Page "Banach algebra" Paragraph 22

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

( In particular, the exponential map can be used to define abstract index groups.

) The formula for the geometric series remains valid in general unital Banach algebras.

The binomial theorem also holds for two commuting elements of a Banach algebra.