naginterfaces.library.specfun.gamma

naginterfaces.library.specfun.gamma(x)

gamma returns the value of the gamma function $\Gamma \left(x\right)$.

For full information please refer to the NAG Library document for s14aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s14aaf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

gxfloat

The value of the gamma function $\Gamma \left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\le ⟨value⟩$.

The argument is too large, the function returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.

(errno $2$)

On entry, $x=⟨value⟩$. The function returns zero.

Constraint: $x\ge ⟨value⟩$.

The argument is too large and negative, the function returns zero.

Warns

NagAlgorithmicWarning

(errno $3$)

On entry, $x=⟨value⟩$.

Constraint: $\left|x\right|\ge ⟨value⟩$.

The argument is too close to zero, the function returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.

(errno $4$)

On entry, $x=⟨value⟩$.

Constraint: $x$ must not be a negative integer.

The argument is a negative integer, at which values $\Gamma \left(x\right)$ is infinite. The function returns a large positive value.

Notes

gamma evaluates an approximation to the gamma function $\Gamma \left(x\right)$. The function is based on the Chebyshev expansion:

$$\Gamma (1+u)=\sum _{r=0}^{\prime }{a}_r{T}_r\left(t\right)$$

where $0\le u<1,t=2u-1\text{,}$ and uses the property $\Gamma (1+x)=x\Gamma \left(x\right)$. If $x=N+1+u$ where $N$ is integral and $0\le u<1$ then it follows that:

for $N>0$,	$\Gamma \left(x\right)=(x-1)(x-2)\cdots (x-N)\Gamma (1+u)$,
for $N=0$,	$\Gamma \left(x\right)=\Gamma (1+u)$,
for $N<0$,	$\Gamma \left(x\right)=\frac{\Gamma (1+u)}{x(x+1)(x+2)\cdots (x-N-1)}$.

There are four possible failures for this function:

1. if $x$ is too large, there is a danger of overflow since $\Gamma \left(x\right)$ could become too large to be represented in the machine;

2. if $x$ is too large and negative, there is a danger of underflow;

3. if $x$ is equal to a negative integer, $\Gamma \left(x\right)$ would overflow since it has poles at such points;

4. if $x$ is too near zero, there is again the danger of overflow on some machines. For small $x$, $\Gamma \left(x\right)\simeq 1/x$, and on some machines there exists a range of nonzero but small values of $x$ for which $1/x$ is larger than the greatest representable value.

References

NIST Digital Library of Mathematical Functions