Editing Theta function (section)

==Jacobi theta function==

There are several closely related functions called Jacobi theta functions, and [[Jacobi theta functions (notational variations)|many different and incompatible systems of notation]] for them. 
One '''Jacobi theta function''' (named after [[Carl Gustav Jacob Jacobi]]) is a function defined for two complex variables {{mvar|z}} and {{mvar|τ}}, where {{mvar|z}} can be any [[complex number]] and {{mvar|τ}} is the [[half-period ratio]], confined to the [[upper half-plane]], which means it has a positive imaginary part. It is given by the formula

:<math>\begin{align} 
\vartheta(z; \tau) &= \sum_{n=-\infty}^\infty \exp \left(\pi i n^2 \tau + 2 \pi i n z\right) \\
&= 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos(2\pi n z) \\
&= \sum_{n=-\infty}^\infty q^{n^2}\eta^n 
\end{align}</math>

where {{math|''q'' {{=}} exp(''πiτ'')}} is the [[nome (mathematics)|nome]] and {{math|''η'' {{=}} exp(2''πiz'')}}. It is a [[Jacobi form]]. The restriction ensures that it is an absolutely convergent series. At fixed {{mvar|τ}}, this is a [[Fourier series]] for a 1-periodic [[entire function]] of {{mvar|z}}. Accordingly, the theta function is 1-periodic in {{mvar|z}}:

:<math>\vartheta(z+1; \tau) = \vartheta(z; \tau).</math>

By [[completing the square]], it is also {{mvar|τ}}-quasiperiodic in {{mvar|z}}, with

:<math>\vartheta(z+\tau;\tau) = \exp\bigl(-\pi i (\tau + 2 z)\bigr) \vartheta(z;\tau).</math>

Thus, in general,

:<math>\vartheta(z+a+b\tau;\tau) = \exp\left(-\pi i b^2 \tau -2 \pi i b z\right) \vartheta(z;\tau)</math>

for any integers {{mvar|a}} and {{mvar|b}}.

For any fixed <math>\tau </math>, the function is an entire function on the complex plane, so by [[Liouville's theorem (complex analysis)|Liouville's theorem]], it cannot be doubly periodic in <math>1, \tau </math> unless it is constant, and so the best we can do is to make it periodic in <math>1 </math> and quasi-periodic in <math>\tau </math>. Indeed, since <math display="block">\left|\frac{\vartheta(z+a+b\tau;\tau)}{\vartheta(z;\tau)}\right| = \exp\left(\pi (b^2 \Im(\tau) + 2b \Im(z)) \right) </math>and <math>\Im(\tau)> 0 </math>, the function <math>\vartheta(z, \tau) </math> is unbounded, as required by Liouville's theorem.

It is in fact the most general entire function with 2 quasi-periods, in the following sense:<ref>{{Cite book |url=http://link.springer.com/10.1007/978-0-8176-4577-9 |title=Tata Lectures on Theta I |date=2007 |publisher=Birkhäuser Boston |isbn=978-0-8176-4572-4 |series=Modern Birkhäuser Classics |location=Boston, MA |pages=4 |language=en |doi=10.1007/978-0-8176-4577-9 }}</ref>
{{Math theorem
| math_statement = If <math>f: \mathbb C \to \mathbb C</math> is entire and nonconstant, and satisfies the functional equations
<math>
\begin{cases}
f(z+1) = f(z)\\
f(z + \tau) = e^{az + 2\pi i b} f(z)
\end{cases}
</math>
for some constant <math>a, b\in \mathbb C</math>.

If <math>a = 0</math>, then <math>b = \tau</math> and <math>f(z) = e^{2\pi i z}</math>. If <math>a = -2\pi i</math>, then <math>f(z) = C \vartheta(z + \frac 1 2 \tau + b, \tau)</math> for some nonzero <math>C\in \mathbb C</math>.
}}[[Image:Complex theta animated1.gif|500px|thumb|center|Theta function {{math|''θ''<sub>1</sub>}} with different nome {{math|''q'' {{=}} ''e''<sup>''iπτ''</sup>}}. The black dot in the right-hand picture indicates how {{mvar|q}} changes with {{mvar|τ}}.]]
[[Image:Complex theta animated2.gif|500px|thumb|center|Theta function {{math|''θ''<sub>1</sub>}} with different nome {{math|''q'' {{=}} ''e''<sup>''iπτ''</sup>}}. The black dot in the right-hand picture indicates how {{mvar|q}} changes with {{mvar|τ}}.]]