Simplifications Involving Absolute value

[andrew-murdza]

Piecewise definition:

• |-x|->|x|
• {match:'|x|',replace:'x',condition:'x\ge0'} (e.g. |x^2|)
• {match:'|x|',replace:'-x',condition:'x\le0'} (e.g. |1-\sqrt{2}|)

Products and Quotients:

• {match:'|xy|',replace:'x|y|',condition:'x\ge0'}
• {match:'|xy|',replace:'-x|y|',condition:'x\le0'}
• {match:'|\frac{x}{y}|',replace:'\frac{x}{|y|}',condition:'x\ge0'}
• {match:'|\frac{x}{y}|',replace:'-\frac{x}{|y|}',condition:'x\le0'}
• |xy|->|x||y| or |x||y|->|xy| (choose 1) helps with stuff like |x|x
• |\frac{x}{y}|->\frac{|x|}{|y|} or \frac{|x|}{|y|}->|\frac{x}{y}| (choose 1)

Powers:

• {match:'|x|^n',replace:'x^n',condition:'n is an even integer'}
• {match:'|x|^{\frac{n}{m}}',replace:'x^{\frac{n}{m}}',condition:'n is an even integer and m is an odd integer'}
• {match:'|x^n|', replace:'|x|^n',condition:'n is not an even integer or a rational with even numerator and odd denominator'}

Even functions:

• \cos(|x|)->\cos(x)
• \sec(|x|)->\sec(x)
• \cosh(|x|)->\cosh(x)
• \sech(|x|)->\sech(x)
• Even better: {match:'f(|x|)',replace:'f(x)',condition:'f is even'}

Odd functions:

• |\sin(x)|->\sin(|x|)
• |\tan(x)|->\tan(|x|)
• |\cot(x)|->\cot(|x|)
• |\csc(x)|->\csc(|x|)
• |\arcsin(x)|->\arcsin(|x|)
• |\arctan(x)|->\arctan(|x|)
• |\arccot(x)|->\arccot(|x|)
• |\arccsc(x)|->\arccsc(|x|)
• |\sinh(x)|->\sinh(|x|)
• |\tanh(x)|->\tanh(|x|)
• |\coth(x)|->\coth(|x|)
• |\csch(x)|->\csch(|x|)
• |\arcsinh(x)|->\arcsinh(|x|)
• |\arctanh(x)|->\arctanh(|x|)
• |\arccoth(x)|->\arccoth(|x|)
• |\arccsch(x)|->\arccsch(|x|)
• Even better: {match:'|f(x)|',replace:'f(|x|)',condition:'f is odd'}