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:

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

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[edit | edit source]

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.

Martinez proved this as follows:

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

Since *m* is nonzero, that would mean *e*^{ni} = 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = \lim_{n → \infty} (1 + \frac{1}{n})^{n} \neq 0}**

Thus, *e*^{ni} ≠ 0. This implies that there is a way to get a product of 0 from two nonzero numbers. This is impossible.

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:

He also defined Ξ_{0} to mean "the value of set Ξ equals 0" and, likewise, Φ_{0} 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

Pondering Φ_{0}, he hypothesized that Φ_{0} ⇔ Θ(*n*, 0). However, he needed a proof of this.

He then made a key observation.

He noted that the second term simply multiplies the complex number *m* by a power of *e*, which would result in either *e*^{a}*m* > *m* or *e*^{a}*m* < *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*^{a}*m* 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