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The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function. Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem .
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
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The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
A fair amount of effort has been made to calculate the numerical value of the Fransén–Robinson constant with high accuracy. The value was computed to 36 decimal places by Herman P. Robinson using 11 point Newton–Cotes quadrature, to 65 digits by A. Fransén using Euler–Maclaurin summation, and to 80 digits by Fransén and S. Wrigge using Taylor series and other methods.
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.