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### 6.11.4 Mathematical functions

`Asymptote`

has built-in versions of the standard `libm`

mathematical
real(real) functions `sin`

, `cos`

, `tan`

, `asin`

,
`acos`

, `atan`

, `exp`

, `log`

, `pow10`

,
`log10`

, `sinh`

, `cosh`

, `tanh`

, `asinh`

,
`acosh`

, `atanh`

, `sqrt`

, `cbrt`

, `fabs`

, `expm1`

,
`log1p`

, as well as the identity function `identity`

.
`Asymptote`

also defines the order `n`

Bessel functions of
the first kind `Jn(int n, real)`

and second kind
`Yn(int n, real)`

, as well as the gamma function `gamma`

,
the error function `erf`

, and the complementary error function
`erfc`

. The standard real(real, real) functions `atan2`

,
`hypot`

, `fmod`

, `remainder`

are also included.

The functions `degrees(real radians)`

and `radians(real degrees)`

can be used to convert between radians and degrees. The function
`Degrees(real radians)`

returns the angle in degrees in the
interval [0,360).
For convenience, `Asymptote`

defines variants `Sin`

,
`Cos`

, `Tan`

, `aSin`

, `aCos`

, and `aTan`

of
the standard trigonometric functions that use degrees rather than radians.
We also define complex versions of the `sqrt`

, `sin`

, `cos`

,
`exp`

, `log`

, and `gamma`

functions.

The functions `floor`

, `ceil`

, and `round`

differ from
their usual definitions in that they all return an int value rather than
a real (since that is normally what one wants).
The functions `Floor`

, `Ceil`

, and `Round`

are
respectively similar, except that if the result cannot be converted
to a valid int, they return `intMax`

for positive arguments and `intMin`

for negative arguments,
rather than generating an integer overflow.
We also define a function `sgn`

, which returns the sign of its
real argument as an integer (-1, 0, or 1).

There is an `abs(int)`

function, as well as an `abs(real)`

function (equivalent to `fabs(real)`

), an `abs(pair)`

function
(equivalent to `length(pair)`

).

Random numbers can be seeded with `srand(int)`

and generated with
the `int rand()`

function, which returns a random integer between 0
and the integer `randMax`

. The `unitrand()`

function returns
a random number uniformly distributed in the interval [0,1].
A Gaussian random number generator
`Gaussrand`

and a collection of statistics routines, including
`histogram`

, are provided in the base file `stats.asy`

.
The functions `factorial(int n)`

, which returns *n!*, and
`choose(int n, int k)`

, which returns *n!/(k!(n-k)!)*, are
also defined.

When configured with the GNU Scientific Library (GSL), available from
http://www.gnu.org/software/gsl/,
`Asymptote`

contains an internal module `gsl`

that
defines the airy functions `Ai(real)`

,
`Bi(real)`

, `Ai_deriv(real)`

, `Bi_deriv(real)`

,
`zero_Ai(int)`

, `zero_Bi(int)`

,
`zero_Ai_deriv(int)`

, `zero_Bi_deriv(int)`

, the Bessel functions
`I(int, real)`

, `K(int, real)`

, `j(int, real)`

,
`y(int, real)`

, `i_scaled(int, real)`

, `k_scaled(int, real)`

,
`J(real, real)`

, `Y(real, real)`

, `I(real, real)`

,
`K(real, real)`

, `zero_J(real, int)`

, the elliptic functions
`F(real, real)`

, `E(real, real)`

, and `P(real, real)`

,
the Jacobi elliptic functions `real[] sncndn(real,real)`

,
the exponential/trigonometric integrals `Ei`

, `Si`

, and `Ci`

,
the Legendre polynomials `Pl(int, real)`

, and the Riemann zeta
function `zeta(real)`

. For example, to compute the sine integral
`Si`

of 1.0:

import gsl; write(Si(1.0));

`Asymptote`

also provides a few general purpose numerical routines:

`real newton(int iterations=100, real f(real), real fprime(real), real x, bool verbose=false);`

Use Newton-Raphson iteration to solve for a root of a real-valued differentiable function

`f`

, given its derivative`fprime`

and an initial guess`x`

. Diagnostics for each iteration are printed if`verbose=true`

. If the iteration fails after the maximum allowed number of loops (`iterations`

),`realMax`

is returned.`real newton(int iterations=100, real f(real), real fprime(real), real x1, real x2, bool verbose=false);`

Use bracketed Newton-Raphson bisection to solve for a root of a real-valued differentiable function

`f`

within an interval [`x1`

,`x2`

] (on which the endpoint values of`f`

have opposite signs), given its derivative`fprime`

. Diagnostics for each iteration are printed if`verbose=true`

. If the iteration fails after the maximum allowed number of loops (`iterations`

),`realMax`

is returned.`real simpson(real f(real), real a, real b, real acc=realEpsilon, real dxmax=b-a)`

returns the integral of

`f`

from`a`

to`b`

using adaptive Simpson integration.

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