Theta function (Martinez)

The theta function, as described by Gilbert Martinez on 29 Aug 2018, allows one to rotate a complex number by any angle. The nomenclature was chosen because mathematicians often use the Greek letter θ (theta) to denote angles. It's for a similar reason that the capital theta is used as the function's symbol.

$\Theta (n, m) = e^{ni}m = (\cos{n}+i\sin{n})(a+bi)$

As can be seen above, the theta function inputs two variables (n representing the angle of rotation and m representing the complex number to be rotated). Conventionally, n is defined in radians.

Theta Function Theorem

Further reading: Gil's Theorems

The same day he created this function, he proved what he called the theta function theorem. This theorem regards when Θ(n, m) equals 0.

$\Theta (n, m) = 0 \iff m = 0.$

Martinez proved this as follows:

Assume Θ(n, m) equals 0 and m ≠ 0. This would mean the following:

$\Theta (n, m) = e^{ni}m = 0.$

Since m is nonzero, that would mean eⁿⁱ = 0, since multiples of 0 must equal 0. Therefore, there must be a way to exponentiate any number to reach 0. However, this is not possible unless the exponentiated number is 0. Thus, e must be equal to 0.

However, this is not true.

Failed to parse (syntax error): {\displaystyle e = \lim_{n → \infty} (1 + \frac{1}{n})^{n} \neq 0}

Thus, eⁿⁱ ≠ 0. This implies that there is a way to get a product of 0 from two nonzero numbers. This is impossible.

$\therefore \Theta (n,m)=0\iff m=0.\square$