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Theta function (Martinez)

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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 [[radian]]s. This function also utilizes [[Euler's formula]].
 
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 [[radian]]s. This function also utilizes [[Euler's formula]].
==Theta Function Theorem==
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==Theta Function Theorems==
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==='''First theorem'''===
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:''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.
 
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.
   
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:<math>\therefore \Theta (n, m) = 0 \iff m = 0 \square</math>
 
:<math>\therefore \Theta (n, m) = 0 \iff m = 0 \square</math>
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==='''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:
 
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:
 
:<math>\Xi = \Theta (n, m) = e^{n}m | n = a+bi, m = c+di, a=0</math>
 
:<math>\Xi = \Theta (n, m) = e^{n}m | n = a+bi, m = c+di, a=0</math>
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By listing out various values of ''e''<sup>''a''</sup>, he realized that it could never equal 0, and so ''e''<sup>''a''</sup>''m'' would always be nonzero if ''m'' ≠ 0. Having already proved that ''e''<sup>''bi''</sup> 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
 
By listing out various values of ''e''<sup>''a''</sup>, he realized that it could never equal 0, and so ''e''<sup>''a''</sup>''m'' would always be nonzero if ''m'' ≠ 0. Having already proved that ''e''<sup>''bi''</sup> 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
:<math>\Phi_{0} \iff \Theta (n, 0)</math>
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:<math>\Phi_{0} \iff \Theta (n, 0) \therefore \Theta (n, m) = 0 \iff m = 0 \square</math>
:<math>\therefore \Theta (n, m) = 0 \iff m = 0 \square</math>
 
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