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 Theorems

**First 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 *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.

- $ 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.

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

**Second theorem**

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 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

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

Pondering Φ_{0}, he hypothesized that Φ_{0} ⇔ 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*^{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

- $ \Phi_{0} \iff \Theta (n, 0) \therefore \Theta (n, m) = 0 \iff m = 0 \square $