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:

  1. import std/complex
  2. from std/math import almostEqual, sqrt
  4. let
  5.   z1 = complex(1.0, 2.0)
  6.   z2 = complex(3.0, -4.0)
  8. assert almostEqual(z1 + z2, complex(4.0, -2.0))
  9. assert almostEqual(z1 - z2, complex(-2.0, 6.0))
  10. assert almostEqual(z1 * z2, complex(11.0, 2.0))
  11. assert almostEqual(z1 / z2, complex(-0.2, 0.4))
  13. assert almostEqual(abs(z1), sqrt(5.0))
  14. assert almostEqual(conjugate(z1), complex(1.0, -2.0))
  16. let (r, phi) = z1.polar
  17. assert almostEqual(rect(r, phi), z1)

Imports

math

Types

  1. Complex[T] = object
  2.   re*, im*: T

A complex number, consisting of a real and an imaginary part. Source Edit 

  1. Complex32 = Complex[float32]

Alias for a complex number using 32-bit floats. Source Edit 

  1. Complex64 = Complex[float64]

Alias for a complex number using 64-bit floats. Source Edit

Procs

  1. func `$`(z: Complex): string

Returns z’s string representation as “(re, im)”.

Example:

  1. doAssert $complex(1.0, 2.0) == "(1.0, 2.0)"

Source Edit 

  1. func `*`[T](x, y: Complex[T]): Complex[T]

Multiplies two complex numbers. Source Edit 

  1. func `*`[T](x: Complex[T]; y: T): Complex[T]

Multiplies a complex number with a real number. Source Edit 

  1. func `*`[T](x: T; y: Complex[T]): Complex[T]

Multiplies a real number with a complex number. Source Edit 

  1. func `*=`[T](x: var Complex[T]; y: Complex[T])

Multiplies x by y. Source Edit 

  1. func `+`[T](x, y: Complex[T]): Complex[T]

Adds two complex numbers. Source Edit 

  1. func `+`[T](x: Complex[T]; y: T): Complex[T]

Adds a complex number to a real number. Source Edit 

  1. func `+`[T](x: T; y: Complex[T]): Complex[T]

Adds a real number to a complex number. Source Edit 

  1. func `+=`[T](x: var Complex[T]; y: Complex[T])

Adds y to x. Source Edit 

  1. func `-`[T](x, y: Complex[T]): Complex[T]

Subtracts two complex numbers. Source Edit 

  1. func `-`[T](x: Complex[T]; y: T): Complex[T]

Subtracts a real number from a complex number. Source Edit 

  1. func `-`[T](x: T; y: Complex[T]): Complex[T]

Subtracts a complex number from a real number. Source Edit 

  1. func `-`[T](z: Complex[T]): Complex[T]

Unary minus for complex numbers. Source Edit 

  1. func `-=`[T](x: var Complex[T]; y: Complex[T])

Subtracts y from x. Source Edit 

  1. func `/`[T](x, y: Complex[T]): Complex[T]

Divides two complex numbers. Source Edit 

  1. func `/`[T](x: Complex[T]; y: T): Complex[T]

Divides a complex number by a real number. Source Edit 

  1. func `/`[T](x: T; y: Complex[T]): Complex[T]

Divides a real number by a complex number. Source Edit 

  1. func `/=`[T](x: var Complex[T]; y: Complex[T])

Divides x by y in place. Source Edit 

  1. func `==`[T](x, y: Complex[T]): bool

Compares two complex numbers for equality. Source Edit 

  1. func abs[T](z: Complex[T]): T

Returns the absolute value of z, that is the distance from (0, 0) to z. Source Edit 

  1. 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 

  1. 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 

  1. func arccos[T](z: Complex[T]): Complex[T]

Returns the inverse cosine of z. Source Edit 

  1. func arccosh[T](z: Complex[T]): Complex[T]

Returns the inverse hyperbolic cosine of z. Source Edit 

  1. func arccot[T](z: Complex[T]): Complex[T]

Returns the inverse cotangent of z. Source Edit 

  1. func arccoth[T](z: Complex[T]): Complex[T]

Returns the inverse hyperbolic cotangent of z. Source Edit 

  1. func arccsc[T](z: Complex[T]): Complex[T]

Returns the inverse cosecant of z. Source Edit 

  1. func arccsch[T](z: Complex[T]): Complex[T]

Returns the inverse hyperbolic cosecant of z. Source Edit 

  1. func arcsec[T](z: Complex[T]): Complex[T]

Returns the inverse secant of z. Source Edit 

  1. func arcsech[T](z: Complex[T]): Complex[T]

Returns the inverse hyperbolic secant of z. Source Edit 

  1. func arcsin[T](z: Complex[T]): Complex[T]

Returns the inverse sine of z. Source Edit 

  1. func arcsinh[T](z: Complex[T]): Complex[T]

Returns the inverse hyperbolic sine of z. Source Edit 

  1. func arctan[T](z: Complex[T]): Complex[T]

Returns the inverse tangent of z. Source Edit 

  1. func arctanh[T](z: Complex[T]): Complex[T]

Returns the inverse hyperbolic tangent of z. Source Edit 

  1. 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 

  1. func complex32(re: float32; im: float32 = 0.0): Complex32 {....raises: [],
  2.     tags: [], forbids: [].}

Returns a Complex32 with real part re and imaginary part im. Source Edit 

  1. func complex64(re: float64; im: float64 = 0.0): Complex64 {....raises: [],
  2.     tags: [], forbids: [].}

Returns a Complex64 with real part re and imaginary part im. Source Edit 

  1. func conjugate[T](z: Complex[T]): Complex[T]

Returns the complex conjugate of z (complex(z.re, -z.im)). Source Edit 

  1. func cos[T](z: Complex[T]): Complex[T]

Returns the cosine of z. Source Edit 

  1. func cosh[T](z: Complex[T]): Complex[T]

Returns the hyperbolic cosine of z. Source Edit 

  1. func cot[T](z: Complex[T]): Complex[T]

Returns the cotangent of z. Source Edit 

  1. func coth[T](z: Complex[T]): Complex[T]

Returns the hyperbolic cotangent of z. Source Edit 

  1. func csc[T](z: Complex[T]): Complex[T]

Returns the cosecant of z. Source Edit 

  1. func csch[T](z: Complex[T]): Complex[T]

Returns the hyperbolic cosecant of z. Source Edit 

  1. func exp[T](z: Complex[T]): Complex[T]

Computes the exponential function (e^z). Source Edit 

  1. func inv[T](z: Complex[T]): Complex[T]

Returns the multiplicative inverse of z (1/z). Source Edit 

  1. func ln[T](z: Complex[T]): Complex[T]

Returns the (principal value of the) natural logarithm of z. Source Edit 

  1. func log2[T](z: Complex[T]): Complex[T]

Returns the logarithm base 2 of z.

See also:

Source Edit 

  1. func log10[T](z: Complex[T]): Complex[T]

Returns the logarithm base 10 of z.

See also:

Source Edit 

  1. 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 

  1. 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:

Source Edit 

  1. func pow[T](x, y: Complex[T]): Complex[T]

x raised to the power of y. Source Edit 

  1. 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 

  1. 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:

Source Edit 

  1. func sec[T](z: Complex[T]): Complex[T]

Returns the secant of z. Source Edit 

  1. func sech[T](z: Complex[T]): Complex[T]

Returns the hyperbolic secant of z. Source Edit 

  1. 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 

  1. func sin[T](z: Complex[T]): Complex[T]

Returns the sine of z. Source Edit 

  1. func sinh[T](z: Complex[T]): Complex[T]

Returns the hyperbolic sine of z. Source Edit 

  1. func sqrt[T](z: Complex[T]): Complex[T]

Computes the (principal) square root of a complex number z. Source Edit 

  1. func tan[T](z: Complex[T]): Complex[T]

Returns the tangent of z. Source Edit 

  1. func tanh[T](z: Complex[T]): Complex[T]

Returns the hyperbolic tangent of z. Source Edit

Templates

  1. template im(arg: float32): Complex32

Returns arg as an imaginary number (complex32(0, arg)). Source Edit 

  1. template im(arg: float64): Complex64

Returns arg as an imaginary number (complex64(0, arg)). Source Edit 

  1. template im(arg: typedesc[float32]): Complex32

Returns the imaginary unit (complex32(0, 1)). Source Edit 

  1. template im(arg: typedesc[float64]): Complex64

Returns the imaginary unit (complex64(0, 1)). Source Edit