Theta functions in terms of the nome
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Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the nome q , where w = exp(πiz ) and q = exp(πi τ). In this form, the functions become
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{\displaystyle {\begin{aligned}\vartheta (w;q)&=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}\quad &\vartheta _{01}(w;q)&=\sum _{n=-\infty }^{\infty }(-1)^{n}(w^{2})^{n}q^{n^{2}}\\\vartheta _{10}(w;q)&=\sum _{n=-\infty }^{\infty }(w^{2})^{\left(n+{\frac {1}{2}}\right)}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w;q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}(w^{2})^{\left(n+{\frac {1}{2}}\right)}q^{\left(n+{\frac {1}{2}}\right)^{2}}\end{aligned}}}
So we see that the Theta functions can also be defined in terms of w and q , without reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers .
x ≈ y .
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{\displaystyle {\begin{aligned}h_{0}&=b_{0}&k_{0}&=1\\h_{1}&=b_{1}b_{0}+a_{1}&k_{1}&=b_{1}\\h_{i+1}&=b_{i+1}h_{i}+a_{i+1}h_{i-1}&k_{i+1}&=b_{i+1}k_{i}+a_{i+1}k_{i-1}\,\end{aligned}}}
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{\displaystyle f(z)=\sum _{n=1}^{\infty }\left(z^{2}+n\right)^{-2}.\,}
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{\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots \,}}}}}}}}}
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{\displaystyle z={\sqrt {a}}\,x+i{\sqrt {b}}\,y.}
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{\displaystyle \int _{a}^{b}x^{2}\,dx}
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{\displaystyle \varphi ={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\+{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\0&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}}
