Built-in Mathematica Symbol

Refine

Refine[expr, assum]
gives the form of expr that would be obtained if symbols in it were replaced by explicit numerical expressions satisfying the assumptions assum.

Refine[expr]
uses default assumptions specified by any enclosing Assuming constructs.

MORE INFORMATION

Assumptions can consist of equations, inequalities, domain specifications such as xIntegers, and logical combinations of these.

Refine can be used on equations, inequalities and domain specifications.

Quantities that appear algebraically in inequalities are always assumed to be real.

Refine is one of the transformations tried by Simplify.

The following options can be given:

	Assumptions	$Assumptions	default assumptions to append to assum
	TimeConstraint	30	for how many seconds to try doing any particular transformation

EXAMPLES

CLOSE ALL

Basic Examples (2)

cannot be simplified for arbitrary complex

:

For explicit positive numeric expressions

evaluates to

:

Refine evaluates

to

, when a symbolic expression

is assumed to be positive:

Weaker assumptions may result in a weaker simplification:

Use Assuming to specify the same assumptions for several Refine calls:

cannot be simplified for arbitrary complex

:

In[1]:=

Out[1]=

For explicit positive numeric expressions

evaluates to

:

In[2]:=

Out[2]=

Refine evaluates

to

, when a symbolic expression

is assumed to be positive:

In[3]:=

Out[3]=

Weaker assumptions may result in a weaker simplification:

In[4]:=

Out[4]=

Use Assuming to specify the same assumptions for several Refine calls:

In[1]:=

Out[1]=

Scope (9)

Nested powers:

Product of powers:

Logarithms:

Trigonometric functions:

Equations and inequalities:

Element statements:

Floor, Ceiling, Round, IntegerPart and FractionalPart:

Mod:

Re, Im, Abs, Arg, Conjugate and Sign:

Options (4)

Assumptions can be given both as an argument and as an option value:

The default value of the Assumptions option is $Assumptions:

When Assumptions is given as an argument, $Assumptions is used as well:

Specifying Assumptions as an option value prevents Refine from using $Assumptions:

Checking whether a condition follows from assumptions may take a long time:

If a condition does not follow from assumptions, checking this may still take a long time:

The time spent on a single condition check is restricted by the value of TimeConstraint:

With a time constraint of 1 second Refine cannot prove that x⁶+y⁶+z⁶-x y z≥0:

Applications (1)

Write code that uses assumptions; find the number of real roots of a x²+b x+c:

Properties & Relations (4)

Refine rules correspond to automatic simplification rules for numeric expressions:

Use Assuming to propagate assumptions:

Use Simplify for more simplification rules:

Use FullSimplify for special function simplification:

Possible Issues (1)

Expressions appearing algebraically in inequality assumptions are assumed to be real: