Theta function (Martinez)

The theta function, as initially described by Gilbert Martinez on 29 Aug 2018 and revised on 19 Sep 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.

The revised version is as follows:

$\Theta (n, m) = e^{n}m = e^{a+bi}(c+di) = e^{bi}[e^{a}(c+di)] = (\cos{b}+i\sin{b})[e^{a}(c+di)]$

It generalizes the original version of the function, which was as follows:

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

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. This function also utilizes Euler's formula.

Theta Function Theorem

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.

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

On 19 Sep 2018, he decided to tackle the general case of Θ(n, m), using the revised version of the theta function he created. To make things easier, he delineated two sets of Θ:

$\Xi = \Theta (n, m) = e^{n}m | n = a+bi, m = c+di, a=0$

$\Phi = \Theta (n, m) = e^{n}m | n = a+bi, m = c+di, a \neq 0$

He also defined Ξ₀ to mean "the value of set Ξ equals 0" and, likewise, Φ₀ to mean "the value of set Φ equals 0."

He quickly observed that the set Ξ is identical to having a point simply rotating around the origin, without any further multiplication involved. Having already proved this, he concluded that

$\Xi_{0} \iff \Theta (n, 0)$

Pondering Φ₀, he hypothesized that Φ₀ ⇔ Θ(n, 0). However, he needed a proof of this.

He then made a key observation.

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

He noted that the second term simply multiplies the complex number m by a power of e, which would result in either e^am > m or e^am < m if a ≠ 0 and m ≠ 0.

By listing out various values of e^a, he realized that it could never equal 0, and so e^am would always be nonzero if m ≠ 0. Having already proved that e^bi would always be nonzero, he used this to show that one could never rotate any nonzero complex point around the origin to 0. Thus, he showed that

$\Phi_{0} \iff \Theta (n, 0)$

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