If[condition, t, f] gives t if condition evaluates to True, and f if it evaluates to False. If[condition, t, f, u] gives u if condition evaluates to neither True nor False.

f' represents the derivative of a function f of one argument. Derivative[n_1, n_2, ...][f] is the general form, representing a function obtained from f by differentiating n_1 ...

CharacteristicPolynomial[m, x] gives the characteristic polynomial for the matrix m. CharacteristicPolynomial[{m, a}, x] gives the generalized characteristic polynomial with ...

Simplifying expressions. Mathematica does not automatically simplify an algebraic expression like this. Simplify performs the simplification.

Zeta[s] gives the Riemann zeta function \[Zeta](s). Zeta[s, a] gives the generalized Riemann zeta function \[Zeta](s, a).

Mathematica's symbolic control objects include options that make it easy to optimize both appearance and functionality in arbitrarily sophisticated interfaces.

Heads is an option for functions which use level specifications that specifies whether heads of expressions should be included.

Functions that are specified by simple algebraic formulas tend to be such that when their input is changed only slightly, their output also changes only slightly. But ...

Cos[z] gives the cosine of z.

Sin[z] gives the sine of z.