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# Expand

### Description

• Expand expands out products and positive integer powers.
• Expand[transformations] expands out products and positive integer powers and applies transformations at each level of expand procedure.

### Examples

Expand a polynomial expressions:

println Expand >> '(x_n + y_n)*(f_m - r_m)'.t

   > x_{n}*f_{m}+f_{m}*y_{n}-r_{m}*y_{n}-x_{n}*r_{m}


println Expand >> '(1 + x)**4'.t

   > x**4+1+4*x**3+6*x**2+4*x


println Expand >> '(x + y)/z'.t

   > x/z+y/z


Expand relabels dummies when necessary:

println Expand >> '(A_m^m + 1)**3'.t

   > 3*A_{m}^{m}*A_{a}^{a}+A_{m}^{m}*A_{a}^{a}*A_{b}^{b}+1+3*A_{b}^{b}


Expand does not go inside functions and denominators; ExpandAll does:

println Expand >> 'f[(x + y)**2]'.t

   > f[(x + y)**2]

println ExpandAll >> 'f[(x + y)**2]'.t

   > f[x**2 + 2*x*y + y**2]


### Details

Expand[transformations] will additionally apply transformations during expand procedure:

println Expand['k_a*k^a = 0'.t] >> '(k_a + t_a)*(k^a + t^a)'.t

   > 2*k_a*t^a + t_a*t^a


Passing additional transformations can significantly improve the performance when expanding huge expressions. For example, when a huge expression involves many metric tensors, one can pass EliminateMetrics in order to reduce the number of processed terms. Consider a random example:

//create random generator, which generates
// random tensors consisting of metric and A_m
RandomTensor randomTensor = new RandomTensor();
randomTensor.clearNamespace()

//loop to warm up JVM
for (def i in 1..1000) {
def a, b
//next random tensor
def t = randomTensor.nextTensorTree(4, 3, 8, '_a'.si)
def simplify = EliminateMetrics & 'A_a*A^a = 1'.t & 'd^i_i = 10'.t

//this will typically 10 times faster
timing {
a = Expand[simplify] >> t
}
//then this
timing {
b = (Expand & simplify) >> t
}

assert a == b
println ''
}

The sample output will looks like:
Time: 10 ms.
Time: 1015 ms.

Time: 7 ms.
Time: 6566 ms.

Time: 66 ms.
Time: 983 ms.
...