Editing Theta function (section)

===Direct power theorems===

For the transformation of the nome<ref>Andreas Dieckmann: ''[http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html#InfinitexProducts Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta.]'' Physikalisches Institut Universität Bonn, Abruf am 1. Oktober 2021.</ref> in the theta functions these formulas can be used:

:<math>\theta_{2}(q^2) = \tfrac{1}{2}\sqrt{2[\theta_{3}(q)^2 - \theta_{4}(q)^2]}</math>
:<math>\theta_{3}(q^2) = \tfrac{1}{2}\sqrt{2[\theta_{3}(q)^2 + \theta_{4}(q)^2]}</math>
:<math>\theta_{4}(q^2) = \sqrt{\theta_{4}(q)\theta_{3}(q)}</math>

The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the [[Pythagorean triple]]s according to the Jacobi Identity. Furthermore, those transformations are valid:

:<math>\theta_{3}(q^4) = \tfrac{1}{2}\theta_{3}(q) + \tfrac{1}{2}\theta_{4}(q)</math>

These formulas can be used to compute the theta values of the cube of the nome:

:<math>27\,\theta_{3}(q^3)^8 - 18\,\theta_{3}(q^3)^4\theta_{3}(q)^4 - \,\theta_{3}(q)^8 = 8\,\theta_{3}(q^3)^2\theta_{3}(q)^2[2\,\theta_{4}(q)^4 - \theta_{3}(q)^4]</math>

:<math>27\,\theta_{4}(q^3)^8 - 18\,\theta_{4}(q^3)^4\theta_{4}(q)^4 - \,\theta_{4}(q)^8 = 8\,\theta_{4}(q^3)^2\theta_{4}(q)^2[2\,\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>

And the following formulas can be used to compute the theta values of the fifth power of the nome:

:<math>[\theta_{3}(q)^2 - \theta_{3}(q^5)^2][5\,\theta_{3}(q^5)^2 - \theta_{3}(q)^2]^5 = 256\,\theta_{3}(q^5)^2\theta_{3}(q)^2\theta_{4}(q)^4 [\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>

:<math>[\theta_{4}(q^5)^2 - \theta_{4}(q)^2][5\,\theta_{4}(q^5)^2 - \theta_{4}(q)^2]^5 = 256\,\theta_{4}(q^5)^2\theta_{4}(q)^2\theta_{3}(q)^4 [\theta_{3}(q)^4 - \theta_{4}(q)^4]</math>