CMath

Module

Trigonometric and transcendental functions for complex numbers.

CMath is a library that provides trigonometric and transcendental functions for complex numbers. The functions in this module accept integers, floating-point numbers or complex numbers as arguments.

Note that the selection of functions is similar, but not identical, to that in module math. The reason for having two modules is that some users aren’t interested in complex numbers, and perhaps don’t even know what they are. They would rather have Math.sqrt(-1) raise an exception than return a complex number.

For more information you can see Complex class.

Usage

To start using this library, simply require cmath library:

require "cmath"

Constants

RealMath

Backup of Math is needed because mathn.rb replaces Math with CMath.

Class Methods

acos(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 281
def acos(z)
  begin
    if z.real? and z >= -1 and z <= 1
      RealMath.acos(z)
    else
      (-1.0).i * log(z + 1.0.i * sqrt(1.0 - z * z))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the arc cosine of z

CMath.acos(1 + 1i) #=> (0.9045568943023813-1.0612750619050357i)

acosh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 346
def acosh(z)
  begin
    if z.real? and z >= 1
      RealMath.acosh(z)
    else
      log(z + sqrt(z * z - 1.0))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the inverse hyperbolic cosine of z

CMath.acosh(1 + 1i) #=> (1.0612750619050357+0.9045568943023813i)

asin(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 265
def asin(z)
  begin
    if z.real? and z >= -1 and z <= 1
      RealMath.asin(z)
    else
      (-1.0).i * log(1.0.i * z + sqrt(1.0 - z * z))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the arc sine of z

CMath.asin(1 + 1i) #=> (0.6662394324925153+1.0612750619050355i)

asinh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 330
def asinh(z)
  begin
    if z.real?
      RealMath.asinh(z)
    else
      log(z + sqrt(1.0 + z * z))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the inverse hyperbolic sine of z

CMath.asinh(1 + 1i) #=> (1.0612750619050357+0.6662394324925153i)

atan(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 297
def atan(z)
  begin
    if z.real?
      RealMath.atan(z)
    else
      1.0.i * log((1.0.i + z) / (1.0.i - z)) / 2.0
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the arc tangent of z

CMath.atan(1 + 1i) #=> (1.0172219678978514+0.4023594781085251i)

atan2(y,x)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 314
def atan2(y,x)
  begin
    if y.real? and x.real?
      RealMath.atan2(y,x)
    else
      (-1.0).i * log((x + 1.0.i * y) / sqrt(x * x + y * y))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the arc tangent of y divided by x using the signs of y and x to determine the quadrant

CMath.atan2(1 + 1i, 0) #=> (1.5707963267948966+0.0i)

atanh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 362
def atanh(z)
  begin
    if z.real? and z >= -1 and z <= 1
      RealMath.atanh(z)
    else
      log((1.0 + z) / (1.0 - z)) / 2.0
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the inverse hyperbolic tangent of z

CMath.atanh(1 + 1i) #=> (0.4023594781085251+1.0172219678978514i)

cbrt(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 157
def cbrt(z)
  z ** (1.0/3)
end

Returns the principal value of the cube root of z

CMath.cbrt(1 + 4i) #=> (1.449461632813119+0.6858152562177092i)

cos(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 182
def cos(z)
  begin
    if z.real?
      RealMath.cos(z)
    else
      Complex(RealMath.cos(z.real) * RealMath.cosh(z.imag),
              -RealMath.sin(z.real) * RealMath.sinh(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the cosine of z, where z is given in radians

CMath.cos(1 + 1i) #=> (0.8337300251311491-0.9888977057628651i)

cosh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 232
def cosh(z)
  begin
    if z.real?
      RealMath.cosh(z)
    else
      Complex(RealMath.cosh(z.real) * RealMath.cos(z.imag),
              RealMath.sinh(z.real) * RealMath.sin(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the hyperbolic cosine of z, where z is given in radians

CMath.cosh(1 + 1i) #=> (0.8337300251311491+0.9888977057628651i)

exp(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 62
def exp(z)
  begin
    if z.real?
      RealMath.exp(z)
    else
      ere = RealMath.exp(z.real)
      Complex(ere * RealMath.cos(z.imag),
              ere * RealMath.sin(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Math::E raised to the z power

CMath.exp(1.i * Math::PI) #=> (-1.0+1.2246467991473532e-16i)

log(z, b=::Math::E)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 82
def log(z, b=::Math::E)
  begin
    if z.real? && z >= 0 && b >= 0
      RealMath.log(z, b)
    else
      Complex(RealMath.log(z.abs), z.arg) / log(b)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the natural logarithm of Complex. If a second argument is given, it will be the base of logarithm.

CMath.log(1 + 4i)     #=> (1.416606672028108+1.3258176636680326i)
CMath.log(1 + 4i, 10) #=> (0.6152244606891369+0.5757952953408879i)

log10(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 114
def log10(z)
  begin
    if z.real? and z >= 0
      RealMath.log10(z)
    else
      log(z) / RealMath.log(10)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the base 10 logarithm of z

CMath.log10(-1) #=> (0.0+1.3643763538418412i)

log2(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 98
def log2(z)
  begin
    if z.real? and z >= 0
      RealMath.log2(z)
    else
      log(z) / RealMath.log(2)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the base 2 logarithm of z

CMath.log2(-1) => (0.0+4.532360141827194i)

sin(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 165
def sin(z)
  begin
    if z.real?
      RealMath.sin(z)
    else
      Complex(RealMath.sin(z.real) * RealMath.cosh(z.imag),
              RealMath.cos(z.real) * RealMath.sinh(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the sine of z, where z is given in radians

CMath.sin(1 + 1i) #=> (1.2984575814159773+0.6349639147847361i)

sinh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 215
def sinh(z)
  begin
    if z.real?
      RealMath.sinh(z)
    else
      Complex(RealMath.sinh(z.real) * RealMath.cos(z.imag),
              RealMath.cosh(z.real) * RealMath.sin(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the hyperbolic sine of z, where z is given in radians

CMath.sinh(1 + 1i) #=> (0.6349639147847361+1.2984575814159773i)

sqrt(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 130
def sqrt(z)
  begin
    if z.real?
      if z < 0
        Complex(0, RealMath.sqrt(-z))
      else
        RealMath.sqrt(z)
      end
    else
      if z.imag < 0 ||
          (z.imag == 0 && z.imag.to_s[0] == '-')
        sqrt(z.conjugate).conjugate
      else
        r = z.abs
        x = z.real
        Complex(RealMath.sqrt((r + x) / 2.0), RealMath.sqrt((r - x) / 2.0))
      end
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the non-negative square root of Complex.

CMath.sqrt(-1 + 0i) #=> 0.0+1.0i

tan(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 199
def tan(z)
  begin
    if z.real?
      RealMath.tan(z)
    else
      sin(z) / cos(z)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the tangent of z, where z is given in radians

CMath.tan(1 + 1i) #=> (0.27175258531951174+1.0839233273386943i)

tanh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 249
def tanh(z)
  begin
    if z.real?
      RealMath.tanh(z)
    else
      sinh(z) / cosh(z)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the hyperbolic tangent of z, where z is given in radians

CMath.tanh(1 + 1i) #=> (1.0839233273386943+0.27175258531951174i)

Instance Methods

acos(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 281
def acos(z)
  begin
    if z.real? and z >= -1 and z <= 1
      RealMath.acos(z)
    else
      (-1.0).i * log(z + 1.0.i * sqrt(1.0 - z * z))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the arc cosine of z

CMath.acos(1 + 1i) #=> (0.9045568943023813-1.0612750619050357i)

acosh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 346
def acosh(z)
  begin
    if z.real? and z >= 1
      RealMath.acosh(z)
    else
      log(z + sqrt(z * z - 1.0))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the inverse hyperbolic cosine of z

CMath.acosh(1 + 1i) #=> (1.0612750619050357+0.9045568943023813i)

asin(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 265
def asin(z)
  begin
    if z.real? and z >= -1 and z <= 1
      RealMath.asin(z)
    else
      (-1.0).i * log(1.0.i * z + sqrt(1.0 - z * z))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the arc sine of z

CMath.asin(1 + 1i) #=> (0.6662394324925153+1.0612750619050355i)

asinh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 330
def asinh(z)
  begin
    if z.real?
      RealMath.asinh(z)
    else
      log(z + sqrt(1.0 + z * z))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the inverse hyperbolic sine of z

CMath.asinh(1 + 1i) #=> (1.0612750619050357+0.6662394324925153i)

atan(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 297
def atan(z)
  begin
    if z.real?
      RealMath.atan(z)
    else
      1.0.i * log((1.0.i + z) / (1.0.i - z)) / 2.0
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the arc tangent of z

CMath.atan(1 + 1i) #=> (1.0172219678978514+0.4023594781085251i)

atan2(y,x)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 314
def atan2(y,x)
  begin
    if y.real? and x.real?
      RealMath.atan2(y,x)
    else
      (-1.0).i * log((x + 1.0.i * y) / sqrt(x * x + y * y))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the arc tangent of y divided by x using the signs of y and x to determine the quadrant

CMath.atan2(1 + 1i, 0) #=> (1.5707963267948966+0.0i)

atanh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 362
def atanh(z)
  begin
    if z.real? and z >= -1 and z <= 1
      RealMath.atanh(z)
    else
      log((1.0 + z) / (1.0 - z)) / 2.0
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

returns the inverse hyperbolic tangent of z

CMath.atanh(1 + 1i) #=> (0.4023594781085251+1.0172219678978514i)

cbrt(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 157
def cbrt(z)
  z ** (1.0/3)
end

Returns the principal value of the cube root of z

CMath.cbrt(1 + 4i) #=> (1.449461632813119+0.6858152562177092i)

cos(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 182
def cos(z)
  begin
    if z.real?
      RealMath.cos(z)
    else
      Complex(RealMath.cos(z.real) * RealMath.cosh(z.imag),
              -RealMath.sin(z.real) * RealMath.sinh(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the cosine of z, where z is given in radians

CMath.cos(1 + 1i) #=> (0.8337300251311491-0.9888977057628651i)

cosh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 232
def cosh(z)
  begin
    if z.real?
      RealMath.cosh(z)
    else
      Complex(RealMath.cosh(z.real) * RealMath.cos(z.imag),
              RealMath.sinh(z.real) * RealMath.sin(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the hyperbolic cosine of z, where z is given in radians

CMath.cosh(1 + 1i) #=> (0.8337300251311491+0.9888977057628651i)

exp(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 62
def exp(z)
  begin
    if z.real?
      RealMath.exp(z)
    else
      ere = RealMath.exp(z.real)
      Complex(ere * RealMath.cos(z.imag),
              ere * RealMath.sin(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Math::E raised to the z power

CMath.exp(1.i * Math::PI) #=> (-1.0+1.2246467991473532e-16i)

log(z, b=::Math::E)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 82
def log(z, b=::Math::E)
  begin
    if z.real? && z >= 0 && b >= 0
      RealMath.log(z, b)
    else
      Complex(RealMath.log(z.abs), z.arg) / log(b)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the natural logarithm of Complex. If a second argument is given, it will be the base of logarithm.

CMath.log(1 + 4i)     #=> (1.416606672028108+1.3258176636680326i)
CMath.log(1 + 4i, 10) #=> (0.6152244606891369+0.5757952953408879i)

log10(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 114
def log10(z)
  begin
    if z.real? and z >= 0
      RealMath.log10(z)
    else
      log(z) / RealMath.log(10)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the base 10 logarithm of z

CMath.log10(-1) #=> (0.0+1.3643763538418412i)

log2(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 98
def log2(z)
  begin
    if z.real? and z >= 0
      RealMath.log2(z)
    else
      log(z) / RealMath.log(2)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the base 2 logarithm of z

CMath.log2(-1) => (0.0+4.532360141827194i)

sin(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 165
def sin(z)
  begin
    if z.real?
      RealMath.sin(z)
    else
      Complex(RealMath.sin(z.real) * RealMath.cosh(z.imag),
              RealMath.cos(z.real) * RealMath.sinh(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the sine of z, where z is given in radians

CMath.sin(1 + 1i) #=> (1.2984575814159773+0.6349639147847361i)

sinh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 215
def sinh(z)
  begin
    if z.real?
      RealMath.sinh(z)
    else
      Complex(RealMath.sinh(z.real) * RealMath.cos(z.imag),
              RealMath.cosh(z.real) * RealMath.sin(z.imag))
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the hyperbolic sine of z, where z is given in radians

CMath.sinh(1 + 1i) #=> (0.6349639147847361+1.2984575814159773i)

sqrt(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 130
def sqrt(z)
  begin
    if z.real?
      if z < 0
        Complex(0, RealMath.sqrt(-z))
      else
        RealMath.sqrt(z)
      end
    else
      if z.imag < 0 ||
          (z.imag == 0 && z.imag.to_s[0] == '-')
        sqrt(z.conjugate).conjugate
      else
        r = z.abs
        x = z.real
        Complex(RealMath.sqrt((r + x) / 2.0), RealMath.sqrt((r - x) / 2.0))
      end
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the non-negative square root of Complex.

CMath.sqrt(-1 + 0i) #=> 0.0+1.0i

tan(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 199
def tan(z)
  begin
    if z.real?
      RealMath.tan(z)
    else
      sin(z) / cos(z)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the tangent of z, where z is given in radians

CMath.tan(1 + 1i) #=> (0.27175258531951174+1.0839233273386943i)

tanh(z)

lib/cmath.rb View on GitHub

          
            # File tmp/rubies/ruby-2.3.8/lib/cmath.rb, line 249
def tanh(z)
  begin
    if z.real?
      RealMath.tanh(z)
    else
      sinh(z) / cosh(z)
    end
  rescue NoMethodError
    handle_no_method_error
  end
end

Returns the hyperbolic tangent of z, where z is given in radians

CMath.tanh(1 + 1i) #=> (1.0839233273386943+0.27175258531951174i)