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### 17.2 Complex Arithmetic

In the descriptions of the following functions, z is the complex number x + i y, where i is defined as `sqrt (-1)`.

— Mapping Function: abs (z)

> Compute the magnitude of z, defined as |z| = `sqrt (x^2 + y^2)`.

> For example,

>               abs (3 + 4i)
>                     5

— Mapping Function: arg (z)
— Mapping Function: angle (z)

> Compute the argument of z, defined as, theta = `atan2 (`y`,` x`)`, in radians.

> For example,

>               arg (3 + 4i)
>                     0.92730

— Mapping Function: conj (z)

> Return the complex conjugate of z, defined as `conj (`z`)` = x - i y.

> See also: real, imag.

— Function File: cplxpair (z)
— Function File: cplxpair (z, tol)
— Function File: cplxpair (z, tol, dim)

> Sort the numbers z into complex conjugate pairs ordered by increasing real part. Place the negative imaginary complex number first within each pair. Place all the real numbers (those with `abs (imag (`z`) /` z`) <` tol`)`) after the complex pairs.

> If tol is unspecified the default value is 100*`eps`.

> By default the complex pairs are sorted along the first non-singleton dimension of z. If dim is specified, then the complex pairs are sorted along this dimension.

> Signal an error if some complex numbers could not be paired. Signal an error if all complex numbers are not exact conjugates (to within tol). Note that there is no defined order for pairs with identical real parts but differing imaginary parts.

>               cplxpair (exp(2i*pi*[0:4]'/5)) == exp(2i*pi*[3; 2; 4; 1; 0]/5)

— Mapping Function: imag (z)

> Return the imaginary part of z as a real number.

> See also: real, conj.

— Mapping Function: real (z)

> Return the real part of z.

> See also: imag, conj.