FullSimplify[expr] tries a wide range of transformations on expr involving elementary and special functions, and returns the simplest form it finds.
FullSimplify[expr, assum] does simplification using assumptions.
FullSimplify will always yield at least as simple a form as Simplify, but may take substantially longer.
The following options can be given:
FullSimplify uses RootReduce on expressions that involve Root objects.
FullSimplify does transformations on most kinds of special functions.
See notes for Simplify.
See The Mathematica Book: Section 1.4.4 and Section 3.3.9.
Implementation Notes: see section A.9.5.
See also: Simplify, Factor, Expand, PowerExpand, ComplexExpand, TrigExpand, Element, FunctionExpand.
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