complex

defmodule Complex do
  @moduledoc """
  *Complex* is a library for types and mathematical functions for complex
  numbers.

  Each complex number is represented as a structure holding the real and
  imaginary part.  There are functions for creation and manipulation of
  them.  Unfortunately since there is n operator overloading in Elixir the
  math functions (add, subtract, etc.) are implemented as add/2, sub/2, etc.

  ## Examples
      iex> Complex.new(3, 4)
      %Complex{im: 4, re: 3}

      iex> Complex.imag()
      %Complex{im: 1.0, re: 0.0}
  """

  @vsn 1

  import Kernel, except: [abs: 1, div: 2]

  @typedoc """
  General type for complex numbers
  """
  @type complex :: %Complex{re: number, im: number}
  defstruct re: 0, im: 0


  @doc """
    Returns a new complex with specified real and imaginary components.  The
    imaginary part defaults to zero so a "real" number can be created with new/1

    #### See also
    [imag/0](#imag/0) [fromPolar/2](#fromPolar/2)

    #### Examples
        iex> Complex.new(3, 4)
        %Complex{im: 4, re: 3}

        iex> Complex.new(2)
        %Complex{im: 0, re: 2}
    """
  @spec new(number, number) :: complex
  def new(re, im \\ 0), do: %Complex{re: re, im: im}

  @doc """
    Returns a new complex representing the pure imaginary number sqrt(-1).

    #### See also
    [new/2](#new/2) [fromPolar/2](#fromPolar/2)

    #### Examples
        iex> Complex.imag()
        %Complex{im: 1.0, re: 0.0}
    """
  @spec imag() :: complex
  def imag(), do: %Complex{re: 0.0, im: 1.0}


  @doc """
    Returns a new complex number described by the supplied polar
    coordinates.  That is, the complex (real and imaginary) will have radius
    (magnitude) r and angle (phase) phi.

    #### See also
    [new/2](#new/2) [imag/0](#imag/0)

    #### Examples
        iex> Complex.fromPolar(1, :math.pi/2)
        %Complex{im: 1.0, re: 6.123233995736766e-17}
    """
  @spec fromPolar(number, number) :: complex
  def fromPolar(r, phi) do
    new(r*:math.cos(phi), r*:math.sin(phi))
  end


  @doc """
    Returns the phase angle of the supplied complex.

    #### See also
    [new/2](#new/2) [fromPolar/2](#fromPolar/2)

    #### Examples
        iex> Complex.phase( Complex.fromPolar(1,:math.pi/2) )
        1.5707963267948966
    """
  @spec phase(complex) :: float
  def phase(z = %Complex{}) do
    :math.atan2(z.im, z.re)
  end


  @doc """
    Returns the polar coordinates of the supplied complex.  That is, the
    returned tuple {r,phi} is the magnitude and phase of z.

    #### See also
    [fromPolar/2](#fromPolar/2)

    #### Examples
        iex> Complex.getPolar( Complex.fromPolar(1,:math.pi/2) )
        {1.0, 1.5707963267948966}
    """
  @spec getPolar(complex) :: {float,float}
  def getPolar(z = %Complex{}) do
    {abs(z), phase(z)}
  end


  @doc """
    Returns a new complex that is the sum of the provided complex numbers.

    #### See also
    [div/2](#div/2), [mult/2](#mult/2), [sub/2](#sub/2)

    #### Examples
        iex> Complex.add( Complex.fromPolar(1, :math.pi/2), Complex.fromPolar(1, :math.pi/2) )
        %Complex{im: 2.0, re: 1.2246467991473532e-16}
    """
  @spec add(complex, complex) :: complex
  def add(%Complex{re: r1, im: i1}, %Complex{re: r2, im: i2}) do
    new(r1+r2, i1+i2)
  end


  @doc """
    Returns a new complex that is the difference of the provided complex numbers.

    #### See also
    [add/2](#add/2), [div/2](#div/2), [mult/2](#mult/2)

    #### Examples
        iex> Complex.sub( Complex.fromPolar(1, :math.pi/2), Complex.fromPolar(1, :math.pi/2) )
        %Complex{im: 0.0, re: 0.0}
    """
  @spec sub(complex, complex) :: complex
  def sub(%Complex{re: r1, im: i1}, %Complex{re: r2, im: i2}) do
    new(r1-r2, i1-i2)
  end


  @doc """
    Returns a new complex that is the product of the provided complex numbers.

    #### See also
    [add/2](#add/2), [div/2](#div/2), [sub/2](#sub/2)

    #### Examples
        iex> Complex.mult( Complex.fromPolar(1, :math.pi/2), Complex.fromPolar(1, :math.pi/2) )
        %Complex{im: 1.2246467991473532e-16, re: -1.0}

        iex> Complex.mult( Complex.imag(), Complex.imag() )
        %Complex{im: 0.0, re: -1.0}
    """
  @spec mult(complex, complex) :: complex
  def mult(%Complex{re: r1, im: i1}, %Complex{re: r2, im: i2}) do
    new(r1*r2 - i1*i2, i1*r2 + r1*i2)
  end


  @doc """
    Returns a new complex that is the ratio (division) of the provided complex
    numbers.

    #### See also
    [add/2](#add/2), [mult/2](#mult/2), [sub/2](#sub/2)

    #### Examples
        iex> Complex.div( Complex.fromPolar(1, :math.pi/2), Complex.fromPolar(1, :math.pi/2) )
        %Complex{im: 0.0, re: 1.0}
    """
  @spec div(complex, complex) :: complex
  def div(%Complex{re: r1, im: i1}, %Complex{re: r2, im: i2}) do
    if Kernel.abs(r2) < Kernel.abs(i2) do
      r = r2 / i2
      den = i2 + r*r2
      new((r1*r + i1)/den, (i1*r - r1)/den)
    else
      r = i2 / r2
      den = r2 + r*i2
      new((r1 + r*i1)/den, (i1 - r*r1)/den)
    end
  end


  @doc """
    Returns a new complex that is the magnitude (length)) of the provided complex
    number.

    #### See also
    [new/2](#new/2), [phase/1](#phase/1)

    #### Examples
        iex> Complex.abs( Complex.fromPolar(1, :math.pi/2) )
        1.0
    """
  @spec abs(complex) :: number
  def abs(%Complex{re: r, im: i}) do
    # optimized by checking special cases (sqrt is expensive)
    x = Kernel.abs(r)
    y = Kernel.abs(i)
    cond do
      x == 0.0 -> y
      y == 0.0 -> x
      x > y    -> x * :math.sqrt(1.0 + (y/x)*(y/x))
      true     -> y * :math.sqrt(1.0 + (x/y)*(x/y))
    end
  end


  @doc """
    Returns a new complex that is the complex conjugate of the provided complex
    number.

    #### See also
    [abs/2](#abs/2), [phase/1](#phase/1)

    #### Examples
        iex> Complex.conjugate( Complex.new(1,2) )
        %Complex{im: -2, re: 1}
    """
  @spec conjugate(complex) :: complex
  def conjugate(%Complex{re: r, im: i}) do
    new(r,-i)
  end


  @doc """
    Returns a new complex that is the complex square root of the provided
    complex number.

    #### See also
    [abs/2](#abs/2), [phase/1](#phase/1)

    #### Examples
        iex> Complex.sqrt( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.4142135623730951, re: 8.659560562354933e-17}
    """
  @spec sqrt(complex) :: complex
  def sqrt(z = %Complex{re: r, im: i}) do
    if z.re == 0.0 and z.im == 0.0 do
      new(z.re,z.im)
    else
      x = Kernel.abs(r)
      y = Kernel.abs(i)
      w = if x >= y do
            :math.sqrt(x) * :math.sqrt(0.5 * (1.0 + :math.sqrt(1.0 + (y/x)*(y/x))))
          else
            :math.sqrt(y) * :math.sqrt(0.5 * ((x/y) + :math.sqrt(1.0 + (x/y)*(x/y))))
          end
      if z.re >= 0.0 do
        new(w, z.im/(2*w))
      else
        i2 = if z.im >= 0.0 do
               w
             else
               -w
             end
        new(z.im/(2*i2), i2)
      end
    end
  end


  @doc """
    Returns a new complex that is the complex exponential of the provided
    complex number.  That is, e raised to the power z.

    #### See also
    [ln/1](#ln/1)

    #### Examples
        iex> Complex.exp( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 3.3147584285483636e-17, re: 0.1353352832366127}
    """
  @spec exp(complex) :: complex
  def exp(z = %Complex{}) do
    rho   = :math.exp(z.re)
    theta = z.im
    new(rho*:math.cos(theta), rho*:math.sin(theta))
  end


  @doc """
    Returns a new complex that is the complex natural log of the provided
    complex number.  That is, log base e of z.

    #### See also
    [exp/1](#exp/1)

    #### Examples
        iex> Complex.ln( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 3.141592653589793, re: 0.6931471805599453}
    """
  @spec ln(complex) :: complex
  def ln(z = %Complex{}) do
    new(:math.log(abs(z)), :math.atan2(z.im,z.re))
  end


  @doc """
    Returns a new complex that is the complex log base 10 of the provided
    complex number.

    #### See also
    [ln/1](#ln/1)

    #### Examples
        iex> Complex.log10( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.3643763538418412, re: 0.30102999566398114}
    """
  @spec log10(complex) :: complex
  def log10(z = %Complex{}) do
    div( ln(z), new(:math.log(10.0),0.0) )
  end


  @doc """
    Returns a new complex that is the complex log base 2 of the provided
    complex number.

    #### See also
    [ln/1](#ln/1), [log10/1](#log10/1)

    #### Examples
        iex> Complex.log2( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 4.532360141827194, re: 1.0}
    """
  @spec log2(complex) :: complex
  def log2(z = %Complex{}) do
    div( ln(z), new(:math.log(2.0),0.0) )
  end


  @doc """
    Returns a new complex that is the provided parameter a raised to the
    complex power b.

    #### See also
    [ln/1](#ln/1), [log10/1](#log10/1)

    #### Examples
        iex> Complex.pow( Complex.fromPolar(2,:math.pi), Complex.imag() )
        %Complex{im: 0.027612020368333014, re: 0.03324182700885666}
    """
  @spec pow(complex, complex) :: complex
  def pow(x = %Complex{}, y = %Complex{}) do
    cond do
    x.re == 0.0  and  x.im == 0.0 ->
      if y.re == 0.0  and  y.im == 0.0 do
        new(1.0,0.0)
      else
        new(0.0,0.0)
      end
    y.re == 1.0  and  y.im == 0.0 ->
      x
    y.re == -1.0  and  y.im == 0.0 ->
      div(new(1.0,0.0),x)
    true ->
      rho   = :math.sqrt(x.re*x.re + x.im*x.im)
      theta = :math.atan2(x.im,x.re)
      s     = :math.pow(rho,y.re) * :math.exp(-y.im*theta)
      r     = y.re*theta + y.im*:math.log(rho)
      new(s*:math.cos(r), s*:math.sin(r))
    end
  end


  @doc """
    Returns a new complex that is the sine of the provided parameter.

    #### See also
    [cos/1](#cos/1), [tan/1](#tan/1)

    #### Examples
        iex> Complex.sin( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -1.0192657827055095e-16, re: -0.9092974268256817}
    """
  @spec sin(complex) :: complex
  def sin(z = %Complex{}) do
    new(:math.sin(z.re)*:math.cosh(z.im),
        :math.cos(z.re)*:math.sinh(z.im))
  end


  @doc """
    Returns a new complex that is the "negation" of the provided parameter.
    That is, the real and imaginary parts are negated.

    #### See also
    [neq/2](#new/2), [imag/0](#imag/0)

    #### Examples
        iex> Complex.neg( Complex.new(3,5) )
        %Complex{im: -5, re: -3}
    """
  @spec neg(complex) :: complex
  def neg(z = %Complex{}) do
    new(-z.re, -z.im)
  end


  @doc """
    Returns a new complex that is the inverse sine (i.e., arcsine) of the
    provided parameter.

    #### See also
    [sin/1](#sin/1)

    #### Examples
        iex> Complex.asin( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.3169578969248164, re: -1.5707963267948963}

        iex> Complex.sin( Complex.asin(Complex.new(2,3)) )
        %Complex{im: 3.000000000000001, re: 1.9999999999999991}
    """
  @spec asin(complex) :: complex
  def asin(z = %Complex{}) do
    i = new(0.0,1.0)
    # result = -i*ln(i*z + sqrt(1.0-z*z))
    # result = -i*ln(t1 + sqrt(t2))
    t1 = mult(i,z)
    t2 = sub( new(1.0,0.0), mult(z,z) )
    mult( neg(i), ln( add(t1, sqrt(t2)) ) )
  end


  @doc """
    Returns a new complex that is the cosine of the provided parameter.

    #### See also
    [sin/1](#sin/1), [tan/1](#tan/1)

    #### Examples
        iex> Complex.cos( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 2.2271363664699914e-16, re: -0.4161468365471424}
    """
  @spec cos(complex) :: complex
  def cos(z = %Complex{}) do
    new(:math.cos(z.re)*:math.cosh(z.im),
        -:math.sin(z.re)*:math.sinh(z.im))
  end


  @doc """
    Returns a new complex that is the inverse cosine (i.e., arccosine) of the
    provided parameter.

    #### See also
    [cos/1](#cos/1)

    #### Examples
        iex> Complex.acos( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.3169578969248164, re: -3.141592653589793}

        iex> Complex.cos( Complex.acos(Complex.new(2,3)) )
        %Complex{im: 3.0, re: 2.0000000000000004}
    """
  @spec acos(complex) :: complex
  def acos(z = %Complex{}) do
    i = new(0.0,1.0)
    one = new(1.0,0.0)
    # result = -i*ln(z + sqrt(z*z-1.0))
    # result = -i*ln(z + sqrt(t1))
    t1 = sub( mult(z,z), one )
    mult( neg(i), ln( add(z, sqrt(t1)) ) )
  end


  @doc """
    Returns a new complex that is the tangent of the provided parameter.

    #### See also
    [sin/1](#sin/1), [cos/1](#cos/1)

    #### Examples
        iex> Complex.tan( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.4143199004457917e-15, re: 2.185039863261519}
    """
  @spec tan(complex) :: complex
  def tan(z = %Complex{}) do
    div(sin(z), cos(z))
  end


  @doc """
    Returns a new complex that is the inverse tangent (i.e., arctangent) of the
    provided parameter.

    #### See also
    [tan/1](#tan/1)

    #### Examples
        iex> Complex.atan( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 0.0, re: -1.1071487177940904}

        iex> Complex.tan( Complex.atan(Complex.new(2,3)) )
        %Complex{im: 3.0, re: 2.0}
    """
  @spec atan(complex) :: complex
  def atan(z = %Complex{}) do
    i = new(0.0,1.0)
    # result = 0.5*i*(ln(1-i*z)-ln(1+i*z))
    t1 = mult(new(0.5,0.0),i)
    t2 = sub( new(1.0,0.0), mult(i,z) )
    t3 = add( new(1.0,0.0), mult(i,z) )
    mult(t1, sub(ln(t2),ln(t3)))
  end


  @doc """
    Returns a new complex that is the cotangent of the provided parameter.

    #### See also
    [sin/1](#sin/1), [cos/1](#cos/1), [tan/1](#tan/1)

    #### Examples
        iex> Complex.cot( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -2.9622992129532336e-16, re: 0.45765755436028577}
    """
  @spec cot(complex) :: complex
  def cot(z = %Complex{}) do
    div(cos(z), sin(z))
  end


  @doc """
    Returns a new complex that is the inverse cotangent (i.e., arccotangent) of
    the provided parameter.

    #### See also
    [cot/1](#cot/1)

    #### Examples
        iex> Complex.acot( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -9.71445146547012e-17, re: -0.46364760900080615}

        iex> Complex.cot( Complex.acot(Complex.new(2,3)) )
        %Complex{im: 2.9999999999999996, re: 1.9999999999999991}
    """
  @spec acot(complex) :: complex
  def acot(z = %Complex{}) do
    i = new(0.0,1.0)
    # result = 0.5*i*(ln(1-i/z)-ln(1+i/z))
    t1 = mult(new(0.5,0.0),i)
    t2 = sub( new(1.0,0.0), div(i,z) )
    t3 = add( new(1.0,0.0), div(i,z) )
    mult(t1, sub(ln(t2),ln(t3)))
  end


  @doc """
    Returns a new complex that is the secant of the provided parameter.

    #### See also
    [sin/1](#sin/1), [cos/1](#cos/1), [tan/1](#tan/1)

    #### Examples
        iex> Complex.sec( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -1.2860374461837126e-15, re: -2.402997961722381}
    """
  @spec sec(complex) :: complex
  def sec(z = %Complex{}) do
    div( new(1.0,0.0), cos(z) )
  end


  @doc """
    Returns a new complex that is the inverse secant (i.e., arcsecant) of
    the provided parameter.

    #### See also
    [sec/1](#sec/1)

    #### Examples
        iex> Complex.asec( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 0.0, re: 2.0943951023931957}

        iex> Complex.sec( Complex.asec(Complex.new(2,3)) )
        %Complex{im: 2.9999999999999987, re: 1.9999999999999984}
    """
  @spec asec(complex) :: complex
  def asec(z = %Complex{}) do
    i = new(0.0,1.0)
    one = new(1.0,0.0)
    # result = -i*ln(i*sqrt(1-1/(z*z))+1/z)
    # result = -i*ln(i*sqrt(1-t2)+t1)
    t1 = div(one,z)
    t2 = div(one, mult(z,z))
    # result = -i*ln(i*sqrt(t3)+t1)
    # result = -i*ln(t4+t1)
    t3 = sub(one,t2)
    t4 = mult(i,sqrt(t3))
    mult(neg(i), ln(add(t4,t1)))
  end


  @doc """
    Returns a new complex that is the cosecant of the provided parameter.

    #### See also
    [sec/1](#sec/1), [sin/1](#sin/1), [cos/1](#cos/1), [tan/1](#tan/1)

    #### Examples
        iex> Complex.csc( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.2327514463765779e-16, re: -1.0997501702946164}
    """
  @spec csc(complex) :: complex
  def csc(z = %Complex{}) do
    one = new(1.0,0.0)
    div(one, sin(z))
  end


  @doc """
    Returns a new complex that is the inverse cosecant (i.e., arccosecant) of
    the provided parameter.

    #### See also
    [sec/1](#sec/1)

    #### Examples
        iex> Complex.acsc( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 0.0, re: -0.5235987755982988}

        iex> Complex.csc( Complex.acsc(Complex.new(2,3)) )
        %Complex{im: 2.9999999999999996, re: 1.9999999999999993}
    """
  @spec acsc(complex) :: complex
  def acsc(z = %Complex{}) do
    i = new(0.0,1.0)
    one = new(1.0,0.0)
    # result = -i*ln(sqrt(1-1/(z*z))+i/z)
    # result = -i*ln(sqrt(1-t2)+t1)
    t1 = div(i,z)
    t2 = div(one, mult(z,z))
    # result = -i*ln(sqrt(t3)+t1)
    # result = -i*ln(t4+t1)
    t3 = sub(one,t2)
    t4 = sqrt(t3)
    mult(neg(i), ln(add(t4,t1)))
  end


  @doc """
    Returns a new complex that is the hyperbolic sine of the provided parameter.

    #### See also
    [cosh/1](#cosh/1), [tanh/1](#tanh/1)

    #### Examples
        iex> Complex.sinh( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 9.214721821703068e-16, re: -3.626860407847019}
    """
  @spec sinh(complex) :: complex
  def sinh(z = %Complex{}) do
    p5 = new(0.5,0.0)
    mult(p5, sub(exp(z),exp(neg(z))))
  end


  @doc """
    Returns a new complex that is the inverse hyperbolic sine (i.e., arcsinh) of
    the provided parameter.

    #### See also
    [sinh/1](#sinh/1)

    #### Examples
        iex> Complex.asinh( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.0953573965284052e-16, re: -1.4436354751788099}

        iex> Complex.sinh( Complex.asinh(Complex.new(2,3)) )
        %Complex{im: 3.0, re: 2.000000000000001}
    """
  @spec asinh(complex) :: complex
  def asinh(z = %Complex{}) do
    one = new(1.0,0.0)
    # result = ln(z+sqrt(z*z+1))
    # result = ln(z+sqrt(t1))
    # result = ln(t2)
    t1 = add(mult(z,z),one)
    t2 = add(z,sqrt(t1))
    ln(t2)
  end


  @doc """
    Returns a new complex that is the hyperbolic cosine of the provided
    parameter.

    #### See also
    [sinh/1](#sinh/1), [tanh/1](#tanh/1)

    #### Examples
        iex> Complex.cosh( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -8.883245978848233e-16, re: 3.7621956910836314}
    """
  @spec cosh(complex) :: complex
  def cosh(z = %Complex{}) do
    p5 = new(0.5,0.0)
    mult(p5, add(exp(z),exp(neg(z))))
  end


  @doc """
    Returns a new complex that is the inverse hyperbolic cosine (i.e., arccosh)
    of the provided parameter.

    #### See also
    [cosh/1](#cosh/1)

    #### Examples
        iex> Complex.acosh( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -3.141592653589793, re: -1.3169578969248164}

        iex> Complex.cosh( Complex.acosh(Complex.new(2,3)) )
        %Complex{im: 3.0, re: 2.0}
    """
  @spec acosh(complex) :: complex
  def acosh(z = %Complex{}) do
    one = new(1.0,0.0)
    # result = ln(z+sqrt(z*z-1))
    # result = ln(z+sqrt(t1))
    # result = ln(t2)
    t1 = sub(mult(z,z),one)
    t2 = add(z,sqrt(t1))
    ln(t2)
  end


  @doc """
    Returns a new complex that is the hyperbolic tangent of the provided
    parameter.

    #### See also
    [sinh/1](#sinh/1), [cosh/1](#cosh/1)

    #### Examples
        iex> Complex.tanh( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.7304461302709572e-17, re: -0.964027580075817}
    """
  @spec tanh(complex) :: complex
  def tanh(z = %Complex{}) do
    div(sinh(z), cosh(z))
  end


  @doc """
    Returns a new complex that is the inverse hyperbolic tangent (i.e., arctanh)
    of the provided parameter.

    #### See also
    [tanh/1](#tanh/1)

    #### Examples
        iex> Complex.atanh( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 1.5707963267948966, re: -0.5493061443340549}

        iex> Complex.tanh( Complex.atanh(Complex.new(2,3)) )
        %Complex{im: 2.999999999999999, re: 1.9999999999999987}
    """
  @spec atanh(complex) :: complex
  def atanh(z = %Complex{}) do
    one = new(1.0,0.0)
    p5 = new(0.5,0.0)
    # result = 0.5*(ln((1+z)/(1-z)))
    # result = 0.5*(ln(t2/t1))
    # result = 0.5*(ln(t3))
    t1 = sub(one,z)
    t2 = add(one,z)
    t3 = div(t2,t1)
    mult(p5,ln(t3))
  end


  @doc """
    Returns a new complex that is the hyperbolic secant of the provided
    parameter.

    #### See also
    [sinh/1](#sinh/1), [cosh/1](#cosh/1), [tanh/1](#tanh/1)

    #### Examples
        iex> Complex.sech( Complex.fromPolar(2,:math.pi) )
        %Complex{im: 6.27608655779184e-17, re: 0.2658022288340797}
    """
  @spec sech(complex) :: complex
  def sech(z = %Complex{}) do
    two = new(2.0,0.0)
    div(two, add(exp(z),exp(neg(z))))
  end


  @doc """
    Returns a new complex that is the inverse hyperbolic secant (i.e., arcsech)
    of the provided parameter.

    #### See also
    [sech/1](#sech/1)

    #### Examples
        iex> Complex.asech( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -2.0943951023931953, re: 0.0}

        iex> Complex.sech( Complex.asech(Complex.new(2,3)) )
        %Complex{im: 2.999999999999999, re: 2.0}
    """
  @spec asech(complex) :: complex
  def asech(z = %Complex{}) do
    one = new(1.0,0.0)
    # result = ln(1/z+sqrt(1/z+1)*sqrt(1/z-1))
    # result = ln(t1+sqrt(t1+1)*sqrt(t1-1))
    # result = ln(t1+t2*t3)
    t1 = div(one,z)
    t2 = sqrt( add(t1,one) )
    t3 = sqrt( sub(t1,one) )
    ln( add(t1,mult(t2,t3)) )
  end


  @doc """
    Returns a new complex that is the hyperbolic cosecant of the provided
    parameter.

    #### See also
    [sinh/1](#sinh/1), [cosh/1](#cosh/1), [tanh/1](#tanh/1)

    #### Examples
        iex> Complex.csch( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -7.00520014334671e-17, re: -0.2757205647717832}
    """
  @spec csch(complex) :: complex
  def csch(z = %Complex{}) do
    two = new(2.0,0.0)
    div(two, sub(exp(z),exp(neg(z))))
  end


  @doc """
    Returns a new complex that is the inverse hyperbolic cosecant (i.e., arccsch)
    of the provided parameter.

    #### See also
    [csch/1](#csch/1)

    #### Examples
        iex> Complex.acsch( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -5.4767869826420256e-17, re: -0.48121182505960336}

        iex> Complex.csch( Complex.acsch(Complex.new(2,3)) )
        %Complex{im: 3.0000000000000018, re: 1.9999999999999982}
    """
  @spec acsch(complex) :: complex
  def acsch(z = %Complex{}) do
    one = new(1.0,0.0)
    # result = ln(1/z+sqrt(1/(z*z)+1))
    # result = ln(t1+sqrt(t2+1))
    # result = ln(t1+t3)
    t1 = div(one,z)
    t2 = div(one,mult(z,z))
    t3 = sqrt( add(t2,one) )
    ln( add(t1,t3) )
  end


  @doc """
    Returns a new complex that is the hyperbolic cotangent of the provided
    parameter.

    #### See also
    [sinh/1](#sinh/1), [cosh/1](#cosh/1), [tanh/1](#tanh/1)

    #### Examples
        iex> Complex.coth( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -1.8619978115303632e-17, re: -1.037314720727548}
    """
  @spec coth(complex) :: complex
  def coth(z = %Complex{}) do
    div(cosh(z), sinh(z))
  end


  @doc """
    Returns a new complex that is the inverse hyperbolic cotangent (i.e., arccoth)
    of the provided parameter.

    #### See also
    [coth/1](#coth/1)

    #### Examples
        iex> Complex.acoth( Complex.fromPolar(2,:math.pi) )
        %Complex{im: -8.164311994315688e-17, re: -0.5493061443340548}

        iex> Complex.coth( Complex.acoth(Complex.new(2,3)) )
        %Complex{im: 2.999999999999999, re: 2.0}
    """
  @spec acoth(complex) :: complex
  def acoth(z = %Complex{}) do
    one = new(1.0,0.0)
    p5 = new(0.5,0.0)
    # result = 0.5*(ln(1+1/z)-ln(1-1/z))
    # result = 0.5*(ln(1+t1)-ln(1-t1))
    # result = 0.5*(ln(t2)-ln(t3))
    t1 = div(one,z)
    t2 = add(one,t1)
    t3 = sub(one,t1)
    mult(p5, sub(ln(t2),ln(t3)))
  end

end