Source Edit

This module implements complex numbers and basic mathematical operations on them.

Complex numbers are currently generic over 64-bit or 32-bit floats.

Example:

import std/complex
from std/math import almostEqual, sqrt

let
  z1 = complex(1.0, 2.0)
  z2 = complex(3.0, -4.0)

assert almostEqual(z1 + z2, complex(4.0, -2.0))
assert almostEqual(z1 - z2, complex(-2.0, 6.0))
assert almostEqual(z1 * z2, complex(11.0, 2.0))
assert almostEqual(z1 / z2, complex(-0.2, 0.4))

assert almostEqual(abs(z1), sqrt(5.0))
assert almostEqual(conjugate(z1), complex(1.0, -2.0))

let (r, phi) = z1.polar
assert almostEqual(rect(r, phi), z1)

Imports

math

Types

Complex[T] = object re*, im*: T: A complex number, consisting of a real and an imaginary part. Source Edit
Complex32 = Complex[float32]: Alias for a complex number using 32-bit floats. Source Edit
Complex64 = Complex[float64]: Alias for a complex number using 64-bit floats. Source Edit

Procs

func `$`(z: Complex): string: Returns z's string representation as "(re, im)".
Example:

doAssert $complex(1.0, 2.0) == "(1.0, 2.0)"
Source Edit
func `*`[T](x, y: Complex[T]): Complex[T]: Multiplies two complex numbers. Source Edit
func `*`[T](x: Complex[T]; y: T): Complex[T]: Multiplies a complex number with a real number. Source Edit
func `*`[T](x: T; y: Complex[T]): Complex[T]: Multiplies a real number with a complex number. Source Edit
func `*=`[T](x: var Complex[T]; y: Complex[T]): Multiplies x by y. Source Edit
func `+`[T](x, y: Complex[T]): Complex[T]: Adds two complex numbers. Source Edit
func `+`[T](x: Complex[T]; y: T): Complex[T]: Adds a complex number to a real number. Source Edit
func `+`[T](x: T; y: Complex[T]): Complex[T]: Adds a real number to a complex number. Source Edit
func `+=`[T](x: var Complex[T]; y: Complex[T]): Adds y to x. Source Edit
func `-`[T](x, y: Complex[T]): Complex[T]: Subtracts two complex numbers. Source Edit
func `-`[T](x: Complex[T]; y: T): Complex[T]: Subtracts a real number from a complex number. Source Edit
func `-`[T](x: T; y: Complex[T]): Complex[T]: Subtracts a complex number from a real number. Source Edit
func `-`[T](z: Complex[T]): Complex[T]: Unary minus for complex numbers. Source Edit
func `-=`[T](x: var Complex[T]; y: Complex[T]): Subtracts y from x. Source Edit
func `/`[T](x, y: Complex[T]): Complex[T]: Divides two complex numbers. Source Edit
func `/`[T](x: Complex[T]; y: T): Complex[T]: Divides a complex number by a real number. Source Edit
func `/`[T](x: T; y: Complex[T]): Complex[T]: Divides a real number by a complex number. Source Edit
func `/=`[T](x: var Complex[T]; y: Complex[T]): Divides x by y in place. Source Edit
func `==`[T](x, y: Complex[T]): bool: Compares two complex numbers for equality. Source Edit
func abs[T](z: Complex[T]): T: Returns the absolute value of z, that is the distance from (0, 0) to z. Source Edit
func abs2[T](z: Complex[T]): T: Returns the squared absolute value of z, that is the squared distance from (0, 0) to z. This is more efficient than abs(z) ^ 2. Source Edit
func almostEqual[T: SomeFloat](x, y: Complex[T]; unitsInLastPlace: Natural = 4): bool: Checks if two complex values are almost equal, using the machine epsilon.

Two complex values are considered almost equal if their real and imaginary components are almost equal.

unitsInLastPlace is the max number of units in the last place difference tolerated when comparing two numbers. The larger the value, the more error is allowed. A 0 value means that two numbers must be exactly the same to be considered equal.

The machine epsilon has to be scaled to the magnitude of the values used and multiplied by the desired precision in ULPs unless the difference is subnormal.
Source Edit
func arccos[T](z: Complex[T]): Complex[T]: Returns the inverse cosine of z. Source Edit
func arccosh[T](z: Complex[T]): Complex[T]: Returns the inverse hyperbolic cosine of z. Source Edit
func arccot[T](z: Complex[T]): Complex[T]: Returns the inverse cotangent of z. Source Edit
func arccoth[T](z: Complex[T]): Complex[T]: Returns the inverse hyperbolic cotangent of z. Source Edit
func arccsc[T](z: Complex[T]): Complex[T]: Returns the inverse cosecant of z. Source Edit
func arccsch[T](z: Complex[T]): Complex[T]: Returns the inverse hyperbolic cosecant of z. Source Edit
func arcsec[T](z: Complex[T]): Complex[T]: Returns the inverse secant of z. Source Edit
func arcsech[T](z: Complex[T]): Complex[T]: Returns the inverse hyperbolic secant of z. Source Edit
func arcsin[T](z: Complex[T]): Complex[T]: Returns the inverse sine of z. Source Edit
func arcsinh[T](z: Complex[T]): Complex[T]: Returns the inverse hyperbolic sine of z. Source Edit
func arctan[T](z: Complex[T]): Complex[T]: Returns the inverse tangent of z. Source Edit
func arctanh[T](z: Complex[T]): Complex[T]: Returns the inverse hyperbolic tangent of z. Source Edit
func complex[T: SomeFloat](re: T; im: T = 0.0): Complex[T]: Returns a Complex[T] with real part re and imaginary part im. Source Edit
func complex32(re: float32; im: float32 = 0.0): Complex32 {....raises: [], tags: [], forbids: [].}: Returns a Complex32 with real part re and imaginary part im. Source Edit
func complex64(re: float64; im: float64 = 0.0): Complex64 {....raises: [], tags: [], forbids: [].}: Returns a Complex64 with real part re and imaginary part im. Source Edit
func conjugate[T](z: Complex[T]): Complex[T]: Returns the complex conjugate of z (complex(z.re, -z.im)). Source Edit
func cos[T](z: Complex[T]): Complex[T]: Returns the cosine of z. Source Edit
func cosh[T](z: Complex[T]): Complex[T]: Returns the hyperbolic cosine of z. Source Edit
func cot[T](z: Complex[T]): Complex[T]: Returns the cotangent of z. Source Edit
func coth[T](z: Complex[T]): Complex[T]: Returns the hyperbolic cotangent of z. Source Edit
func csc[T](z: Complex[T]): Complex[T]: Returns the cosecant of z. Source Edit
func csch[T](z: Complex[T]): Complex[T]: Returns the hyperbolic cosecant of z. Source Edit
func exp[T](z: Complex[T]): Complex[T]: Computes the exponential function (e^z). Source Edit
func inv[T](z: Complex[T]): Complex[T]: Returns the multiplicative inverse of z (1/z). Source Edit
func ln[T](z: Complex[T]): Complex[T]: Returns the (principal value of the) natural logarithm of z. Source Edit
func log2[T](z: Complex[T]): Complex[T]: Returns the logarithm base 2 of z.

See also:

ln func

Source Edit
func log10[T](z: Complex[T]): Complex[T]: Returns the logarithm base 10 of z.

See also:

ln func

Source Edit
func phase[T](z: Complex[T]): T: Returns the phase (or argument) of z, that is the angle in polar representation.
result = arctan2(z.im, z.re)
Source Edit
func polar[T](z: Complex[T]): tuple[r, phi: T]: Returns z in polar coordinates.
result.r = abs(z)
result.phi = phase(z)
See also:

rect func for the inverse operation

Source Edit
func pow[T](x, y: Complex[T]): Complex[T]: x raised to the power of y. Source Edit
func pow[T](x: Complex[T]; y: T): Complex[T]: The complex number x raised to the power of the real number y. Source Edit
func rect[T](r, phi: T): Complex[T]: Returns the complex number with polar coordinates r and phi.
result.re = r * cos(phi)
result.im = r * sin(phi)
See also:

polar func for the inverse operation

Source Edit
func sec[T](z: Complex[T]): Complex[T]: Returns the secant of z. Source Edit
func sech[T](z: Complex[T]): Complex[T]: Returns the hyperbolic secant of z. Source Edit
func sgn[T](z: Complex[T]): Complex[T]: Returns the phase of z as a unit complex number, or 0 if z is 0. Source Edit
func sin[T](z: Complex[T]): Complex[T]: Returns the sine of z. Source Edit
func sinh[T](z: Complex[T]): Complex[T]: Returns the hyperbolic sine of z. Source Edit
func sqrt[T](z: Complex[T]): Complex[T]: Computes the (principal) square root of a complex number z. Source Edit
func tan[T](z: Complex[T]): Complex[T]: Returns the tangent of z. Source Edit
func tanh[T](z: Complex[T]): Complex[T]: Returns the hyperbolic tangent of z. Source Edit

Templates

template im(arg: float32): Complex32: Returns arg as an imaginary number (complex32(0, arg)). Source Edit
template im(arg: float64): Complex64: Returns arg as an imaginary number (complex64(0, arg)). Source Edit
template im(arg: typedesc[float32]): Complex32: Returns the imaginary unit (complex32(0, 1)). Source Edit
template im(arg: typedesc[float64]): Complex64: Returns the imaginary unit (complex64(0, 1)). Source Edit