lyng/docs/Complex.md

Complex Numbers (lyng.complex)

lyng.complex adds a pure-Lyng Complex type backed by Real components.

Import it when you want ordinary complex arithmetic:

import lyng.complex

Construction

Use any of these:

import lyng.complex

val a = Complex(1.0, 2.0)
val b = complex(1.0, 2.0)
val c = 2.i
val d = 3.re

assertEquals(Complex(1.0, 2.0), 1 + 2.i)

Convenience extensions:

• Int.re, Real.re: embed a real value into the complex plane
• Int.i, Real.i: create a pure imaginary value
• cis(angle): shorthand for cos(angle) + i sin(angle)

Core Operations

Complex supports:

• +
• -
• *
• /
• unary -
• conjugate
• magnitude
• phase

Mixed arithmetic with Int and Real is enabled through lyng.operators, so both sides work naturally:

import lyng.complex

assertEquals(Complex(1.0, 2.0), 1 + 2.i)
assertEquals(Complex(1.5, 2.0), 1.5 + 2.i)
assertEquals(Complex(2.0, 2.0), 2.i + 2)

Mixed equality with built-in numeric types is intentionally not promised yet. Keep equality checks in the Complex domain for now.

Transcendental Functions

For now, use member-style calls:

import lyng.complex

val z = 1 + π.i
val w = z.exp()
val s = z.sin()
val r = z.sqrt()

This is deliberate. Lyng already has built-in top-level real-valued functions such as exp(x) and sin(x), and imported modules do not currently replace those root bindings. So plain exp(z) is not yet the right extension mechanism for complex math.

Design Scope

This module intentionally uses Complex with Real parts, not Complex<T>.

Reasons:

• the existing math runtime is Real-centric
• the operator interop registry works with concrete runtime classes
• transcendental functions (exp, sin, ln, sqrt) are defined over the Real math backend here

If Lyng later gets a more general numeric-trait or callable-overload registry, a generic algebraic Complex<T> can be revisited on firmer ground.