Complex numbers
A complex is a scalar, independent of any lattice.
complex
complex(re, im) --> complex
The constructor. Both arguments are numbers.
A complex number is a value, not a container: like a Lua number it is used whole and never modified. There is no way to set its real or imaginary part, and z.real = 1 raises. z:real() and z:imag() read the parts out; to change one, build a new number from the parts you want.
local z = complex(1, 2)
local w = complex(3, z:imag()) -- not z:real(3)
Complex values take part in ordinary arithmetic; see Arithmetic.
complex:real
z:real() --> number
complex:imag
z:imag() --> number
complex:conj
z:conj() --> complex
complex:abs
z:abs() --> number
The modulus.
complex:arg
z:arg() --> number
The argument, in radians.
complex:norm2
z:norm2() --> number
The squared modulus — re*re + im*im, with no square root.
complex:sqrt
z:sqrt() --> complex
The principal square root: the one whose real part is non-negative, so the result lies in the right half-plane with the positive imaginary axis included. The branch cut is the negative real axis, approached from above: for a negative real x, complex(x, 0):sqrt() is +i·√|x|, and the value just below the cut is its negative. sqrt is discontinuous across that line.
complex:exp
z:exp() --> complex
e^z. Entire — no cut.
complex:log
z:log() --> complex
The principal logarithm: the real part is log|z| and the imaginary part is z:arg(), which lies in (−π, π]. The branch cut is again the negative real axis, approached from above, so complex(-1, 0):log() is +iπ while a point just below the cut gives nearly −iπ. log is undefined at zero.
Both cuts are the C library's, inherited unchanged; arg is atan2(im, re) and carries the same convention.
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