__(two _characters) or BlankSequence[] is a pattern object that can stand for any sequence of one or more Mathematica expressions. __h or BlankSequence[h] can stand for any ...

p : v is a pattern object that represents an expression of the form p, which, if omitted, should be replaced by v.

file.tm is a MathLink template file.

TrigExpand[expr] expands out trigonometric functions in expr.

CompileEvaluate[expr] compiles expr and then evaluates the resulting compiled code.

ExistsRealQ[ineqs, {x_1, x_2, ...}] tests whether there exist real values of the x_i for which the inequalities and equations ineqs are satisfied.

ForAllRealQ[ineqs, {x_1, x_2, ...}] tests whether for all real values of the x_i the inequalities and equations ineqs are satisfied.

ImpliesRealQ[ineqs_1, ineqs_2] tests whether the inequalities and equations ineqs_1 imply the ineqs_2 for all real values of all variables.

 

Everything that Mathematica does can be thought of as derived from its ability to apply general transformation rules to arbitrary symbolic expressions. The Mathematica ...