#

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# complex.rb -

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# $Release Version: 0.5 $

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# $Revision: 1.3 $

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# $Date: 1998/07/08 10:05:28 $

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# by Keiju ISHITSUKA(SHL Japan Inc.)

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#

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# ----

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#

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# complex.rb implements the Complex class for complex numbers. Additionally,

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# some methods in other Numeric classes are redefined or added to allow greater

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# interoperability with Complex numbers.

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#

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# Complex numbers can be created in the following manner:

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# - <tt>Complex(a, b)</tt>

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# - <tt>Complex.polar(radius, theta)</tt>

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#

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# Additionally, note the following:

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# - <tt>Complex::I</tt> (the mathematical constant <i>i</i>)

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# - <tt>Numeric#im</tt> (e.g. <tt>5.im -> 0+5i</tt>)

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#

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# The following +Math+ module methods are redefined to handle Complex arguments.

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# They will work as normal with non-Complex arguments.

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# sqrt exp cos sin tan log log10

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# cosh sinh tanh acos asin atan atan2 acosh asinh atanh

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#

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#

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# Numeric is a built-in class on which Fixnum, Bignum, etc., are based. Here

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# some methods are added so that all number types can be treated to some extent

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# as Complex numbers.

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#

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class Numeric

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#

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# Returns a Complex number <tt>(0,<i>self</i>)</tt>.

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#

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def im

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Complex(0, self)

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end

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#

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# The real part of a complex number, i.e. <i>self</i>.

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#

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def real

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self

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end

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#

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# The imaginary part of a complex number, i.e. 0.

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#

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def image

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0

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end

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alias imag image

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#

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# See Complex#arg.

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#

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def arg

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Math.atan2!(0, self)

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end

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alias angle arg

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#

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# See Complex#polar.

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#

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def polar

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return abs, arg

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end

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#

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# See Complex#conjugate (short answer: returns <i>self</i>).

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#

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def conjugate

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self

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end

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alias conj conjugate

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end

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#

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# Creates a Complex number. +a+ and +b+ should be Numeric. The result will be

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# <tt>a+bi</tt>.

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#

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def Complex(a, b = 0)

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if b == 0 and (a.kind_of?(Complex) or defined? Complex::Unify)

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a

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else

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Complex.new( a.real-b.imag, a.imag+b.real )

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end

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end

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#

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# The complex number class. See complex.rb for an overview.

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#

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class Complex < Numeric

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@RCS_ID='-$Id: complex.rb,v 1.3 1998/07/08 10:05:28 keiju Exp keiju $-'

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undef step

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undef div, divmod

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undef floor, truncate, ceil, round

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def Complex.generic?(other) # :nodoc:

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other.kind_of?(Integer) or

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other.kind_of?(Float) or

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(defined?(Rational) and other.kind_of?(Rational))

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end

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#

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# Creates a +Complex+ number in terms of +r+ (radius) and +theta+ (angle).

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#

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def Complex.polar(r, theta)

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Complex(r*Math.cos(theta), r*Math.sin(theta))

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end

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#

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# Creates a +Complex+ number <tt>a</tt>+<tt>b</tt><i>i</i>.

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#

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def Complex.new!(a, b=0)

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new(a,b)

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end

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def initialize(a, b)

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raise TypeError, "non numeric 1st arg `#{a.inspect}'" if !a.kind_of? Numeric

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raise TypeError, "`#{a.inspect}' for 1st arg" if a.kind_of? Complex

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raise TypeError, "non numeric 2nd arg `#{b.inspect}'" if !b.kind_of? Numeric

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raise TypeError, "`#{b.inspect}' for 2nd arg" if b.kind_of? Complex

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@real = a

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@image = b

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end

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#

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# Addition with real or complex number.

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#

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def + (other)

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if other.kind_of?(Complex)

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re = @real + other.real

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im = @image + other.image

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Complex(re, im)

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elsif Complex.generic?(other)

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Complex(@real + other, @image)

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else

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x , y = other.coerce(self)

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x + y

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end

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end

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#

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# Subtraction with real or complex number.

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#

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def - (other)

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if other.kind_of?(Complex)

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re = @real - other.real

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im = @image - other.image

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Complex(re, im)

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elsif Complex.generic?(other)

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Complex(@real - other, @image)

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else

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x , y = other.coerce(self)

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x - y

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end

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end

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#

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# Multiplication with real or complex number.

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#

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def * (other)

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if other.kind_of?(Complex)

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re = @real*other.real - @image*other.image

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im = @real*other.image + @image*other.real

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Complex(re, im)

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elsif Complex.generic?(other)

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Complex(@real * other, @image * other)

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else

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x , y = other.coerce(self)

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x * y

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end

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end

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#

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# Division by real or complex number.

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#

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def / (other)

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if other.kind_of?(Complex)

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self*other.conjugate/other.abs2

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elsif Complex.generic?(other)

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Complex(@real/other, @image/other)

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else

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x, y = other.coerce(self)

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x/y

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end

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end

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def quo(other)

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Complex(@real.quo(1), @image.quo(1)) / other

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end

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#

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# Raise this complex number to the given (real or complex) power.

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#

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def ** (other)

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if other == 0

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return Complex(1)

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end

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if other.kind_of?(Complex)

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r, theta = polar

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ore = other.real

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oim = other.image

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nr = Math.exp!(ore*Math.log!(r) - oim * theta)

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ntheta = theta*ore + oim*Math.log!(r)

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Complex.polar(nr, ntheta)

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elsif other.kind_of?(Integer)

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if other > 0

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x = self

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z = x

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n = other - 1

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while n != 0

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while (div, mod = n.divmod(2)

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mod == 0)

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x = Complex(x.real*x.real - x.image*x.image, 2*x.real*x.image)

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n = div

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end

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z *= x

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n -= 1

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end

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z

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else

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if defined? Rational

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(Rational(1) / self) ** -other

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else

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self ** Float(other)

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end

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end

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elsif Complex.generic?(other)

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r, theta = polar

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Complex.polar(r**other, theta*other)

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else

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x, y = other.coerce(self)

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x**y

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end

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end

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#

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# Remainder after division by a real or complex number.

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#

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def % (other)

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if other.kind_of?(Complex)

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Complex(@real % other.real, @image % other.image)

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elsif Complex.generic?(other)

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Complex(@real % other, @image % other)

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else

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x , y = other.coerce(self)

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x % y

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end

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end

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#--

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# def divmod(other)

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# if other.kind_of?(Complex)

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# rdiv, rmod = @real.divmod(other.real)

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# idiv, imod = @image.divmod(other.image)

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# return Complex(rdiv, idiv), Complex(rmod, rmod)

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# elsif Complex.generic?(other)

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# Complex(@real.divmod(other), @image.divmod(other))

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# else

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# x , y = other.coerce(self)

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# x.divmod(y)

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# end

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# end

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#++

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#

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# Absolute value (aka modulus): distance from the zero point on the complex

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# plane.

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#

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def abs

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Math.hypot(@real, @image)

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end

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#

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# Square of the absolute value.

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#

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def abs2

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@real*@real + @image*@image

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end

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#

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# Argument (angle from (1,0) on the complex plane).

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#

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def arg

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Math.atan2!(@image, @real)

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end

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alias angle arg

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#

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# Returns the absolute value _and_ the argument.

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#

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def polar

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return abs, arg

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end

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#

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# Complex conjugate (<tt>z + z.conjugate = 2 * z.real</tt>).

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#

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def conjugate

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Complex(@real, -@image)

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end

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alias conj conjugate

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#

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# Compares the absolute values of the two numbers.

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#

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def <=> (other)

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self.abs <=> other.abs

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end

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#

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# Test for numerical equality (<tt>a == a + 0<i>i</i></tt>).

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#

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def == (other)

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if other.kind_of?(Complex)

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@real == other.real and @image == other.image

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elsif Complex.generic?(other)

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@real == other and @image == 0

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else

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other == self

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end

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end

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#

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# Attempts to coerce +other+ to a Complex number.

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#

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def coerce(other)

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if Complex.generic?(other)

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return Complex.new!(other), self

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else

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super

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end

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end

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#

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# FIXME

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#

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def denominator

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@real.denominator.lcm(@image.denominator)

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end

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#

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# FIXME

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#

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def numerator

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cd = denominator

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Complex(@real.numerator*(cd/@real.denominator),

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@image.numerator*(cd/@image.denominator))

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end

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#

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# Standard string representation of the complex number.

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#

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def to_s

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if @real != 0

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if defined?(Rational) and @image.kind_of?(Rational) and @image.denominator != 1

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if @image >= 0

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@real.to_s+"+("+@image.to_s+")i"

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else

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@real.to_s+"-("+(-@image).to_s+")i"

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end

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else

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if @image >= 0

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@real.to_s+"+"+@image.to_s+"i"

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else

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@real.to_s+"-"+(-@image).to_s+"i"

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end

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end

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else

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if defined?(Rational) and @image.kind_of?(Rational) and @image.denominator != 1

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"("+@image.to_s+")i"

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else

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@image.to_s+"i"

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end

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end

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end

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#

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# Returns a hash code for the complex number.

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#

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def hash

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@real.hash ^ @image.hash

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end

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#

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# Returns "<tt>Complex(<i>real</i>, <i>image</i>)</tt>".

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#

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def inspect

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sprintf("Complex(%s, %s)", @real.inspect, @image.inspect)

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end

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#

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# +I+ is the imaginary number. It exists at point (0,1) on the complex plane.

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#

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I = Complex(0,1)

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# The real part of a complex number.

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attr :real

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# The imaginary part of a complex number.

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attr :image

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alias imag image

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end

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class Integer

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unless defined?(1.numerator)

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def numerator() self end

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def denominator() 1 end

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def gcd(other)

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min = self.abs

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max = other.abs

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while min > 0

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tmp = min

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min = max % min

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max = tmp

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end

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max

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end

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def lcm(other)

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if self.zero? or other.zero?

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0

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else

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(self.div(self.gcd(other)) * other).abs

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end

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end

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end

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end

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module Math

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alias sqrt! sqrt

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alias exp! exp

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alias log! log

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alias log10! log10

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alias cos! cos

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alias sin! sin

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alias tan! tan

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alias cosh! cosh

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alias sinh! sinh

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alias tanh! tanh

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alias acos! acos

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alias asin! asin

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alias atan! atan

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alias atan2! atan2

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alias acosh! acosh

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alias asinh! asinh

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alias atanh! atanh

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# Redefined to handle a Complex argument.

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def sqrt(z)

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if Complex.generic?(z)

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if z >= 0

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sqrt!(z)

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else

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Complex(0,sqrt!(-z))

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end

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else

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if z.image < 0

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sqrt(z.conjugate).conjugate

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else

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r = z.abs

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x = z.real

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Complex( sqrt!((r+x)/2), sqrt!((r-x)/2) )

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end

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end

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end

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# Redefined to handle a Complex argument.

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def exp(z)

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if Complex.generic?(z)

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exp!(z)

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else

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Complex(exp!(z.real) * cos!(z.image), exp!(z.real) * sin!(z.image))

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end

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end

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# Redefined to handle a Complex argument.

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def cos(z)

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if Complex.generic?(z)

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cos!(z)

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else

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Complex(cos!(z.real)*cosh!(z.image),

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-sin!(z.real)*sinh!(z.image))

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end

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end

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# Redefined to handle a Complex argument.

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