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Function: expands - symbolic version of expand

Calling sequences:

expands(expr);

Parameters:

expr - the expression to be expanded.

Description:

This procedure distributes powers over sums and applies expansion formulas for many functions. The main difference between this and the built-in procedure expand is that expands applies the formulas even in cases when they are not known to be valid. Thus this is a "symbolic" expansion, analogous to the symbolic simplification of simplify(..., symbolic) .

expand is the procedure normally used to expand an expression. However, the expansion formulas for certain functions are only valid under restrictive assumptions, and expand will not apply them unless Maple knows that these assumptions are valid. For example,

is not always valid (e.g. it is false if

are negative reals). Therefore expand will only apply it in case

is known to be a positive real. This can be made known to Maple by using assume .

However, in many cases it is inconvenient to make the necessary assumptions, or the assumptions are excessively restrictive (e.g. the expansion formula for ln given above is also valid when

, but expand will not use it).

To go in the reverse direction, there is the combine procedure. Note that some of the sub-procedures of this have symbolic options, e.g. combine(..., power, symbolic) .

expands will also expand the arctan function, which expand will not. This uses the formulas arctan((a+b)/(1-a*b)) = arctan(a)+arctan(b)

There is one case where expand will produce an expansion which may be undesirable, and expands will not: when

is assumed to be real, expand(exp(I*a)) results in (-1)^a

, while expands(exp(I*a)) leaves it as

This function is part of the Maple Advisor Database library.

Examples:

Both expand and expands produce many of the same expansions.

> expands((x+1)*(x+2));

> expands(sin(x+y));

The expansion of the logarithm of a product. This is not always valid, as there may be an additional term where is an integer.

> expand(ln(x*y)),expands(ln(x*y));

The square root of a product. This is not always valid, as there may be a factor of .

> expand(sqrt(x*y)),expands(sqrt(x*y));

A power of a power. Again, not always valid (e.g. try it for ).

> expand((x^(3/2))^(2/3)), expands((x^(3/2))^(2/3));

An expandable arctan. expand won't do anything with this, even under conditions that

guarantee its validity.

> assume(x>0,x<1/4,y>0,y<1/4);
expand(arctan((x+y)/(1-x*y))), expands(arctan((x+y)/(1-x*y)));

A case where expand expands but expands does not. I think it is better to use exp rather than a power of here, to avoid complications with multivalued functions.

> assume(x,real); expand(exp(2*Pi*I*x)), expands(exp(2*Pi*I*x));

See also:

assume , combine , expand

Maple Advisor Database R. Israel 2000