 The expansion of tan was used by both Lambert and Legendre to prove that π is irrational.The Bessel function formula_57 can be written

from which it follows

These formulas are also valid for every complex "z".Since formula_60, formula_61

With some manipulation, this can be used to prove the simple continued fraction representation of
"e",

The error function erf ("z"), given by

can also be computed in terms of Kummer's hypergeometric function