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# Mathematical Functions

This chapter describes basic mathematical functions. Some of these functions are present in system libraries, but the alternative versions given here can be used as a substitute when the system functions are not available.

The functions and macros described in this chapter are defined in the header file ``gsl_math.h'`.

## Mathematical Constants

The library ensures that the standard BSD mathematical constants are defined. For reference here is a list of the constants.

`M_E`
  The base of exponentials, e
`M_LOG2E`
  The base-2 logarithm of e, \log_2 (e)
`M_LOG10E`
  The base-10 logarithm of e, \log_10 (e)
`M_SQRT2`
  The square root of two, \sqrt 2
`M_SQRT1_2`
  The square root of one-half, \sqrt{1/2}
`M_SQRT3`
  The square root of three, \sqrt 3
`M_PI`
  The constant pi, \pi
`M_PI_2`
  Pi divided by two, \pi/2
`M_PI_4`
  Pi divided by four, \pi/4
`M_SQRTPI`
  The square root of pi, \sqrt\pi
`M_2_SQRTPI`
  Two divided by the square root of pi, 2/\sqrt\pi
`M_1_PI`
  The reciprocal of pi, 1/\pi
`M_2_PI`
  Twice the reciprocal of pi, 2/\pi
`M_LN10`
  The natural logarithm of ten, \ln(10)
`M_LN2`
  The natural logarithm of two, \ln(2)
`M_LNPI`
  The natural logarithm of pi, \ln(\pi)
`M_EULER`
  Euler's constant, \gamma

## Infinities and Not-a-number

Macro: GSL_POSINF
  This macro contains the IEEE representation of positive infinity, +\infty. It is computed from the expression `+1.0/0.0`.

Macro: GSL_NEGINF
  This macro contains the IEEE representation of negative infinity, -\infty. It is computed from the expression `-1.0/0.0`.

Macro: GSL_NAN
  This macro contains the IEEE representation of the Not-a-Number symbol, `NaN`. It is computed from the ratio `0.0/0.0`.

Function: int gsl_isnan (const double x)
  This function returns 1 if x is not-a-number.

Function: int gsl_isinf (const double x)
  This function returns +1 if x is positive infinity, -1 if x is negative infinity and 0 otherwise.

Function: int gsl_finite (const double x)
  This function returns 1 if x is a real number, and 0 if it is infinite or not-a-number.

## Elementary Functions

The following routines provide portable implementations of functions found in the BSD math library. When native versions are not available the functions described here can be used instead. The substitution can be made automatically if you use `autoconf` to compile your application (see section Portability functions).

Function: double gsl_log1p (const double x)
  This function computes the value of \log(1+x) in a way that is accurate for small x. It provides an alternative to the BSD math function `log1p(x)`.

Function: double gsl_expm1 (const double x)
  This function computes the value of \exp(x)-1 in a way that is accurate for small x. It provides an alternative to the BSD math function `expm1(x)`.

Function: double gsl_hypot (const double x, const double y)
  This function computes the value of \sqrt{x^2 + y^2} in a way that avoids overflow. It provides an alternative to the BSD math function `hypot(x,y)`.

Function: double gsl_acosh (const double x)
  This function computes the value of \arccosh(x). It provides an alternative to the standard math function `acosh(x)`.

Function: double gsl_asinh (const double x)
  This function computes the value of \arcsinh(x). It provides an alternative to the standard math function `asinh(x)`.

Function: double gsl_atanh (const double x)
  This function computes the value of \arctanh(x). It provides an alternative to the standard math function `atanh(x)`.

## Small integer powers

A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides a simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.

Function: double gsl_pow_int (double x, int n)
  This routine computes the power x^n for integer n. The power is computed using the minimum number of multiplications. For example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. A version of this function which also computes the numerical error in the result is available as `gsl_sf_pow_int_e`.

Function: double gsl_pow_2 (const double x)
Function: double gsl_pow_3 (const double x)
Function: double gsl_pow_4 (const double x)
Function: double gsl_pow_5 (const double x)
Function: double gsl_pow_6 (const double x)
Function: double gsl_pow_7 (const double x)
Function: double gsl_pow_8 (const double x)
Function: double gsl_pow_9 (const double x)
  These functions can be used to compute small integer powers x^2, x^3, etc. efficiently. The functions will be inlined when possible so that use of these functions should be as efficient as explicitly writing the corresponding product expression.

    #include <gsl/gsl_math.h>
    double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */

## Testing the Sign of Numbers

Macro: GSL_SIGN (x)
  This macro returns the sign of x. It is defined as `((x) >= 0 ? 1 : -1)`. Note that with this definition the sign of zero is positive (regardless of its IEEE sign bit).

## Testing for Odd and Even Numbers

Macro: GSL_IS_ODD (n)
  This macro evaluates to 1 if n is odd and 0 if n is even. The argument n must be of integer type.

Macro: GSL_IS_EVEN (n)
  This macro is the opposite of `GSL_IS_ODD(n)`. It evaluates to 1 if n is even and 0 if n is odd. The argument n must be of integer type.

## Maximum and Minimum functions

Macro: GSL_MAX (a, b)
  This macro returns the maximum of a and b. It is defined as `((a) > (b) ? (a):(b))`.

Macro: GSL_MIN (a, b)
  This macro returns the minimum of a and b. It is defined as `((a) < (b) ? (a):(b))`.

Function: extern inline double GSL_MAX_DBL (double a, double b)
  This function returns the maximum of the double precision numbers a and b using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro `GSL_MAX` will be automatically substituted.

Function: extern inline double GSL_MIN_DBL (double a, double b)
  This function returns the minimum of the double precision numbers a and b using an inline function. The use of a function allows for type checking of the arguments as an extra safety feature. On platforms where inline functions are not available the macro `GSL_MIN` will be automatically substituted.

Function: extern inline int GSL_MAX_INT (int a, int b)
Function: extern inline int GSL_MIN_INT (int a, int b)
  These functions return the maximum or minimum of the integers a and b using an inline function. On platforms where inline functions are not available the macros `GSL_MAX` or `GSL_MIN` will be automatically substituted.

Function: extern inline long double GSL_MAX_LDBL (long double a, long double b)
Function: extern inline long double GSL_MIN_LDBL (long double a, long double b)
  These functions return the maximum or minimum of the long doubles a and b using an inline function. On platforms where inline functions are not available the macros `GSL_MAX` or `GSL_MIN` will be automatically substituted.

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