gamma
Gamma Function (and Its Derivatives and Logarithm)

### Description

Describes the special mathematics function gamma and its derivatives and logarithma.

### Usage

    gamma(x)
    lgamma(x)
    digamma(x)
    trigamma(x)
    psigamma(x, deriv = 0)

### Arguments

x | for gamma and lgamma, a numeric or complex object. for digamma, a numeric or complex value, or a vector. For trigamma, a numeric or complex value, or a vector. It can be real or complex, but it must have a positive real part. Missing values (NA) are allowed.

### Details

gamma and lgamma:

- For positive integral values, gamma(x) is (x-1)!.
- For negative integral values, gamma(x) is undefined.
- gamma(x) increases very rapidly with x. Use lgamma to avoid overflow.

For more details about the gamma function, see http://en.wikipedia.org/wiki/Gamma_function.
digamma, trigamma and psigamma:

- The digamma function is the derivative of the log gamma function. (Abramowitz and Stegun (1965, p. 258.) digamma(x) is equal to psigamma(x, 0).
- trigamma uses an asymptotic expansion where Re(x) > 5 and a recurrence formula to such a case where Re(x) <= 5. Accuracy is good. This function is based upon the function trigamma in Venables and Ripley (1997).
- The trigamma function is the second derivative of the log gamma function. trigamma(x) is equal to psigamma(x, 1)
- The psigamma function (also called polygamma function) is the higher derivative of the log gamma function. The level is defined by the fs"deriv" argument.

gamma, lgamma, digamma and trigamma are members of the Math group of generic functions.
NAs are returned when evaluation would cause numerical problems, except that when an overflow occurs, an infinity is returned.
Value

gamma | returns the gamma function evaluated for each value in x.
lgamma | returns natural log of the gamma function evaluated for each value in x.
digamma | returns the derivative of the log gamma function evaluated for each value in x.
trigamma | returns the second derivative of the log gamma function evaluated for each value in x.
psigamma | returns the (deriv+1)-th derivative of the log gamma function evaluated for each value in x.

Classes
gamma is used as the default method for classes that do not inherit a specific method for the function or for the Math group of functions. The result retains the class and the attributes. If this behavior is not appropriate, the designer of the class should provide a method for the function or for the Math group.
References
Abramowitz, M., and Stegun, I. A. (Eds.). (1965). Handbook of Mathematical Functions. New York: Dover.
Venables, W. N. and Ripley, B. D. (1997). Modern Applied Statistics with S-PLUS, Second Ed. New York: Springer.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (For gamma and lgamma.)
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.
http://en.wikipedia.org/wiki/Gamma_function.
http://en.wikipedia.org/wiki/Beta_function.
Note
See family for the family generating function Gamma used with the glm and gam functions. See GAMMA for the functions related to the gamma distribution: dgamma (density), pgamma (probability), qgamma (quantile), rgamma (sample).
See Also
Special, GAMMA, family, exp, sum, cumsum
Examples

    gamma(6)   # same as 5 factorial
    (gamma(5+1.0e-6)-gamma(5))/1.0e-6/gamma(5)  # Gives: 1.506119
    lgamma(20) # natural log of gamma(20)
    (lgamma(5+1.0e-6)-lgamma(5))/1.0e-6  # Gives: 1.506118
    digamma(6) # derivative
    digamma(5)  # Gives: 1.506118
    trigamma(6)# second derivative
    x <- seq(1, 2, .1)
    matrix(c(x, trigamma(x)), ncol=2)
    identical(digamma(x), psigamma(x, 0))  # Gives: TRUE
    identical(trigamma(x), psigamma(x, 1))  # Gives: TRUE

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