Module complex 

Complex arithmetic on interleaved [re, im] pairs.

The element type is [f64; 2], which is the layout numpy’s complex128, C99’s double _Complex and a pair of Arrow Float64 children all agree on, so a complex buffer crosses every boundary without a conversion.

Functions here are the mathematics only; the languages’ type rules and diagnostics live in verb.rs.

Constants

I

ONE

ZERO

Functions

abs

acos

_2 o. y: the arcsine’s complement.

acosh

_6 o. y: i * arccos y.

add

arg

The argument, in radians; arg(0) is 0.

asin

_1 o. y: -i ln(iy + sqrt(1 - y^2)).

asinh

_5 o. y: ln(y + sqrt(y^2 + 1)).

atan

_3 o. y: (i/2)(ln(1 - iy) - ln(1 + iy)), the two-logarithm form, which puts the branch cuts where both references put them.

atanh

_7 o. y: (ln(1 + y) - ln(1 - y)) / 2, again as two logarithms.

ceil

The ceiling is the floor reflected through the origin.

circle

The circle function k on a complex argument. None for a k the table does not define.

conj

cos

cosh

div

Division, with J’s rule for a zero divisor carried onto both parts: 0 % 0 is 0 and anything else over zero is a signed infinity.

exp

floor

McDonnell’s complex floor: the Gaussian integer at or below y, chosen so that the residue keeps a magnitude below one. Published in the J dictionary’s account of <.; both references answer with it.

from_degrees

The unit complex at degrees, exact on the quadrant boundaries — both references answer 2ad90 with 0j2, not with a cosine’s rounding of it.

from_radians

The complex of the given magnitude at the given angle in radians.

from_real

gcd

The Gaussian-integer greatest common divisor, by Euclid with the nearest Gaussian integer as the quotient. gcd(0, 0) is 0.

lcm

x *. y: the least common multiple, (x * y) % gcd.

ln

The principal logarithm; ln 0 is negative infinity, as on the reals.

log

x ^. y: the logarithm of y to base x.

mul

neg

pow

x ^ y. An integer exponent is repeated multiplication, which keeps 0j1 ^ 2 exactly _1 rather than a rounded neighbour of it.

recip

residue

x | y: y reduced modulo x, with the complex floor doing the rounding.

root

x %: y: the x-th root of y.

signum

y % | y: the unit complex in y’s direction, and 0 at the origin.

sin

sinh

sqrt

The principal square root, by the algebraic form: sqrt _4 has to be exactly 0j2, which halving the argument and taking a cosine does not give.

sub

tan

tanh

Type Aliases

Cx

One complex number: [real, imaginary].