evenfuncs=F where F is a function name or a list or set of function names

oddfuncs=G where G is a function name or a list or set of function names

The verify(expr1, expr2, sign) calling sequence returns true if by means of applying any combination of the following transformations to subexpressions of expr1 and expr2 they can eventually be transformed into the same target expression:

$\left(-x\right)y$ $\mapsto$ $-xy$ if $x$ is of type $\mathrm{`+`}$

${\left(-a\right)}^n$ $\mapsto$ $a^n$ if $n$ is an even integer

${\left(-a\right)}^n$ $\mapsto$ $-a^n$ if $n$ is an odd integer

$f\left(-a\,\mathrm{...}\right)$ $\mapsto$ $f\left(a\,\mathrm{...}\right)$ if $f$ is one of the even functions given in the option evenfuncs=F

$g\left(-a\,\mathrm{...}\right)$ $\mapsto$ $-g\left(a\,\mathrm{...}\right)$ if $g$ is one of the odd functions given in the option oddfuncs=G

and returns false otherwise.

This verification not only ensures the expr1 and expr2 are mathematically equal but also that they have similar forms, which is a property that is not ensured by verifying with normal.

The arguments expr1 and expr2 need not be algebraic.

The symmetries defined by the optional arguments evenfuncs=F and oddfuncs=G are assumed to apply to the first argument of each given function, except in the case of piecewise functions where factoring out a sign means factoring it out of each branch (i.e. each of the value arguments) simultaneously.

This verification is symmetric.

$\mathrm{verify}\left(\frac{x}{a-b}\,\frac{x}{b-a}\,\mathrm{sign}\right)$

$\mathrm{false}$

$\mathrm{verify}\left(-\frac{x}{a-b}\,\frac{x}{b-a}\,\mathrm{sign}\right)$

$\mathrm{true}$

$\mathrm{verify}\left(\cos \left(x\left(a-b\right)\right)\,\cos \left(x\left(b-a\right)\right)\,'\mathrm{sign}\left(\mathrm{evenfuncs}=\cos \right)'\right)$

$\mathrm{true}$

$\mathrm{verify}\left(-a\left(\mathrm{Int}\left(\sin \left(\frac{1}{1-x}\right)\,x\right)\right)\,a\left(\mathrm{Int}\left(\sin \left(\frac{1}{x-1}\right)\,x\right)\right)\,'\mathrm{sign}\left(\mathrm{oddfuncs}=\left\{\mathrm{Int}\,\sin \right\}\right)'\right)$

$\mathrm{true}$

$p≔\frac{3{\left(x-y\right)}^2\left(x-2y-xy\right)\left(y-3x\right)\left(x{y}^2+2x-2y\right)}{3x-y}$

$p≔\frac{3{\left(x-y\right)}^2\left(-xy+x-2y\right)\left(y-3x\right)\left(x{y}^2+2x-2y\right)}{3x-y}$

$q≔-3{\left(y-x\right)}^2\left(xy+2y-x\right)\left(2y-x{y}^2-2x\right)$

$q≔-3{\left(y-x\right)}^2\left(xy-x+2y\right)\left(-x{y}^2-2x+2y\right)$

$\mathrm{verify}\left(p\,q\,\mathrm{sign}\right)$

$\mathrm{false}$

$\mathrm{verify}\left(p\,q\,\mathrm{normal}\right)$

$\mathrm{true}$

The verify/sign command was updated in Maple 2023.