# complex

##### Complex Numbers and Basic Functionality

Basic functions which support complex arithmetic in R, in addition to
the arithmetic operators `+`

, `-`

, `*`

, `/`

, and `^`

.

- Keywords
- complex

##### Usage

```
complex(length.out = 0, real = numeric(), imaginary = numeric(), modulus = 1, argument = 0)
as.complex(x, ...)
is.complex(x)
Re(z)
Im(z)
Mod(z)
Arg(z)
Conj(z)
```

##### Arguments

- length.out
- numeric. Desired length of the output vector, inputs being recycled as needed.
- real
- numeric vector.
- imaginary
- numeric vector.
- modulus
- numeric vector.
- argument
- numeric vector.
- x
- an object, probably of mode
`complex`

. - z
- an object of mode
`complex`

, or one of a class for which a methods has been defined. - ...
- further arguments passed to or from other methods.

##### Details

Complex vectors can be created with `complex`

. The vector can be
specified either by giving its length, its real and imaginary parts, or
modulus and argument. (Giving just the length generates a vector of
complex zeroes.)

`as.complex`

attempts to coerce its argument to be of complex
type: like `as.vector`

it strips attributes including
names. Up to R versions 3.2.x, all forms of `NA`

and `NaN`

were coerced to a complex `NA`

, i.e., the `NA_complex_`

constant, for which both the real and imaginary parts are `NA`

.
Since R 3.3.0, typically only objects which are `NA`

in parts
are coerced to complex `NA`

, but others with `NaN`

parts,
are *not*. As a consequence, complex arithmetic where only
`NaN`

's (but no `NA`

's) are involved typically will
*not* give complex `NA`

but complex numbers with real or
imaginary parts of `NaN`

.

Note that `is.complex`

and `is.numeric`

are never both
`TRUE`

.

The functions `Re`

, `Im`

, `Mod`

, `Arg`

and
`Conj`

have their usual interpretation as returning the real
part, imaginary part, modulus, argument and complex conjugate for
complex values. The modulus and argument are also called the *polar
coordinates*. If $z = x + i y$ with real $x$ and $y$, for
$r = Mod(z) = \sqrt(x^2 + y^2)$,
and $\phi = Arg(z)$, $x = r*cos(\phi)$ and
$y = r*sin(\phi)$. They are all
internal generic primitive functions: methods can be
defined for them
individually or *via* the `Complex`

group generic.

In addition to the arithmetic operators (see Arithmetic)
`+`

, `-`

, `*`

, `/`

, and `^`

, the elementary
trigonometric, logarithmic, exponential, square root and hyperbolic
functions are implemented for complex values.

Matrix multiplications (`%*%`

, `crossprod`

,
`tcrossprod`

) are also defined for complex matrices
(`matrix`

), and so are `solve`

,
`eigen`

or `svd`

.

Internally, complex numbers are stored as a pair of double
precision numbers, either or both of which can be `NaN`

(including `NA`

, see `NA_complex_`

and above) or
plus or minus infinity.

##### Note

Operations and functions involving complex `NaN`

mostly
rely on the C library's handling of double complex arithmetic,
which typically returns `complex(re=NaN, im=NaN)`

(but we have
not seen a guarantee for that).
For `+`

and `-`

, R's own handling works strictly
“coordinate wise”.

Operations involving complex `NA`

, i.e., `NA_complex_`

, return
`NA_complex_`

.

##### S4 methods

`as.complex`

is primitive and can have S4 methods set. `Re`

, `Im`

, `Mod`

, `Arg`

and `Conj`

constitute the S4 group generic
`Complex`

and so S4 methods can be
set for them individually or via the group generic.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole.

##### See Also

`Arithmetic`

; `polyroot`

finds all $n$
complex roots of a polynomial of degree $n$.

##### Examples

`library(base)`

```
require(graphics)
0i ^ (-3:3)
matrix(1i^ (-6:5), nrow = 4) #- all columns are the same
0 ^ 1i # a complex NaN
## create a complex normal vector
z <- complex(real = stats::rnorm(100), imaginary = stats::rnorm(100))
## or also (less efficiently):
z2 <- 1:2 + 1i*(8:9)
## The Arg(.) is an angle:
zz <- (rep(1:4, len = 9) + 1i*(9:1))/10
zz.shift <- complex(modulus = Mod(zz), argument = Arg(zz) + pi)
plot(zz, xlim = c(-1,1), ylim = c(-1,1), col = "red", asp = 1,
main = expression(paste("Rotation by "," ", pi == 180^o)))
abline(h = 0, v = 0, col = "blue", lty = 3)
points(zz.shift, col = "orange")
showC <- function(z) noquote(sprintf("(R = %g, I = %g)", Re(z), Im(z)))
## The exact result of this *depends* on the platform, compiler, math-library:
(NpNA <- NaN + NA_complex_) ; str(NpNA) # *behaves* as 'cplx NA' ..
stopifnot(is.na(NpNA), is.na(NA_complex_), is.na(Re(NA_complex_)), is.na(Im(NA_complex_)))
showC(NpNA)# but not always is {shows '(R = NaN, I = NA)' on some platforms}
## and this is not TRUE everywhere:
identical(NpNA, NA_complex_)
showC(NA_complex_) # always == (R = NA, I = NA)
```

*Documentation reproduced from package base, version 3.3.0, License: Part of R 3.3.0*