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# 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 raise an exception than return a complex number.

## Usage

To start using this library, simply require cmath library:

``````require "cmath"
``````
Methods
A
C
E
L
S
T
Included Modules
Constants
 RealMath = Math # :nodoc: Backup of Math is needed because mathn.rb replaces Math with CMath.
Class Public methods
acos(z)

Returns the arc cosine of `z`

``````CMath.acos(1 + 1i) #=> (0.9045568943023813-1.0612750619050357i)
``````
```# File 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```
acosh(z)

returns the inverse hyperbolic cosine of `z`

``````CMath.acosh(1 + 1i) #=> (1.0612750619050357+0.9045568943023813i)
``````
```# File 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```
asin(z)

Returns the arc sine of `z`

``````CMath.asin(1 + 1i) #=> (0.6662394324925153+1.0612750619050355i)
``````
```# File 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```
asinh(z)

returns the inverse hyperbolic sine of `z`

``````CMath.asinh(1 + 1i) #=> (1.0612750619050357+0.6662394324925153i)
``````
```# File 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```
atan(z)

Returns the arc tangent of `z`

``````CMath.atan(1 + 1i) #=> (1.0172219678978514+0.4023594781085251i)
``````
```# File 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```
atan2(y,x)

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)
``````
```# File 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```
atanh(z)

returns the inverse hyperbolic tangent of `z`

``````CMath.atanh(1 + 1i) #=> (0.4023594781085251+1.0172219678978514i)
``````
```# File 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```
cbrt(z)

Returns the principal value of the cube root of `z`

``````CMath.cbrt(1 + 4i) #=> (1.449461632813119+0.6858152562177092i)
``````
```# File lib/cmath.rb, line 157
def cbrt(z)
z ** (1.0/3)
end```
cos(z)

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

``````CMath.cos(1 + 1i) #=> (0.8337300251311491-0.9888977057628651i)
``````
```# File 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```
cosh(z)

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

``````CMath.cosh(1 + 1i) #=> (0.8337300251311491+0.9888977057628651i)
``````
```# File 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```
exp(z)

Math::E raised to the `z` power

``````CMath.exp(1.i * Math::PI) #=> (-1.0+1.2246467991473532e-16i)
``````
```# File 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```
log(z, b=::Math::E)

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)
``````
```# File 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```
log10(z)

Returns the base 10 logarithm of `z`

``````CMath.log10(-1) #=> (0.0+1.3643763538418412i)
``````
```# File 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```
log2(z)

Returns the base 2 logarithm of `z`

``CMath.log2(-1) => (0.0+4.532360141827194i)``
```# File 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```
sin(z)

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

``````CMath.sin(1 + 1i) #=> (1.2984575814159773+0.6349639147847361i)
``````
```# File 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```
sinh(z)

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

``````CMath.sinh(1 + 1i) #=> (0.6349639147847361+1.2984575814159773i)
``````
```# File 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```
sqrt(z)

Returns the non-negative square root of Complex.

``````CMath.sqrt(-1 + 0i) #=> 0.0+1.0i
``````
```# File 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 == '-')
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```
tan(z)

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

``````CMath.tan(1 + 1i) #=> (0.27175258531951174+1.0839233273386943i)
``````
```# File 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```
tanh(z)

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

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