R: Incomplete Gamma Function

gammainc {expint} | R Documentation

## Incomplete Gamma Function

### Description

The incomplete gamma function G(a, x).

### Usage

    gammainc(a, x)

### Arguments

`a` | vector of real numbers.
`x` | vector of non-negative real numbers.

### Details

As defined in 6.5.3 of Abramowitz and Stegun (1972), the incomplete gamma function is

G(a, x) = int_x^Inf t^(a - 1) exp(-t) dt

for a real and x ≥ 0.

For non-negative values of a, we have

G(a, x) = Gamma(a) (1 - P(a, x)),

where Gamma(a) is the function implemented by R's `gamma()` and P(a, x) is the cumulative distribution function of the gamma distribution (with scale equal to one) implemented by R's `pgamma()`.

Also, G(0, x) = E_1(x), x > 0, where E_1(x) is the exponential integral implemented in `expint`.

### Value

The value of the incomplete gamma function.

Invalid arguments will result in return value `NaN`, with a warning.

### Note

The C implementation is based on code from the GNU Software Library https://www.gnu.org/software/gsl/.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover.

### See Also

`expint`

### Examples

    ## a > 0
    x <- c(0.2, 2.5, 5, 8, 10)
    a <- 1.2
    gammainc(a, x)
    gamma(a) * pgamma(x, a, 1, lower = FALSE) # same

    ## a = 0
    a <- 0
    gammainc(a, x)
    expint(x)                                 # same

    ## a < 0
    a <- c(-0.25, -1.2, -2)
    sapply(a, gammainc, x = x)

[Package expint version 0.1-7 Index]