Is there a conventional term for the $e^{i\theta}$ thing in $z=\rho e^{i\theta}$?

Question

In the polar representation of a complex number, is there a conventional term for the $e^{i\theta}$ thing in $z=\rho e^{i\theta}$? I know $|z|=\rho$ is typically called the modulus, and $\theta$ is typically called the argument. $|z|\cos\theta=\Re z$ is called the real part, and $|z|\sin\theta=\Im z$ is called the imaginary part. But I don't recall there being any standard term for the expression $e^{i\theta}$ as a whole.

Answer 1

From linear algebra, I think it would be reasonable to call that factor the "direction", but I don't think that's common when working with the complex plane like this.

Answer 2

It has a bit of physics ring to it, but, at least in a mixed company (mingling with physics majors) I often call it the phase factor. Wikipedia seems to give me some support.

Maybe I got used to it when working in telcomm? They do waves! I think I used it before my stint already :-)

Answer 3

This is often referred to as cis (for cosine plus i sine) See Wolfram Mathworld, for example. Thus,

$$ \text{cis}(\theta)=e^{i\theta}=\cos\theta+i\sin\theta $$

Beyond that, $e^{i\theta}$ is literally an anticlockwise rotation of $\theta$ radians.