 Finally, putting these together, the analytic function

satisfies formula_12 for all integers formula_7.Since formula_14 for all complex numbers formula_15, this function also provides an extension of the Fibonacci sequence to the entire complex plane.Hence we can calculate the generalized Fibonacci function of a complex variable, for example, 

The term "Fibonacci sequence" is also applied more generally to any function formula_17 from the integers to a field for which formula_18.