Advanced Mathematical Functions
===============================

Calc can compute a variety of less common functions that arise in
various branches of mathematics.  All of the functions described in
this section allow arbitrary complex arguments and, except as noted,
will work to arbitrarily large precisions.  They can not at present
handle error forms or intervals as arguments.

NOTE: These functions are still experimental.  In particular, their
accuracy is not guaranteed in all domains.  It is advisable to set the
current precision comfortably higher than you actually need when using
these functions.  Also, these functions may be impractically slow for
some values of the arguments.

The `f g' (`calc-gamma') [`gamma'] command computes the Euler gamma
function.  For positive integer arguments, this is related to the
factorial function: `gamma(n+1) = fact(n)'.  For general complex
arguments the gamma function can be defined by the following definite
integral: `gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)'.  (The actual
implementation uses far more efficient computational methods.)

The `f G' (`calc-inc-gamma') [`gammaP'] command computes the
incomplete gamma function, denoted `P(a,x)'.  This is defined by the
integral, `gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)'.
This implies that `gammaP(a,inf) = 1' for any `a' (see the definition
of the normal gamma function).

Several other varieties of incomplete gamma function are defined.  The
complement of `P(a,x)', called `Q(a,x) = 1-P(a,x)' by some authors, is
computed by the `I f G' [`gammaQ'] command.  You can think of this as
taking the other half of the integral, from `x' to infinity.

The functions corresponding to the integrals that define `P(a,x)' and
`Q(a,x)' but without the normalizing `1/gamma(a)' factor are called
`g(a,x)' and `G(a,x)', respectively (where `g' and `G' represent the
lower- and upper-case Greek letter gamma).  You can obtain these using
the `H f G' [`gammag'] and `H I f G' [`gammaG'] commands.

The `f b' (`calc-beta') [`beta'] command computes the Euler beta
function, which is defined in terms of the gamma function as
`beta(a,b) = gamma(a) gamma(b) / gamma(a+b)', or by `beta(a,b) =
integ(t^(a-1) (1-t)^(b-1), t, 0, 1)'.

The `f B' (`calc-inc-beta') [`betaI'] command computes the incomplete
beta function `I(x,a,b)'.  It is defined by `betaI(x,a,b) =
integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)'.  Once again, the `H'
(hyperbolic) prefix gives the corresponding un-normalized version
[`betaB'].

The `f e' (`calc-erf') [`erf'] command computes the error function
`erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)'.  The
complementary error function `I f e' (`calc-erfc') [`erfc'] is the
corresponding integral from `x' to infinity; the sum `erf(x) + erfc(x)
= 1'.

The `f j' (`calc-bessel-J') [`besJ'] and `f y' (`calc-bessel-Y')
[`besY'] commands compute the Bessel functions of the first and second
kinds, respectively.  In `besJ(n,x)' and `besY(n,x)' the "order"
parameter `n' is often an integer, but is not required to be one.
Calc's implementation of the Bessel functions currently limits the
precision to 8 digits, and may not be exact even to that precision.
Use with care!