##
# = 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.
#
# == Usage
#
# To start using this library, simply require cmath library:
#
# require "cmath"
#
# And after call any CMath function. For example:
#
# CMath.sqrt(-9) #=> 0+3.0i
# CMath.exp(0 + 0i) #=> 1.0+0.0i
# CMath.log10(-5.to_c) #=> (0.6989700043360187+1.3643763538418412i)
#
#
# For more information you can see Complec class.
module CMath
include Math
alias exp! exp
alias log! log
alias log2! log2
alias log10! log10
alias sqrt! sqrt
alias cbrt! cbrt
alias sin! sin
alias cos! cos
alias tan! tan
alias sinh! sinh
alias cosh! cosh
alias tanh! tanh
alias asin! asin
alias acos! acos
alias atan! atan
alias atan2! atan2
alias asinh! asinh
alias acosh! acosh
alias atanh! atanh
##
# Math::E raised to the +z+ power
#
# CMath.exp(2i) #=> (-0.4161468365471424+0.9092974268256817i)
def exp(z)
begin
if z.real?
exp!(z)
else
ere = exp!(z.real)
Complex(ere * cos!(z.imag),
ere * sin!(z.imag))
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)
def log(*args)
begin
z, b = args
unless b.nil? || b.kind_of?(Numeric)
raise TypeError, "Numeric Number required"
end
if z.real? and z >= 0 and (b.nil? or b >= 0)
log!(*args)
else
a = Complex(log!(z.abs), z.arg)
if b
a /= log(b)
end
a
end
rescue NoMethodError
handle_no_method_error
end
end
##
# returns the base 2 logarithm of +z+
#
# CMath.log2(-1) => (0.0+4.532360141827194i)
def log2(z)
begin
if z.real? and z >= 0
log2!(z)
else
log(z) / log!(2)
end
rescue NoMethodError
handle_no_method_error
end
end
##
# returns the base 10 logarithm of +z+
#
# CMath.log10(-1) #=> (0.0+1.3643763538418412i)
def log10(z)
begin
if z.real? and z >= 0
log10!(z)
else
log(z) / log!(10)
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
def sqrt(z)
begin
if z.real?
if z < 0
Complex(0, sqrt!(-z))
else
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(sqrt!((r + x) / 2.0), sqrt!((r - x) / 2.0))
end
end
rescue NoMethodError
handle_no_method_error
end
end
##
# returns the principal value of the cube root of +z+
#
# CMath.cbrt(1 + 4i) #=> (1.449461632813119+0.6858152562177092i)
def cbrt(z)
z ** (1.0/3)
end
##
# returns the sine of +z+, where +z+ is given in radians
#
# CMath.sin(1 + 1i) #=> (1.2984575814159773+0.6349639147847361i)
def sin(z)
begin
if z.real?
sin!(z)
else
Complex(sin!(z.real) * cosh!(z.imag),
cos!(z.real) * 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)
def cos(z)
begin
if z.real?
cos!(z)
else
Complex(cos!(z.real) * cosh!(z.imag),
-sin!(z.real) * sinh!(z.imag))
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)
def tan(z)
begin
if z.real?
tan!(z)
else
sin(z) / cos(z)
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)
def sinh(z)
begin
if z.real?
sinh!(z)
else
Complex(sinh!(z.real) * cos!(z.imag),
cosh!(z.real) * 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)
def cosh(z)
begin
if z.real?
cosh!(z)
else
Complex(cosh!(z.real) * cos!(z.imag),
sinh!(z.real) * sin!(z.imag))
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)
def tanh(z)
begin
if z.real?
tanh!(z)
else
sinh(z) / cosh(z)
end
rescue NoMethodError
handle_no_method_error
end
end
##
# returns the arc sine of +z+
#
# CMath.asin(1 + 1i) #=> (0.6662394324925153+1.0612750619050355i)
def asin(z)
begin
if z.real? and z >= -1 and z <= 1
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 cosine of +z+
#
# CMath.acos(1 + 1i) #=> (0.9045568943023813-1.0612750619050357i)
def acos(z)
begin
if z.real? and z >= -1 and z <= 1
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 tangent of +z+
#
# CMath.atan(1 + 1i) #=> (1.0172219678978514+0.4023594781085251i)
def atan(z)
begin
if z.real?
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 +y+ divided by +x+ using the signs of +y+ and
# +x+ to determine the quadrant
#
# CMath.atan2(1 + 1i, 0) #=> (1.5707963267948966+0.0i)
def atan2(y,x)
begin
if y.real? and x.real?
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 inverse hyperbolic sine of +z+
#
# CMath.asinh(1 + 1i) #=> (1.0612750619050357+0.6662394324925153i)
def asinh(z)
begin
if z.real?
asinh!(z)
else
log(z + sqrt(1.0 + z * z))
end
rescue NoMethodError
handle_no_method_error
end
end
##
# returns the inverse hyperbolic cosine of +z+
#
# CMath.acosh(1 + 1i) #=> (1.0612750619050357+0.9045568943023813i)
def acosh(z)
begin
if z.real? and z >= 1
acosh!(z)
else
log(z + sqrt(z * z - 1.0))
end
rescue NoMethodError
handle_no_method_error
end
end
##
# returns the inverse hyperbolic tangent of +z+
#
# CMath.atanh(1 + 1i) #=> (0.4023594781085251+1.0172219678978514i)
def atanh(z)
begin
if z.real? and z >= -1 and z <= 1
atanh!(z)
else
log((1.0 + z) / (1.0 - z)) / 2.0
end
rescue NoMethodError
handle_no_method_error
end
end
module_function :exp!
module_function :exp
module_function :log!
module_function :log
module_function :log2!
module_function :log2
module_function :log10!
module_function :log10
module_function :sqrt!
module_function :sqrt
module_function :cbrt!
module_function :cbrt
module_function :sin!
module_function :sin
module_function :cos!
module_function :cos
module_function :tan!
module_function :tan
module_function :sinh!
module_function :sinh
module_function :cosh!
module_function :cosh
module_function :tanh!
module_function :tanh
module_function :asin!
module_function :asin
module_function :acos!
module_function :acos
module_function :atan!
module_function :atan
module_function :atan2!
module_function :atan2
module_function :asinh!
module_function :asinh
module_function :acosh!
module_function :acosh
module_function :atanh!
module_function :atanh
module_function :frexp
module_function :ldexp
module_function :hypot
module_function :erf
module_function :erfc
module_function :gamma
module_function :lgamma
private
def handle_no_method_error # :nodoc:
if $!.name == :real?
raise TypeError, "Numeric Number required"
else
raise
end
end
module_function :handle_no_method_error
end
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