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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. Other fractional arguments can be approximated through efficient infinite products, infinite series ...
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.
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 ...
The values are organized in a format that makes them readable by a thermodynamic calculation program or for use in a spreadsheet. For example, the Excel -based thermodynamic database FREED [1] creates the following type of datafile, here for a standard pressure of 1 atm.
Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in spreadsheet or computer algebra systems. In Microsoft Excel, for example, the complete beta function can be computed with the GammaLn function (or special.gammaln in Python's SciPy package):
The gamma distribution generalizes the Erlang distribution by allowing k to be any positive real number, using the gamma function instead of the factorial function. That is: if k is an integer and X ∼ Gamma ( k , λ ) , {\displaystyle X\sim \operatorname {Gamma} (k,\lambda ),} then X ∼ Erlang ( k , λ ) {\displaystyle X\sim ...
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 ...
Similarly, pixel values in standard image file formats are usually gamma-compensated, either for sRGB gamma (or equivalent, an approximation of typical of legacy monitor gammas), or according to some gamma specified by metadata such as an ICC profile. If the encoding gamma does not match the reproduction system's gamma, further correction may ...