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| {{for|other θ functions|Theta function (disambiguation)}}
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| [[image:Complex theta minus0point1times e i pi 0point1.jpg|400px|thumb|upright=1.2|Jacobi's original theta function <math> \theta_1 </math> with <math> u = i \pi z</math> and with nome <math> q = e^{i \pi \tau}= 0.1 e^{0.1 i \pi}</math>. Conventions are (Mathematica):<math> \theta_1(u;q) = 2 q^{1/4} \sum_{n=0}^\infty (-1)^n q^{n(n+1)} \sin(2n+1)u </math> this is:
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| <math>\theta_1(u;q) = \sum_{n=-\infty}^{n=\infty} (-1)^{n-1/2} q^{(n+1/2)^2} e^{(2n+1)i u} </math>]]
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| In [[mathematics]], '''theta functions''' are [[special function]]s of [[several complex variables]]. They are important in many areas, including the theories of [[abelian variety|abelian varieties]] and [[moduli space]]s, and of [[quadratic form]]s. They have also been applied to [[soliton]] theory. When generalized to a [[Grassmann algebra]], they also appear in [[quantum field theory]].
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| The most common form of theta function is that occurring in the theory of [[elliptic function]]s. With respect to one of the complex variables (conventionally called ''z''), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a [[quasiperiodic function]]. In the abstract theory this comes from a [[line bundle]] condition of [[descent (category theory)|descent]].
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| ==Jacobi theta function==
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| There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them.
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| One '''Jacobi theta function''' (named after [[Carl Gustav Jacob Jacobi]]) is a function defined for two complex variables ''z'' and τ, where ''z'' can be any complex number and τ is confined to the [[upper half-plane]], which means it has positive imaginary part. It is given by the formula
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| :<math>
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| \vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)
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| = 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = \sum_{n=-\infty}^\infty q^{n^2}\eta^n
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| </math>
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| where ''q'' = exp(π''i''τ) and η = exp(2π''iz''). It is a [[Jacobi form]].
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| If τ is fixed, this becomes a [[Fourier series]] for a periodic [[entire function]] of ''z'' with period 1; in this case, the theta function satisfies the identity
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| :<math>\vartheta(z+1; \tau) = \vartheta(z; \tau).</math>
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| The function also behaves very regularly with respect to its quasi-period τ and satisfies the functional equation
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| :<math>\vartheta(z+a+b\tau;\tau) = \exp(-\pi i b^2 \tau -2 \pi i b z)\,\vartheta(z;\tau)</math>
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| where ''a'' and ''b'' are integers.
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| [[image:Complex theta animated1.gif|500px|thumb|center|Theta function <math>\theta_1 </math> with different nome <math>q = e^{i \pi \tau}</math>. The black dot in the right-hand picture indicates how <math>\tau</math> is changing.]]
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| [[image:Complex theta animated2.gif|500px|thumb|center|Theta function <math>\theta_1 </math> with different nome <math>q = e^{i \pi \tau}</math>. The black dot in the right-hand picture indicates how <math>\tau</math> is changing.]]
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| ==Auxiliary functions==
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| The Jacobi theta function defined above is sometimes considered along with three
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| auxiliary theta functions, in which case it is written with a double 0 subscript:
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| :<math>\vartheta_{00}(z;\tau) = \vartheta(z;\tau)</math>
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| The auxiliary (or half-period) functions are defined by
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| :<math>
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| \begin{align}
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| \vartheta_{01}(z;\tau)& = \vartheta\!\left(z+{\textstyle\frac{1}{2}};\tau\right)\\[3pt]
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| \vartheta_{10}(z;\tau)& = \exp\!\left({\textstyle\frac{1}{4}}\pi i \tau + \pi i z\right)
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| \vartheta\!\left(z + {\textstyle\frac{1}{2}}\tau;\tau\right)\\[3pt]
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| \vartheta_{11}(z;\tau)& = \exp\!\left({\textstyle\frac{1}{4}}\pi i \tau + \pi i\!\left(z+{\textstyle
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| \frac{1}{2}}\right)\right)\vartheta\!\left(z+{\textstyle\frac{1}{2}}\tau + {\textstyle\frac{1}{2}};\tau\right).
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| \end{align}
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| </math>
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| This notation follows [[Bernhard Riemann|Riemann]] and [[David Mumford|Mumford]]; [[Carl Gustav Jacobi|Jacobi]]'s original formulation was in terms of the [[nome (mathematics)|nome]] ''q'' = exp(''πiτ'') rather than τ. In Jacobi's notation the θ-functions are written:
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| :<math>
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| \begin{align}
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| \theta_1(z;q) &= -\vartheta_{11}(z;\tau)\\
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| \theta_2(z;q) &= \vartheta_{10}(z;\tau)\\
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| \theta_3(z;q) &= \vartheta_{00}(z;\tau)\\
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| \theta_4(z;q) &= \vartheta_{01}(z;\tau)
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| \end{align}
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| </math>
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| The above definitions of the Jacobi theta functions are by no means unique. See [[Jacobi theta functions – notational variations]] for further discussion.
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| If we set ''z'' = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane (sometimes called theta constants.) These can be used to define a variety of [[modular forms]], and to parametrize certain curves; in particular, the '''Jacobi identity''' is
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| :<math>
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| \vartheta_{00}(0;\tau)^4 = \vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4
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| </math>
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| which is the [[Fermat's last theorem|Fermat curve]] of degree four.
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| ==Jacobi identities==
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| Jacobi's identities describe how theta functions transform under the [[modular group]], which is generated by τ ↦ τ+1 and τ ↦ -1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (n is congruent to n squared modulo 2). For the second, let
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| :<math> | |
| \alpha = (-i \tau)^{\frac{1}{2}} \exp\!\left(\frac{\pi}{\tau} i z^2 \right).\,
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| </math>
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| Then
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| :<math>
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| \begin{align}
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| \vartheta_{00}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = \alpha\,\vartheta_{00}(z; \tau)\quad&
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| \vartheta_{01}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = \alpha\,\vartheta_{10}(z; \tau)\\[3pt]
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| \vartheta_{10}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = \alpha\,\vartheta_{01}(z; \tau)\quad&
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| \vartheta_{11}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = -i\alpha\,\vartheta_{11}(z; \tau).
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| \end{align}
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| </math>
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| ==Theta functions in terms of the nome==
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| Instead of expressing the Theta functions in terms of <math>z\,</math> and <math>\tau \,</math>, we may express them in terms of arguments <math>w\,</math> and the [[nome (mathematics)|nome]] q, where <math>w=e^{\pi {\mathrm{i}}z}\,</math> and <math>q=e^{\pi {\mathrm{i}}\tau}\,</math> . In this form, the functions become
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| :<math>
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| \begin{align}
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| \vartheta_{00}(w, q)& = \sum_{n=-\infty}^\infty (w^2)^n q^{n^2}\quad&
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| \vartheta_{01}(w, q)& = \sum_{n=-\infty}^\infty (-1)^n (w^2)^n q^{n^2}\\[3pt]
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| \vartheta_{10}(w, q)& = \sum_{n=-\infty}^\infty (w^2)^{\left(n+1/2\right)}
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| q^{\left(n + 1/2\right)^2}\quad&
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| \vartheta_{11}(w, q)& = i \sum_{n=-\infty}^\infty (-1)^n (w^2)^{\left(n+1/2\right)}
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| q^{\left(n + 1/2\right)^2}.
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| \end{align}
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| </math>
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| We see that the Theta functions can also be defined in terms of ''w'' and ''q'', without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other [[field (mathematics)|fields]] where the exponential function might not be everywhere defined, such as fields of [[p-adic number]]s.
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| ==Product representations==
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| The [[Jacobi triple product]] tells us that for complex numbers ''w'' and ''q'' with |''q''| < 1 and ''w'' ≠ 0 we have
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| :<math>\prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 + w^{2}q^{2m-1}\right)
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| \left( 1 + w^{-2}q^{2m-1}\right)
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| = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}.
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| </math>
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| It can be proven by elementary means, as for instance in Hardy and Wright's ''An Introduction to the Theory of Numbers''.
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| If we express the theta function in terms of the nome <math>q = \exp(\pi i \tau)</math> and <math>w = \exp(\pi i z)</math> then
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| :<math>\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp(\pi i \tau n^2) \exp(\pi i z 2n) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}. </math>
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| We therefore obtain a product formula for the theta function in the form
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| :<math>\vartheta(z; \tau) = \prod_{m=1}^\infty
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| \left( 1 - \exp(2m \pi i \tau)\right)
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| \left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right)
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| \left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right).
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| </math>
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| In terms of ''w'' and ''q'':
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| :<math>\vartheta(z; \tau) = \prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 + q^{2m-1}w^2\right)
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| \left( 1 + q^{2m-1}/w^2\right)
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| </math>
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| :<math> = (q^2;q^2)_\infty\,(-w^2q;q^2)_\infty\,(-q/w^2;q^2)_\infty </math>
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| :<math> = (q^2;q^2)_\infty\,\theta(-w^2q;q^2)</math>
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| where <math>(\cdot \cdot)_\infty</math> is the [[q-Pochhammer symbol]] and <math>\theta(\cdot \cdot)</math> is the [[q-theta function]].
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| Expanding terms out, the Jacobi triple product can also be written
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| :<math>\prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 + (w^{2}+w^{-2})q^{2m-1}+q^{4m-2}\right),</math>
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| which we may also write as
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| :<math>\vartheta(z|q) = \prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).</math>
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| This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are
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| :<math>\vartheta_{01}(z|q) = \prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).</math>
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| :<math>\vartheta_{10}(z|q) = 2 q^{1/4}\cos(\pi z)\prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).</math>
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| :<math>\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty
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| \left( 1 - q^{2m}\right)
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| \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).</math>
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| ==Integral representations==
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| The Jacobi theta functions have the following integral representations:
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| :<math>\vartheta_{00} (z; \tau) = -i
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| \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2}
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| \cos (2 u z + \pi u) \over \sin (\pi u)} du</math>
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| :<math>\vartheta_{01} (z; \tau) = -i
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| \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2}
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| \cos (2 u z) \over \sin (\pi u)} du.</math>
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| :<math>\vartheta_{10} (z; \tau) = -i e^{iz + i \pi \tau / 4}
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| \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2}
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| \cos (2 u z + \pi u + \pi \tau u) \over \sin (\pi u)} du</math>
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| :<math>\vartheta_{11} (z; \tau) = e^{iz + i \pi \tau / 4}
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| \int_{i - \infty}^{i + \infty} {e^{i \pi \tau u^2}
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| \cos (2 u z + \pi \tau u) \over \sin (\pi u)} du</math>
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| ==Explicit values==
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| See <ref name="Yi">{{Citation
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| | last = Jinhee
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| | first = Yi
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| | coauthors =
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| | title = Theta-function identities and the explicit formulas for theta-function and their applications
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| | journal = Journal of Mathematical Analysis and Applications
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| | pages = 381–400
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| | volume = 292
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| | doi=10.1016/j.jmaa.2003.12.0091
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| | year = 2004
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| | postscript = .}}
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| </ref>
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| :<math>
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| \varphi(e^{-\pi x}) = \vartheta(0; {\mathrm{i}}x) = \theta_3(0;e^{-\pi x}) = \sum_{n=-\infty}^\infty e^{-x \pi n^2}
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| </math>
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| :<math>
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| \varphi\left(e^{-\pi} \right) = \frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}
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| </math>
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| :<math>
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| \varphi\left(e^{-2\pi} \right) = \frac{\sqrt[4]{6\pi+4\sqrt2\pi}}{2\Gamma(\frac{3}{4})}
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| </math>
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| :<math>
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| \varphi\left(e^{-3\pi}\right) = \frac{\sqrt[4]{27\pi+18\sqrt3\pi}}{3\Gamma(\frac{3}{4})}
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| </math>
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| :<math>
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| \varphi\left(e^{-4\pi}\right) =\frac{\sqrt[4]{8\pi}+2\sqrt[4]{\pi}}{4\Gamma(\frac{3}{4})}
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| </math>
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| :<math>
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| \varphi\left(e^{-5\pi} \right) =\frac{\sqrt[4]{225\pi+ 100\sqrt5 \pi}}{5\Gamma(\frac{3}{4})}
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| </math>
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| :<math>
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| \varphi\left(e^{-6\pi}\right) = \frac{\sqrt[3]{3\sqrt{2}+3\sqrt[4]{3}+2\sqrt{3}-\sqrt[4]{27}+\sqrt[4]{1728}-4}\cdot \sqrt[8]{243{\pi}^2}}{6\sqrt[6]{1+\sqrt6-\sqrt2-\sqrt3}{\Gamma(\frac{3}{4})}}
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| </math>
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| ==Some series identities==
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| The next two series identities were proved by István Mező
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| :<ref name="Mezo2">{{Citation
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| | last = Mező
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| | first = István
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| | title = Duplication formulae involving Jacobi theta functions and Gosper's ''q''-trigonometric functions
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| | journal = Proceedings of the American Mathematical Society
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| | pages = 2401–2410
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| | volume = 141
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| | issue = 7
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| | year = 2013
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| }}</ref>
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| : <math>
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| \vartheta_4^2(q)=iq^{\frac14}\sum_{k=-\infty}^\infty q^{2k^2-k}\vartheta_1\left(\frac{2k-1}{2i}\ln q,q\right),
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| </math>
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| : <math>
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| \vartheta_4^2(q)=\sum_{k=-\infty}^\infty q^{2k^2}\vartheta_4\left(\frac{k\ln q}{i},q\right).
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| </math>
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| These relations hold for all 0 < ''q'' < 1. Specializing the values of ''q'', we have the next parameter free sums
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| :<math>
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| \sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma^2\left(\frac34\right)}=i\sum_{k=-\infty}^\infty e^{\pi(k-2k^2)}\vartheta_1\left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right),
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| </math>
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| and
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| :<math>
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| \sqrt{\frac{\pi}{2}}\frac{1}{\Gamma^2\left(\frac34\right)}=\sum_{k=-\infty}^\infty\frac{\vartheta_4(ik\pi,e^{-\pi})}{e^{2\pi k^2}}
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| </math>
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| ==Zeros of the Jacobi theta functions==
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| All zeros of the Jacobi theta functions are simple zeros and are given by the following:
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| :<math> \vartheta(z,\tau) = \vartheta_3(z,\tau) = 0 \quad \Longleftrightarrow \quad z = m + n \tau + \frac{1}{2} + \frac{\tau}{2} </math>
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| :<math> \vartheta_1(z,\tau) = 0 \quad \Longleftrightarrow \quad z = m + n \tau </math>
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| :<math> \vartheta_2(z,\tau) = 0 \quad \Longleftrightarrow \quad z = m + n \tau + \frac{1}{2} </math>
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| :<math> \vartheta_4(z,\tau) = 0 \quad \Longleftrightarrow \quad z = m + n \tau + \frac{\tau}{2} </math>
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| where ''m,n'' are arbitrary integers.
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| ==Relation to the Riemann zeta function==
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| The relation
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| :<math>\vartheta(0;-1/\tau)=(-i\tau)^{1/2} \vartheta(0;\tau)</math>
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| was used by [[Riemann]] to prove the functional equation for the [[Riemann zeta function]], by means of the integral
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| :<math>\Gamma\left(\frac{s}{2}\right) \pi^{-s/2} \zeta(s) =
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| \frac{1}{2}\int_0^\infty\left[\vartheta(0;it)-1\right]
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| t^{s/2}\frac{dt}{t}</math>
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| which can be shown to be invariant under substitution of ''s'' by 1 − ''s''. The corresponding integral for ''z'' not zero is given in the article on the [[Hurwitz zeta function]].
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| ==Relation to the Weierstrass elliptic function==
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| The theta function was used by Jacobi to construct (in a form adapted to easy calculation) [[Jacobi's elliptic functions|his elliptic functions]] as the quotients of the above four theta functions, and could have been used by him to construct [[Weierstrass's elliptic functions]] also, since
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| :<math>\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c</math>
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| where the second derivative is with respect to z and the constant c is defined so that the [[Laurent expansion]] of <math>\wp(z)</math> at ''z'' = 0 has zero constant term.
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| ==Relation to the ''q''-gamma function==
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| The fourth theta function – and thus the others too – is intimately connected to the [[q-gamma function|Jackson ''q''-gamma function]] via the relation<ref name = 'Mezo'>{{Cite journal| last1=Mező | first1=István | title=A q-Raabe formula and an integral of the fourth Jacobi theta function | year=2012 | journal=Journal of Number Theory | volume=130 | issue=2 | pages=360–369}}</ref>
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| : <math> \left(\Gamma_{q^2}(x)\Gamma_{q^2}(1-x)\right)^{-1}=\frac{q^{2x(1-x)}}{(q^{-2};q^{-2})^3_\infty(q^2-1)}\vartheta_4\left(\frac{1}{2i}(1-2x)\log q,\frac{1}{q}\right). </math>
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| ==Some relations to modular forms==
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| Let η be the [[Dedekind eta function]]. Then
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| :<math>\vartheta(0;\tau)=\frac{\eta^2\left(\tfrac{1}{2}(\tau+1)\right)}{\eta(\tau+1)}.</math>
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| ==A solution to heat equation==
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| The Jacobi theta function is the unique solution to the one-dimensional [[heat equation]] with periodic boundary conditions at time zero. This is most easily seen by taking ''z'' = ''x'' to be real, and taking τ = ''it'' with ''t'' real and positive. Then we can write
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| :<math>\vartheta (x,it)=1+2\sum_{n=1}^\infty \exp(-\pi n^2 t) \cos(2\pi nx)</math>
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| which solves the heat equation
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| :<math>\frac{\partial}{\partial t} \vartheta(x,it)=\frac{1}{4\pi} \frac{\partial^2}{\partial x^2} \vartheta(x,it).</math>
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| That this solution is unique can be seen by noting that at ''t'' = 0, the theta function becomes the [[Dirac comb]]:
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| :<math>\lim_{t\rightarrow 0} \vartheta(x,it)=\sum_{n=-\infty}^\infty \delta(x-n)</math>
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| where δ is the [[Dirac delta function]]. Thus, general solutions can be specified by convolving the (periodic) boundary condition at ''t'' = 0 with the theta function.
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| ==Relation to the Heisenberg group==
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| The Jacobi theta function is invariant under the action of a discrete subgroup of the [[Heisenberg group]]. This invariance is presented in the article on the [[theta representation]] of the Heisenberg group.
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| ==Generalizations==
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| If ''F'' is a [[quadratic form]] in ''n'' variables, then the theta function associated with ''F'' is
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| :<math>\theta_F (z)= \sum_{m\in Z^n} \exp(2\pi izF(m))</math> | |
| with the sum extending over the [[lattice (group)|lattice]] of integers '''Z'''<sup>''n''</sup>. This theta function is a [[modular form]] of weight ''n''/2 (on an appropriately defined subgroup) of the [[modular group]]. In the Fourier expansion,
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| :<math>\widehat{\theta}_F (z) = \sum_{k=0}^\infty R_F(k) \exp(2\pi ikz),</math> | |
| the numbers ''R''<sub>F</sub>(''k'') are called the ''representation numbers'' of the form.
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| ===Ramanujan theta function===
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| {{see|Ramanujan theta function|mock theta function}}
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| ===Riemann theta function===
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| Let
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| :<math>\mathbb{H}_n=\{F\in M(n,\mathbb{C}) \; \mathrm{s.t.}\, F=F^T \;\textrm{and}\; \mbox{Im} F >0 \}</math>
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| be set of [[symmetric]] square [[matrix (mathematics)|matrices]] whose imaginary part is [[Positive-definite matrix|positive definite]]. ''H''<sub>''n''</sub> is called the [[Siegel upper half-space]] and is the multi-dimensional analog of the [[upper half-plane]]. The ''n''-dimensional analogue of the [[modular group]] is the [[symplectic group]] Sp(2n,'''Z'''); for ''n'' = 1, Sp(2,'''Z''') = SL(2,'''Z'''). The ''n''-dimensional analog of the [[congruence subgroup]]s is played by <math>\textrm{Ker} \{\textrm{Sp}(2n,\mathbb{Z})\rightarrow \textrm{Sp}(2n,\mathbb{Z}/k\mathbb{Z}) \}</math>.
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| Then, given <math>\tau\in \mathbb{H}_n</math>, the '''Riemann theta function''' is defined as
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| :<math>\theta (z,\tau)=\sum_{m\in Z^n} \exp\left(2\pi i
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| \left(\frac{1}{2} m^T \tau m +m^T z \right)\right). </math>
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| Here, <math>z\in \mathbb{C}^n</math> is an ''n''-dimensional complex vector, and the superscript ''T'' denotes the [[transpose]]. The Jacobi theta function is then a special case, with ''n'' = 1 and <math>\tau \in \mathbb{H}</math> where <math>\mathbb{H}</math> is the upper half-plane. | |
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| The Riemann theta converges absolutely and uniformly on compact subsets of <math>\mathbb{C}^n\times \mathbb{H}_n.</math>
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| The functional equation is
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| :<math>\theta (z+a+\tau b, \tau) = \exp 2\pi i
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| \left(-b^Tz-\frac{1}{2}b^T\tau b\right) \theta (z,\tau)</math>
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| which holds for all vectors <math>a,b \in \mathbb{Z}^n</math>, and for all <math>z \in \mathbb{C}^n</math> and <math>\tau \in \mathbb{H}_n</math>.
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| ===Poincaré series===
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| The [[Poincaré series (modular form)|Poincaré series]] generalizes the theta series to automorphic forms with respect to arbitrary [[Fuchsian group]]s.
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| *{{Citation |first=Milton |last=Abramowitz |lastauthoramp=yes |first2=Irene A. |last2=Stegun |title=[[Abramowitz and Stegun|Handbook of Mathematical Functions]] |year=1964 |publisher=Dover Publications |location=New York |isbn=0-486-61272-4 }}. ''(See section 16.27ff.)''
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| *{{Citation |first=Naum Illyich |last=Akhiezer |title=Elements of the Theory of Elliptic Functions |origyear=1970 |series=AMS Translations of Mathematical Monographs |volume=79 |year=1990 |publisher=AMS |location=Providence, RI |isbn=0-8218-4532-2 }}.
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| *{{Citation |first=Hershel M. |last=Farkas |lastauthoramp=yes |first2=Irwin |last2=Kra |title=Riemann Surfaces |year=1980 |publisher=Springer-Verlag |location=New York |isbn=0-387-90465-4 }}. ''(See Chapter 6 for treatment of the Riemann theta)''
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| *{{Citation |first=G. H. |last=Hardy |authorlink=G. H. Hardy |lastauthoramp=yes |first2=E. M. |last2=Wright |authorlink2=E. M. Wright |title=An Introduction to the Theory of Numbers |edition=Fourth |year=1959 |publisher=Clarendon Press |location=Oxford |isbn= }}.
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| *{{Citation |first=David |last=Mumford |authorlink=David Mumford |title=Tata Lectures on Theta I |year=1983 |publisher=Birkhauser |location=Boston |isbn=3-7643-3109-7 }}.
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| *{{Citation |authorlink=James Pierpont (mathematician) |first=James |last=Pierpont |title=Functions of a Complex Variable |location=New York |publisher=Dover |year=1959 |isbn= }}.
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| *{{Citation |first=Harry E. |last=Rauch |lastauthoramp=yes |first2=Hershel M. |last2=Farkas |title=Theta Functions with Applications to Riemann Surfaces |year=1974 |publisher=Williams & Wilkins |location=Baltimore |isbn=0-683-07196-3 }}.
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| *{{dlmf|first=William P. |last=Reinhardt|first2=Peter L. |last2=Walker|id=20|title=Theta Functions}}
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| *{{Citation |authorlink=E. T. Whittaker |first=E. T. |last=Whittaker |lastauthoramp=yes |first2=G. N. |last2=Watson |authorlink2=G. N. Watson |title=A Course in Modern Analysis |edition=Fourth |publisher=Cambridge University Press |location=Cambridge |year=1927 }}. ''(See chapter XXI for the history of Jacobi's θ functions)''
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| ==Further reading==
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| * {{cite book | editor1-first=Krishnaswami | editor1-last=Alladi | title=Surveys in Number Theory | series=Developments in Mathematics | volume=17 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-0-387-78509-7 | first=Hershel M. |last=Farkas | chapter=Theta functions in complex analysis and number theory | pages=57–87 | zbl=1206.11055 }}
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| * {{cite book | first=Bruno | last=Schoeneberg | title=Elliptic modular functions | series=Die Grundlehren der mathematischen Wissenschaften | volume=203 | publisher=[[Springer-Verlag]] | year=1974 | isbn=3-540-06382-X | chapter=IX. Theta series | pages=203–226 }}
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| ==External links==
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| *[http://elliptic.googlecode.com/ Matlab code for theta function evaluation] by elliptic project
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| {{PlanetMath attribution|id=6262|title=Integral representations of Jacobi theta functions}}
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| [[Category:Theta functions| ]]
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| [[Category:Elliptic functions]]
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| [[Category:Modular forms]]
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| [[Category:Q-analogs]]
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| [[Category:Riemann surfaces]]
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| [[Category:Analytic functions]]
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| [[Category:Several complex variables]]
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