The Gamma Function

This program lets you calculate the gamma function of a number and the natural log of gamma. The program is designed to recognize out-of-range results and to compute

ln |gamma|

instead of gamma when gamma is too large.

Introduction

The gamma function is defined by the improper integral:

G = gamma(x) = ∫tx-1e-1dt over the range 0 to ∞ (infinity)

which converges for noninteger values of x < 1 and all values of x > 0. The gamma function is related to the factorial function: gamma( x+1 ) = x!.

The approximate input range of gamma is: -70.064 ≤ x ≤ 70.957, where x is not an integer less than 1.

Although the calculator cannot display gamma for results whose exponents exceed 99, you can still determine the magnitude of gamma from the ln |gamma| using the laws of logarithms. The mantissa of gamma is

Antilog( Frac( lnG / ln10 ) )

and the exponent of scientific notation for gamma is

Intg( lnG / ln10 ).

Reference

Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, National Bureau of Standards, 1972.

Using the Program

To use the Gamma Function program:
  1. Select { GAM } from the MATHEMATICS menu.
    The calculator displays the GAMMA FUNCTION menu.
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    { G } - Calculates the gamma function of the displayed value or the natural log of gamma if the result is too large.
    { lnG } - Calculates the natural log of the gamma function of the displayed value.
  2. Enter the value and press the applicable key.
    The program displays the result. For example, with a value of 0.5 entered in the display, pressing { G } will produce the following displayed result:
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  3. Repeat step 2 for other values.

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