I am trying to interpret some notation for functions on the extended plane involving the Riemann Sphere & complex plane but cannot find an explanation in the same form (Wikipedia & other websites mostly have different set of calculations). Note that in this case the south pole of the sphere is tangent to the plane.
$z = re^{i\theta} \neq 0$
So in the above, I realise that this states that $z$ on the complex plane is equal to the rest when not equal to 0.
But I am unsure about the rest:
- $r$ is the radius of the Riemann Sphere (1 for unit sphere)?
- $e$ is Euler's number or something else?
- $^{i\theta}$ is the angle from the equator? In radians or degrees?
There is another function which shares the notation but introduces nothing novel:
$w = (1/r)e^{-i\theta}$
The following webpage has similar notation although I was reading from Mathematics: Form & Function by Saunders Mac Lane (chapter 4, section 11, 'Stereographic Projection and Infinity').
http://mathfaculty.fullerton.edu/mathews/c2003/ComplexFunReciprocalMod.html
An explanation of the above would be great; examples including numbers would also be useful, I am learning from a basic level.
Thanks in advance!
Best Answer
$z=re^{i\theta}$ is a way to represent a complex number $z$ in terms of its polar coordinates: $r$ is the magnitude of $z$ and $\theta$ is the angle (in radians) between the real axis and the line that passes through $z$ and $0+i0$.
And yes, $e$ is just Euler's number: the exponential function $e^z$ is extended in the complex domain by defining it over the imaginary numbers as
$e^{i\alpha}=cos(\alpha)+isin(\alpha)$
To understand why, I suggest watching this video (if you have a basic understanding of exponentiation or derivatives. If you don't, then the channel is still a gold mine to get started)
In a way, $z=re^{i\theta}$ holds true even when it's equal to $0$, because the magnitude becomes zero; the only problem is that $\theta$ becomes undefined at that point.
Now, the second equation just means that $w$ is equal to $z^{-1}$ (its magnitude is inverted and the angle is opposite, and because the multiplication of two complex numbers yields a number which has the angle equal to the sum of the angles and the magnitude equal to the product of the magnitudes, if you multiply the two together you get $1$). Hope I helped clear things up, I wish you good luck in your learning process!