GAMMA

This function evaluates the complete gamma function.

Required Arguments

X — Argument for which the complete gamma function is desired. (Input)

FORTRAN 90 Interface

Generic: GAMMA (X)

Specific: The specific interface names are S_GAMMA, D_GAMMA, and C_GAMMA.

FORTRAN 77 Interface

Single: GAMMA (X)

Double: The double precision function name is DGAMMA.

Complex: The complex name is CGAMMA.

Description

The gamma function, Γ(z), is defined to be

For ℜ(z) < 0, the above definition is extended by analytic continuation.

z must not be so close to a negative integer that the result is less accurate than half precision. If ℜ(z) is too small, then the result will underflow. Users who need such values should use the log gamma function ALNGAM. When ℑ(z) ≈ 0, ℜ(z) should be greater than xmin so that the result does not underflow, and ℜ(z) should be less than xmax so that the result does not overflow. xmin and xmax are available by

CALL R9GAML (XMIN, XMAX)

Note that z must not be too far from the real axis because the result will underflow.

Figure 1, Plot of Γ(x) and 1/Γ(x)

Comments

Informational Errors

Type	Code	Description
2	3	The function underflows because X is too small.
3	2	Result is accurate to less than one‑half precision because X is too near a negative integer.

Examples

Example 1

In this example, Γ(5.0) is computed and printed.

USE GAMMA_INT

USE UMACH_INT

IMPLICIT NONE

! Declare variables

INTEGER NOUT

REAL VALUE, X

! Compute

X = 5.0

VALUE = GAMMA(X)

! Print the results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) X, VALUE

99999 FORMAT (' GAMMA(', F6.3, ') = ', F6.3)

END

Output

GAMMA( 5.000) = 24.000

Example 2

In this example, Γ(1.4 + 3i) is computed and printed.

USE GAMMA_INT

USE UMACH_INT

IMPLICIT NONE

! Declare variables

INTEGER NOUT

COMPLEX VALUE, Z

! Compute

Z = (1.4, 3.0)

VALUE = GAMMA(Z)

! Print the results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) Z, VALUE

99999 FORMAT (' GAMMA(', F6.3, ',', F6.3, ') = (', &

F6.3, ',', F6.3, ')')

END

Output

GAMMA( 1.400, 3.000) = (-0.001, 0.061)