Editing Theta function (section)

==Generalizations==

If {{mvar|F}} is a [[quadratic form]] in {{mvar|n}} variables, then the theta function associated with {{mvar|F}} is

:<math>\theta_F (z)= \sum_{m\in \Z^n} e^{2\pi izF(m)}</math>

with the sum extending over the [[lattice (group)|lattice]] of integers <math>\mathbb{Z}^n</math>. This theta function is a [[modular form]] of weight {{math|{{sfrac|''n''|2}}}} (on an appropriately defined subgroup) of the [[modular group]].  In the Fourier expansion,

:<math>\hat{\theta}_F (z) = \sum_{k=0}^\infty R_F(k) e^{2\pi ikz},</math>

the numbers {{math|''R<sub>F</sub>''(''k'')}} are called the ''representation numbers'' of the form.

===Theta series of a Dirichlet character===

For {{mvar|χ}} a primitive [[Dirichlet character]] modulo {{mvar|q}} and {{math|''ν'' {{=}} {{sfrac|1 − ''χ''(−1)|2}}}} then

:<math>\theta_\chi(z) = \frac12\sum_{n=-\infty}^\infty \chi(n) n^\nu e^{2i \pi n^2 z}</math>

is a weight {{math|{{sfrac|1|2}} + ''ν''}} modular form of level {{math|4''q''<sup>2</sup>}} and character
:<math>\chi(d) \left(\frac{-1}{d}\right)^\nu,</math>
which means<ref>Shimura, On modular forms of half integral weight</ref>

:<math>\theta_\chi\left(\frac{az+b}{cz+d}\right) = \chi(d) \left(\frac{-1}{d}\right)^\nu \left(\frac{\theta_1\left(\frac{az+b}{cz+d}\right)}{\theta_1(z)}\right)^{1+2\nu}\theta_\chi(z)</math>

whenever

:<math>a,b,c,d\in \Z^4, ad-bc=1,c \equiv 0 \bmod 4 q^2.</math>

===Ramanujan theta function===
{{further|Ramanujan theta function|mock theta function}}

===Riemann theta function===
Let

:<math>\mathbb{H}_n=\left\{F\in M(n,\Complex) \,\big|\, F=F^\mathsf{T} \,,\, \operatorname{Im} F >0 \right\}</math>

be the set of [[symmetric]] square [[matrix (mathematics)|matrices]] whose imaginary part is [[Positive-definite matrix|positive definite]]. <math>\mathbb{H}_n</math> is called the [[Siegel upper half-space]] and is the multi-dimensional analog of the [[upper half-plane]]. The {{mvar|n}}-dimensional analogue of the [[modular group]] is the [[symplectic group]] {{math|Sp(2''n'',<math>\mathbb{Z}</math>)}}; for {{math|''n'' {{=}} 1}}, {{math|Sp(2,<math>\mathbb{Z}</math>) {{=}} SL(2,<math>\mathbb{Z}</math>)}}. The {{mvar|n}}-dimensional analogue of the [[congruence subgroup]]s is played by

:<math>\ker \big\{\operatorname{Sp}(2n,\Z)\to \operatorname{Sp}(2n,\Z/k\Z) \big\}.</math>

Then, given {{math|''τ'' ∈ <math>\mathbb{H}_n</math>}}, the '''Riemann theta function''' is defined as

:<math>\theta (z,\tau)=\sum_{m\in \Z^n} \exp\left(2\pi i \left(\tfrac12 m^\mathsf{T} \tau m +m^\mathsf{T} z \right)\right). </math>

Here, {{math|''z'' ∈ <math>\mathbb{C}^n</math>}} is an {{mvar|n}}-dimensional complex vector, and the superscript '''T''' denotes the [[transpose]]. The Jacobi theta function is then a special case, with {{math|''n'' {{=}} 1}} and {{math|''τ'' ∈ <math>\mathbb{H}</math>}} where {{math|<math>\mathbb{H}</math>}} is the [[upper half-plane]]. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact [[Riemann surface]]s, as well as other auxiliary objects that figure prominently in their function theory, by taking {{mvar|τ}} to be the period matrix with respect to a canonical basis for its first [[Homology (mathematics)|homology group]].

The Riemann theta converges absolutely and uniformly on compact subsets of <math>\mathbb{C}^n \times \mathbb{H}_n</math>.

The functional equation is

:<math>\theta (z+a+\tau b, \tau) = \exp\left( 2\pi i \left(-b^\mathsf{T}z-\tfrac12 b^\mathsf{T}\tau b\right)\right) \theta (z,\tau)</math>

which holds for all vectors {{math|''a'', ''b'' ∈ <math>\mathbb{Z}^n</math>}}, and for all {{math|''z'' ∈ <math>\mathbb{C}^n</math>}} and {{math|''τ'' ∈ <math>\mathbb{H}_n</math>}}.

===Poincaré series===
The [[Poincaré series (modular form)|Poincaré series]] generalizes the theta series to automorphic forms with respect to arbitrary [[Fuchsian group]]s.