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# Mathematical Functions

Contents:

1. Mathematical Constants

   1. Infinities and Not-a-number

   2. Constants

2. Module functions

3. Elementary Functions

4. Small Integer Powers

5. Testing the Sign of Numbers

6. Testing for Odd and Even Numbers

7. Maximum and Minimum functions

8. Approximate Comparison of Floating Point Numbers

## Mathematical Constants

- GSL::M_E

  The base of exponentials, e

- GSL::M_LOG2E

  The base-2 logarithm of e, log_2(e)

- GSL::M_LOG10E

  The base-10 logarithm of e, log_10(e)

- GSL::M_SQRT2

  The square root of two, sqrt(2)

- GSL::M_SQRT1_2

  The square root of one-half, sqrt(1/2)

- GSL::M_SQRT3

  The square root of three, sqrt(3)

- GSL::M_PI

  The constant pi

- GSL::M_PI_2

  Pi divided by two

- GSL::M_PI_4

  Pi divided by four

- GSL::M_SQRTPI

  The square root of pi

- GSL::M_2_SQRTPI

  Two divided by the square root of pi

- GSL::M_1_PI

  The reciprocal of pi, 1/pi

- GSL::M_2_PI

  Twice the reciprocal of pi, 2/pi

- GSL::M_LN10

  The natural logarithm of ten, ln(10)

- GSL::M_LN2

  The natural logarithm of ten, ln(2)

- GSL::M_LNPI

  The natural logarithm of ten, ln(pi)

- GSL::M_EULER

  Euler's constant

## Infinities and Not-a-number

### Constants

- GSL::POSINF

  The IEEE representation of positive infinity, computed from the expression +1.0/0.0.

- GSL::NEGINF

  The IEEE representation of negative infinity, computed from the expression -1.0/0.0.

- GSL::NAN

  The IEEE representation of the Not-a-Number symbol, computed from the ratio 0.0/0.0.

### Module functions

- GSL::isnan(x)

  This returns 1 if `x` is not-a-number.

- GSL::isnan?(x)

  This returns `true` if `x` is not-a-number, and `false` otherwise.

- GSL::isinf(x)

  This returns +1 if `x` is positive infinity, -1 if `x` is negative infinity and 0 otherwise. NOTE: In Darwin9.5.0-gcc4.0.1, this method returns 1 for -inf.

- GSL::isinf?(x)

  This returns `true` if `x` is positive or negative infinity, and `false` otherwise.

- GSL::finite(x)

  This returns 1 if `x` is a real number, and 0 if it is infinite or not-a-number.

- GSL::finite?(x)

  This returns `true` if `x` is a real number, and `false` if it is infinite or not-a-number.

## Elementary Functions

- GSL::log1p(x)

  This method 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).

- GSL::expm1(x)

  This method 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).

- GSL::hypot(x, y)

  This method computes the value of sqrt{x^2 + y^2} in a way that avoids overflow.

- GSL::hypot3(x, y, z)

  Computes the value of sqrt{x^2 + y^2 + z^2} in a way that avoids overflow.

- GSL::acosh(x)

  This method computes the value of arccosh(x).

- GSL::asinh(x)

  This method computes the value of arcsinh(x).

- GSL::atanh(x)

  This method computes the value of arctanh(x).

  These methods above can take argument `x` of Integer, Float, Array, Vector or Matrix.

- GSL::ldexp(x)

  This method computes the value of x * 2^e.

- GSL::frexp(x)

  This method splits the number `x` into its normalized fraction f and exponent e, such that x = f * 2^e and 0.5 <= f < 1. The method returns f and the exponent e as an array, [f, e]. If `x` is zero, both f and e are set to zero.

## Small Integer Powers

- GSL::pow_int(x, n)

  This routine computes the power `x^n` for integer `n`. The power is computed efficiently -- for example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications.

- GSL::pow_2(x)

- GSL::pow_3(x)

- GSL::pow_4(x)

- GSL::pow_5(x)

- GSL::pow_6(x)

- GSL::pow_7(x)

- GSL::pow_8(x)

- GSL::pow_9(x)

  These methods can be used to compute small integer powers x^2, x^3, etc. efficiently.

## Testing the Sign of Numbers

- GSL::SIGN(x)

- GSL::sign(x)

  Return 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

- GSL::is_odd(n)

- GSL::IS_ODD(n)

  Evaluate to 1 if `n` is odd and 0 if `n` is even. The argument `n` must be of Fixnum type.

- GSL::is_odd?(n)

- GSL::IS_ODD?(n)

  Return `true` if `n` is odd and `false` if even.

- GSL::is_even(n)

- GSL::IS_EVEN(n)

  Evaluate to 1 if `n` is even and 0 if `n` is odd. The argument `n` must be of Fixnum type.

- GSL::is_even?(n)

- GSL::IS_even?(n)

  Return `true` if `n` is even and `false` if odd.

## Maximum and Minimum functions

- GSL::max(a, b)

- GSL::MAX(a, b)

- GSL::min(a, b)

- GSL::MIN(a, b)

## Approximate Comparison of Floating Point Numbers

- GSL::fcmp(a, b, epsilon = 1e-10)

  This method determines whether `x` and `y` are approximately equal to a relative accuracy `epsilon`.

- GSL::equal?(a, b, epsilon = 1e-10)

## Module Constants

- GSL::VERSION

  GSL version

- GSL::RB_GSL_VERSION

- GSL::RUBY_GSL_VERSION

  Ruby/GSL version

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