 Applying the recurrence relation to the second term, we have
which with simple rearrangement gives

Second, consider the reflection formula applied to formula_107.Formulas for other values of formula_31 for which the real part is integer or half-integer quickly follow by induction using the recurrence relation in the positive and negative directions.Perhaps the best-known value of the gamma function at a non-integer argument is

which can be found by setting formula_111 in the reflection or duplication formulas, by using the relation to the beta function given below with formula_112, or simply by making the substitution formula_113 in the integral definition of the gamma function, resulting in a Gaussian integral.