# Gamma, Bessel and Associated Functions

As a general reference to the functions described in this section (and much more), we refer the reader to Whittaker and Watson [[Whittaker and Watson, 1915](../../references.md#cite-whit15)].

## `Gamma(f): RngSerElt -> RngSerElt`

Return the Gamma function $\Gamma(f)$ of the series $f$. The series $f$ must be defined over the free real or complex field, the valuation of $f$ must be 0 and the constant term of $f$ must be 1.

## `Gamma(r): FldReElt -> FldReElt`

## `Gamma(r): FldComElt -> FldComElt`

Given a real or complex number $s$ (not equal to $0, -1, -2, \ldots$), calculate the value $\Gamma(s)$ of the *gamma function* at $s$. For $s$ with positive real part this is the value of

$$
\Gamma(s)=\int_0^{\infty}u^{s-1}e^{-u}du.
$$

For other $s$ (not a non-positive integer) the function is defined by analytic continuation, and it satisfies the product formula

$$
{1\over s\Gamma(s)}=e^{\gamma s}\prod_{n=1}^\infty (1+{s\over n})e^{-s/n}.
$$

The function $\Gamma$ also satisfies

$$
\Gamma(s)\Gamma(1-s)={\pi\over{\sin(\pi s)}},
$$

and

$$
\Gamma(s+1)=s\Gamma(s).
$$

## `Gamma(s, t): FldReElt, FldReElt -> FldReElt`

```magma
Complementary: BoolElt                     Default: false
Gamma        : FldReElt                    Default: 
```

For real numbers $s, t$ this returns the value of the incomplete gamma function

$$
\gamma(s, t)=\int_0^t u^{s-1}e^{-u}du.
$$

The optional argument `Complementary` can be used to find the complement

$$
\int_t^\infty  u^{s-1}e^{-u}du
$$

instead. There is a second optional argument that may be used in the computation of the incomplete gamma value; the free real value of `Gamma` should be the value of $\Gamma(s)$, in which case $\gamma(s, t)$ may be computed as the difference between the given value for $\Gamma(s)$ and that of the complementary $\gamma$ at $s, t$. Pari is used here.

## `GammaD(s): FldReElt -> FldReElt`

For free real $s$ (such that $s+{1\over2}$ is not a non-positive integer) this returns the value of $\Gamma(s+{1\over2})$. For integer values of $s$ this is faster than `Gamma(s+(1/2))`, because Legendre’s doubling formula

$$
\Gamma(s+{1\over2})=2^{1-2s}\sqrt{\pi}{\Gamma(2s)\over\Gamma(s)}
$$

is used. Pari is used here.

## `LogGamma(f): RngSerElt -> RngSerElt`

Return the Log-Gamma function ${\rm Log}(\Gamma(f))$ of the series $f$. $f$ must be defined over a real or complex field, the valuation of $f$ must be 0 and the constant term of $f$ must be 1.

## `LogGamma(r): FldReElt -> FldReElt`

## `LogGamma(r): FldComElt -> FldComElt`

For real or complex $s$ (not a non-positive integer) return the value of the principal branch of the logarithm of the gamma function of $s$.

## `LogDerivative(s): FldReElt -> FldReElt`

## `LogDerivative(s): FldComElt -> FldComElt`

## `Psi(s): FldReElt -> FldReElt`

## `Psi(s): FldComElt -> FldComElt`

For real or complex $s$ (not a non-positive integer) return the principal value of the logarithmic derivative

$$
\Psi(s)={d \log\Gamma(s)\over ds}={\Gamma'(s)\over\Gamma(s)},
$$

of the gamma function, which allows the expansion

$$
\Psi(s)=-\gamma-{1\over s}+s\sum_{n=1}^\infty{1\over n(s+n)};
$$

here $\gamma$ is Euler’s gamma. Pari is used here.

## `BesselFunction(n, r): RngIntElt, FldReElt -> FldReElt`

Given a small integer $n$ and a real number $r$, calculate the value of the *Bessel function* $y$ = $J_n(r)$, of the first kind of order $n$. Results for negative arguments are defined by: $J_{-n}(r) = J_n(-r) = (-1)^n J_n(r)$. The Bessel function of the first kind of order $n$ is defined by

$$
J_n(x)={1\over2\pi{\mathrm{i}}}\left({z\over2}\right)^n\int_{-\infty}^{0^+}
   u^{-n-1}e^{u-{z^2\over4t}}du,
$$

and satisfies

$$
J_n(x)=\sum_{k=0}^\infty {(-1)^kz^{n+2k}\over 2^{n+2k}k!\Gamma(n+k+1)}.
$$

## `BesselFunctionSecondKind(n, r): RngIntElt, FldReElt -> FldReElt`

Given a small integer $n$ and a real number $r$, calculate the value of the *Bessel function* $y$ = $Y_n(r)$, of the second kind of order $n$. Results for negative arguments are defined by: $Y_{-n}(r) = -(-1)^n Y_n(r)$, $Y_n(-r)$ is not a real number. The Bessel function of the second kind of order $n$ satisfies the Bessel differential equation.

## `JBessel(n, s): RngIntElt, FldReElt -> FldReElt`

## `JBessel(n, s): FldReElt, FldReElt -> FldReElt`

Given a small integer $n$ and a real number $s$, calculate the value of the *Bessel function* of the first kind of half integral index $n+{1\over2}$, $J_{n+{1\over2}}$, defined as above. Pari is used here.

## `KBessel(n, s): FldReElt, FldReElt -> FldReElt`

## `KBessel(n, s): FldComElt, FldReElt -> FldComElt`

## `KBessel2(n, s): FldReElt, FldReElt -> FldReElt`

## `KBessel2(n, s): FldComElt, FldReElt -> FldComElt`

Given a complex $n$ and a positive real $s$, compute the value of the *modified Bessel function of the second kind* $K_n(s)$, which may be defined by

$$
K_n(s)={\pi\over 2}\left({\mathrm{i}}^nJ_{-n}({\mathrm{i}}s)-{\mathrm{i}}^{-n}J_n(s)\right)\cot(n\pi).
$$

The function `KBessel2` is an alternative (often faster) implementation of this function. Pari is used here.