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Homeproof of functional equation for the theta function

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# proof of functional equation for the theta function

All sums are over all integers unless otherwise specified. Thus the theta function is

$\theta(x)=\sum_{n}e^{{-\pi n^{2}x}}.$ |

Using the Jacobi’s identity for $\vartheta$ functions with $z=0$ and $\tau=i/x$, so that $-1/\tau=ix$ gives

$\theta_{3}(0\mid ix)=(1/x)^{{1/2}}\theta_{3}(0\mid i/x).$ |

Using the definition of $\theta_{3}$ we have that the left hand side is

$\sum_{{n}}e^{{-\pi xn^{2}}}=\theta(x)$ |

while the right hand side is

$(1/x)^{{1/2}}\sum_{{n}}e^{{i\pi(i/x)n^{2}}}$ |

which is

$(1/x)^{{1/2}}\sum_{{n}}e^{{-\pi n^{2}/x}}=\frac{1}{\sqrt{x}}\theta(1/x)$ |

so the identity is established.

The identity is attributed to Poisson by Jacobi [1]. Jacobi writes: M. Poisson, dans ses savantes recheches sur les intégrales définies, a fait connaître plusieurs propriétés de la fonction $\Theta(x)$. Les méthodes délicates, propres à cet illustre géomètre, trouvent une belle vérification dans la théorie des fonctions elliptiques. Par exemple, M. Poisson démontre dans dix-neuvième cahier du Journal de l’école polytechnique la formule remarquable:

$\sqrt{\frac{1}{x}}=\frac{1+2e^{{-\pi x}}+2e^{{-4\pi x}}+2e^{{-9\pi x}}+2e^{{-1% 6\pi x}}+\cdots}{1+2e^{{-\frac{\pi}{x}}}+2e^{{-\frac{4\pi}{x}}}+2e^{{-\frac{9% \pi}{x}}}+2e^{{-\frac{16\pi}{x}}}+\cdots}$ |

# References

- 1
M.C.G.J Jacobi,
*Notices sur Les Fonctions Elliptiques*, in Jacobi’s Gesammelte Werke, Band 1, Berlin, 1881, page 260.

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11M06*no label found*

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## Comments

## Entry is incomplete

Mathprof,

This entry is incomplete. It was left incomplete by the original author. Now you have adopted this entry and the entry has not changed a bit. Are you planning on improving it?

A

## Re: Entry is incomplete

yes, of course.

## Adopting entries (Re: Entry is incomplete)

> Alvaro said: This entry is incomplete. It was left incomplete by the original author. Now you have adopted this entry and the entry has not changed a bit. Are you planning on improving it?

> Mathprof replied: yes, of course.

I certainly hope so. I've realized that you are adopting a large number of those entries that go to adoption. As the system says "Mantaining an object is work!" (or similar). Even though I appreciate that you adopt the entries and take care of the pending corrections, many of these entries that go to adoption require a lot of effort to bring them up to shape. In many cases, the orphaned entries are in such state that they may require a total overhaul. And in all cases, the new owner should be knowledgeable about the topic of the entry (and hopefully an expert on that topic), and willing to improve it significantly. So, I would ask you to allow some reasonable time until you adopt an entry, in order to let others who may be more knowledgeable have a chance to adopt the entry. Others may be able to improve the entry in better ways.

Alvaro