Search results
Results From The WOW.Com Content Network
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 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.
Suppose we wish to generate random variables from Gamma(n + δ, 1), where n is a non-negative integer and 0 < δ < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then −ln U is ...
Thus computing the gamma function becomes a matter of evaluating only a small number of elementary functions and multiplying by stored constants. The Lanczos approximation was popularized by Numerical Recipes , according to which computing the gamma function becomes "not much more difficult than other built-in functions that we take for granted ...
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
Plot of 1 / Γ(x) along the real axis Reciprocal gamma function 1 / Γ(z) in the complex plane, plotted using domain coloring. In mathematics, the reciprocal gamma function is the function = (), where Γ(z) denotes the gamma function.
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding:
In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper. [1] The formula is a modification of Stirling's approximation, and has the form