Editing Theta function (section)

==Nome power theorems==

===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>

=== Transformation at the cube root of the nome ===
The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:

: <math>\biggl[\frac{\theta_{3}(q^{1/3})^2}{\theta_{3}(q)^2} - \frac{3\,\theta_{3}(q^{3})^2}{\theta_{3}(q)^2}\biggr]^2 = 4 - 4\biggl[\frac{2\,\theta_{2}(q)^2 \theta_{4}(q)^2}{\theta_{3}(q)^4}\biggr]^{2/3} </math>

: <math>\biggl[\frac{3\,\theta_{4}(q^{3})^2}{\theta_{4}(q)^2} - \frac{\theta_{4}(q^{1/3})^2}{\theta_{4}(q)^2}\biggr]^2 = 4 + 4\biggl[\frac{2\,\theta_{2}(q)^2 \theta_{3}(q)^2}{\theta_{4}(q)^4}\biggr]^{2/3} </math>

=== Transformation at the fifth root of the nome ===
The [[Rogers-Ramanujan continued fraction]] can be defined in terms of the '''Jacobi theta function''' in the following way:

: <math>R(q) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta _{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} </math>

: <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{2/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{1}{2} - \frac{\theta_{4}(q)^2}{2\,\theta_{4}(q^5)^2}\biggr]\biggr\}^{1/5} </math>

: <math>R(q^2) = \tan\biggl\{\frac{1}{2}\arctan\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5} \tan\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} </math>

The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:

: <math>S(q) = \frac{R(q^4)}{R(q^2)R(q)} = \tan\biggl\{\frac{1}{2}\arctan\biggl [\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{1/5} \cot\biggl\{\frac{1}{2}\arccot\biggl[\frac{\theta_{3}(q)^2}{2\,\theta_{3}(q^5)^2} - \frac{1}{2}\biggr]\biggr\}^{2/5}</math>

The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:

: <math>\frac{\theta_{3}(q^{1/5})}{\theta_{3}(q^5)} - 1 = \frac{1}{S(q)}\bigl[S(q)^2 + R(q^2)\bigr]\bigl[1 + R(q^2)S(q)\bigr] </math>
: <math>1 - \frac{\theta_{4}(q^{1/5})}{\theta_{4}(q^5)} = \frac{1}{R(q)}\bigl[R(q^2) + R(q)^2\bigr]\bigl[1 - R(q^2)R(q)\bigr] </math>
: <math>\theta_{3}(q^{1/5})^2 - \theta_{3}(q)^2 = \bigl[\theta_{3}(q)^2 - \theta_{3}(q^5)^2\bigr]\biggl[1+\frac{1}{R(q^2)S(q)}+R(q^2)S(q)+\frac{1}{R(q^2)^2}+R(q^2)^2+\frac{1}{S(q)}-S(q)\biggr] </math>
: <math>\theta_{4}(q)^2 - \theta_{4}(q^{1/5})^2 = \bigl[\theta_{4}(q^5)^2 - \theta_{4}(q)^2\bigr]\biggl[1-\frac{1}{R(q^2)R(q)}-R(q^2)R(q)+\frac{1}{R(q^2)^2}+R(q^2)^2-\frac{1}{R(q)}+R(q)\biggr] </math>

=== Modulus dependent theorems===

In combination with the elliptic modulus, the following formulas can be displayed:

These are the formulas for the square of the elliptic nome:

:<math>\theta_{4}[q(k)] = \theta_{4}[q(k)^2]\sqrt[8]{1 - k^2}</math>
:<math>\theta_{4}[q(k)^2] = \theta_{3}[q(k)]\sqrt[8]{1 - k^2}</math>
:<math>\theta_{3}[q(k)^2] = \theta_{3}[q(k)]\cos[\tfrac{1}{2}\arcsin(k)]</math>

And this is an efficient formula for the cube of the nome:

:<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(t^3)\bigr]\bigr\}^3 \biggr\rangle = 
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(t^3)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{2\sqrt{t^4 - t^2 + 1} - t^2 + 2} + \sqrt{t^2 + 1}\,\bigr)^{1/2} </math>

For all real values <math> t \in \R </math> the now mentioned formula is valid.

And for this formula two examples shall be given:

First calculation example with the value <math> t = 1 </math> inserted:

:{| class="wikitable"
|
<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\}^3 \biggr\rangle = 
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2} </math>
|-
|
<math> \theta_{4}\bigl[\exp(-3\sqrt{2}\,\pi)\bigr] = 
\theta_{4}\bigl[\exp(-\sqrt{2}\,\pi)\bigr] \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2} </math>
|}

Second calculation example with the value <math> t = \Phi^{-2} </math> inserted:

:{| class="wikitable"
|
<math> \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(\Phi^{-6})\bigr]\bigr\}^3 \biggr\rangle = 
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(\Phi^{-6})\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{2\sqrt{\Phi^{-8} - \Phi^{-4} + 1} - \Phi^{-4} + 2} + \sqrt{\Phi^{-4} + 1}\,\bigr)^{1/2} </math>
|-
|
<math> \theta_{4}\bigl[\exp(-3\sqrt{10}\,\pi)\bigr] = 
\theta_{4}\bigl[\exp(-\sqrt{10}\,\pi)\bigr] \,3^{-1/2} \bigl(\sqrt{2\sqrt{\Phi^{-8} - \Phi^{-4} + 1} - \Phi^{-4} + 2} + \sqrt{\Phi^{-4} + 1}\,\bigr)^{1/2} </math>
|}

The constant <math> \Phi </math> represents the [[Golden ratio]] number <math> \Phi = \tfrac{1}{2}(\sqrt{5} + 1)</math> exactly.