A complex number is expressed in the form z = a + b * i
, where:
a
is the real part (a real number),
b
is the imaginary part (also a real number), and
i
is the imaginary unit satisfying i^2 = -1
.
The conjugate of the complex number z = a + b * i
is given by:
zc = a - b * i
The absolute value (or modulus) of z
is defined as:
|z| = sqrt(a^2 + b^2)
The square of the absolute value is computed as the product of z
and its conjugate zc
:
|z|^2 = z * zc = a^2 + b^2
The sum of two complex numbers z1 = a + b * i
and z2 = c + d * i
is computed by adding their real and imaginary parts separately:
z1 + z2 = (a + b * i) + (c + d * i)
= (a + c) + (b + d) * i
The difference of two complex numbers is obtained by subtracting their respective parts:
z1 - z2 = (a + b * i) - (c + d * i)
= (a - c) + (b - d) * i
The product of two complex numbers is defined as:
z1 * z2 = (a + b * i) * (c + d * i)
= (a * c - b * d) + (b * c + a * d) * i
The reciprocal of a non-zero complex number is given by:
1 / z = 1 / (a + b * i)
= a / (a^2 + b^2) - b / (a^2 + b^2) * i
The division of one complex number by another is given by:
z1 / z2 = z1 * (1 / z2)
= (a + b * i) / (c + d * i)
= (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i
Raising e (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:
e^(a + b * i) = e^a * e^(b * i)
= e^a * (cos(b) + i * sin(b))
Given that you should not use built-in support for complex numbers, implement the following operations:
This exercise requires you to use Closure-based objects. You can read more about this approach to object orientation in Lua in this article.
This exercise also requires you to use operator overloading via metatable metamethods. You can read more about this in Programming in Lua.
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