Complex Vectors

Description

These are basic functions which support complex arithmetic in R. 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.

Usage

complex(length.out = 0, real = numeric(), imaginary = numeric(),
        modulus = 1, argument = 0)
as.complex(z)
is.complex(z)

Re(z)
Im(z)
Mod(z)
Arg(z)
Conj(z)

Details

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. Modulus and argument are also called the polar coordinates. If z = x + i y with real x and y, Mod(z) = sqrt{x^2 + y^2}, and for phi= Arg(z), x = cos(phi) and y = sin(phi).

In addition, the elementary trigonometric, logarithmic and exponential functions are available for complex values.

Examples

( z <- 0i ^ (-3:3) )
all(Re(z) == 0 ^ (-3:3))
matrix(1i^ (-6:5), nr=4)#- all columns are the same
0 ^ 1i # a complex NaN

## create a complex normal vector
z <- complex(real = rnorm(100), imag = rnorm(100))
## or also (less efficiently):
z2 <- 1:2 + 1i*(8:9)

all(Mod ( 1 -  sin(z) / ( (exp(1i*z)-exp(-1i*z))/(2*1i) ))
    < 100*.Machine$double.eps)
## 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")