 The gamma function is the most useful solution in practice, being analytic (except at the non-positive integers), and it can be defined in several equivalent ways.However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as , will give another function with that property.A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function,

for equal to any positive real number.