic
simplifications as well as these extended, or “unsafe”, simplifications.
Use this only if you know the values in your formula lie in the
restricted ranges for which these simplifications are valid. You can
use Extended Simplification mode (‘m E’) to have these simplifications
done automatically.
The symbolic integrator uses these extended simplifications; one
effect of this is that the integrator’s results must be used with
caution. Where an integral table will often attach conditions like “for
positive ‘a’ only,” Calc (like most other symbolic integration programs)
will simply produce an unqualified result.
Because ‘a e’’s simplifications are unsafe, it is sometimes better to
type ‘C-u -3 a v’, which does extended simplification only on the top
level of the formula without affecting the sub-formulas. In fact, ‘C-u
-3 j v’ allows you to target extended simplification to any specific
part of a formula.
The variable ‘ExtSimpRules’ contains rewrites to be applied when the
extended simplifications are used. These are applied in addition to
‘EvalRules’ and ‘AlgSimpRules’. (The ‘a r AlgSimpRules’ step described
above is simply followed by an ‘a r ExtSimpRules’ step.)
Following is a complete list of the “unsafe” simplifications.
Inverse trigonometric or hyperbolic functions, called with their
corresponding non-inverse functions as arguments, are simplified. For
example, ‘arcsin(sin(x))’ changes to ‘x’. Also, ‘arcsin(cos(x))’ and
‘arccos(sin(x))’ both change to ‘pi/2 - x’. These simplifications are
unsafe because they are valid only for values of ‘x’ in a certain range;
outside that range, values are folded down to the 360-degree range that
the inverse trigonometric functions always produce.
Powers of powers ‘(x^a)^b’ are simplified to ‘x^(a b)’ for all ‘a’
and ‘b’. These results will be valid only in a restricted range of ‘x’;
for example, in ‘(x^2)^1:2’ the powers cancel to get ‘x’, which is valid
for positive values of ‘x’ but not for negative or complex values.
Similarly, ‘sqrt(x^a)’ and ‘sqrt(x)^a’ are both simplified (possibly
unsafely) to ‘x^(a/2)’.
Forms like ‘sqrt(1 - sin(x)^2)’ are simplified to, e.g., ‘cos(x)’.
Calc has identities of this sort for ‘sin’, ‘cos’, ‘tan’, ‘sinh’, and
‘cosh’.
Arguments of square roots are partially factored to look for squared
terms that can be extracted. For example, ‘sqrt(a^2 b^3 + a^3 b^2)’
simplifies to ‘a b sqrt(a+b)’.
The simplifications of ‘ln(exp(x))’, ‘ln(e^x)’, and ‘log10(10^x)’ to
‘x’ are also unsafe because of problems with principal values (although
these simplifications are safe if ‘x’ is known to be real).
Common factors are canceled from products on both sides of an
equation, even if those factors may be zero: ‘a x / b x’ to ‘a / b’.
Such factors are never canceled from inequalities: Even the extended
simplifications are not bold enough to reduce ‘a x < b x’ to ‘a < b’ (or
‘a > b’, depending on whether you believe ‘x’ is positive or negative).
The ‘a M /’ command can be used to divide a factor out of both sides of
an inequality.
Info Catalog
calc: Algebraic Simplifications
calc: Simplifying Formulas
calc: Simplification of Units