Theta function explained

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.^[1]

The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called), 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 quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.

One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".^[2]

Throughout this article,

(e ^{\pi} ) ^{\alpha} should be interpreted as e ^{\alpha} (in order to resolve issues of choice of branch).^[3]

Jacobi theta function

There are several closely related functions called Jacobi theta functions, and 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 and, where can be any complex number and is the half-period ratio, confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula

\begin{align} \vartheta(z; \tau) & = \sum ^{inf ty} _{n =-inf ty} \exp \left(\pi i n ^{2} \tau + 2 \pi i n z\right) \\ & = 1 + 2 \sum ^{inf ty} _{n =1} q ^{n ^{2}} \cos( 2 \pi n z) \\ & = \sum ^{inf ty} _{n =-inf ty} q ^{n ^{2}} η ^{n} \end{align}

where is the nome and . It is a Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed, this is a Fourier series for a 1-periodic entire function of . Accordingly, the theta function is 1-periodic in :

\vartheta(z +1 ; \tau) = \vartheta(z; \tau).

By completing the square, it is also -quasiperiodic in, with

\vartheta(z + \tau;\tau) = \exp l ( - \pi i (\tau + 2 z) r ) \vartheta(z;\tau).

Thus, in general,

\vartheta(z + a + b\tau;\tau) = \exp\left( - \pi i b ^{2} \tau -2 \pi i b z\right) \vartheta(z;\tau)

for any integers and .

For any fixed

\tau , the function is an entire function on the complex plane, so by Liouville's theorem, it cannot be doubly periodic in 1 , \tau unless it is constant, and so the best we could do is to make it periodic in 1 and quasi-periodic in \tau . Indeed, since $$\backslash left|\backslash frac\backslash right|\; =\; \backslash exp\backslash left(\backslash pi\; (b^2\; \backslash Im(\backslash tau)\; +\; 2b\; \backslash Im(z))\; \backslash right)$$and \Im(\tau)> 0 , the function \vartheta(z, \tau) 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:^[4]

Auxiliary functions

The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:

\vartheta _{00} (z;\tau) = \vartheta(z;\tau)

The auxiliary (or half-period) functions are defined by

\begin{align} \vartheta _{01} (z;\tau)& = \vartheta \left(z + \tfrac 12 ;\tau\right)\\[ 3 pt] \vartheta _{10} (z;\tau)& = \exp\left(\tfrac 14 \pi i \tau + \pi i z\right)\vartheta\left(z + \tfrac 12 \tau;\tau\right)\\[ 3 pt] \vartheta _{11} (z;\tau)& = \exp\left(\tfrac 14 \pi i \tau + \pi i\left(z + \tfrac 12 \right)\right)\vartheta\left(z + \tfrac 12 \tau + \tfrac 12 ;\tau\right). \end{align}

This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome rather than . In Jacobi's notation the -functions are written:

\begin{align} \theta _{1(z;q)} & = \theta _{1(\pi} z,q) = - \vartheta _{11} (z;\tau)\\ \theta _{2(z;q)} & = \theta _{2(\pi} z,q) = \vartheta _{10} (z;\tau)\\ \theta _{3(z;q)} & = \theta _{3(\pi} z,q) = \vartheta _{00} (z;\tau)\\ \theta _{4(z;q)} & = \theta _{4(\pi} z,q) = \vartheta _{01} (z;\tau) \end{align}

The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions (notational variations) for further discussion.

If we set in the above theta functions, we obtain four functions of only, defined on the upper half-plane. These functions are called Theta Nullwert functions, based on the German term for zero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of only, defined on the unit disk

|q|< 1 . They are sometimes called theta constants:^[5] \begin{align} \vartheta _{11} ( 0 ;\tau)& =- \theta _{1(q)=-\sum} ^{inf ty} ^{n =-inf ty} ( -1 ) ^{n-1/2} q ^{(n +1 / 2 ) ^{2}} \\ \vartheta _{10} ( 0 ;\tau)& = \theta _{2(q)=\sum} ^{inf ty} ^{n =-inf ty} q ^{(n +1 / 2 ) ^{2}} \\ \vartheta _{00} ( 0 ;\tau)& = \theta _{3(q)=\sum} ^{inf ty} ^{n =-inf ty} q ^{n ^{2}} \\ \vartheta _{01} ( 0 ;\tau)& = \theta _{4(q)=\sum} ^{inf ty} ^{n =-inf ty} ( -1 ) ^{n} q ^{n ^{2}} \end{align}

with the nome . Observe that

\theta _{1(q)=0} . These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is\theta ^{4} _{2 (q)} + \theta ^{4} _{4 (q)} = \theta ^{4} _{3 (q)}

or equivalently,

\vartheta _{01} ( 0 ;\tau) ^{4} + \vartheta _{10} ( 0 ;\tau) ^{4} = \vartheta _{00} ( 0 ;\tau) ^{4}

which is the Fermat curve of degree four.

Jacobi identities

Jacobi's identities describe how theta functions transform under the modular group, which is generated by and . Equations for the first transform are easily found since adding one to in the exponent has the same effect as adding to . For the second, let

\alpha = ( - i \tau) ^{(12)/(\exp\left( (\pi)/(\tau))} i z ^{2} \right).

Then

\begin{align} \vartheta _{00} \left( (z)/(\tau) ; (-1)/(\tau) \right)& = \alpha\vartheta _{00} (z; \tau) & \vartheta _{01} \left( (z)/(\tau) ; (-1)/(\tau) \right)& = \alpha\vartheta _{10} (z; \tau)\\[ 3 pt] \vartheta _{10} \left( (z)/(\tau) ; (-1)/(\tau) \right)& = \alpha\vartheta _{01} (z; \tau) & \vartheta _{11} \left( (z)/(\tau) ; (-1)/(\tau) \right)& = - i\alpha\vartheta _{11} (z; \tau). \end{align}

Theta functions in terms of the nome

Instead of expressing the Theta functions in terms of and, we may express them in terms of arguments and the nome, where and . In this form, the functions become

\begin{align} \vartheta _{00} (w, q)& = \sum ^{inf ty} _{n =-inf ty} \left(w ^{2\right)} ^{n} q ^{n ^{2}} & \vartheta _{01} (w, q)& = \sum ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} \left(w ^{2\right)} ^{n} q ^{n ^{2}} \\[ 3 pt] \vartheta _{10} (w, q)& = \sum ^{inf ty} _{n =-inf ty} \left(w ^{2\right)} ^{n + (12)/()} q ^{\left(n + (12 \right))/(^{2})} & \vartheta _{11} (w, q)& = i \sum ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} \left(w ^{2\right)} ^{n + (12)/()} q ^{\left(n + (12 \right))/(^{2})} . \end{align}

We see that the theta functions can also be defined in terms of and, without a direct 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 -adic numbers.

Product representations

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers and with and we have

\prod ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right) \left( 1 + w ^{2} q ^{2m-1} \right) \left( 1 + w ^{-2} q ^{2m-1} \right) = \sum ^{inf ty} _{n =-inf ty} w ^{2n} q ^{n ^{2}} .

It can be proven by elementary means, as for instance in Hardy and Wright's An Introduction to the Theory of Numbers.

If we express the theta function in terms of the nome (noting some authors instead set) and take then

\vartheta(z; \tau) = \sum ^{inf ty} _{n =-inf ty} \exp(\pi i \tau n ^{2)} \exp( 2 \pi i z n) = \sum ^{inf ty} _{n =-inf ty} w ^{2n} q ^{n ^{2}} .

We therefore obtain a product formula for the theta function in the form

\vartheta(z; \tau) = \prod ^{inf ty} _{m =1} ( 1 - \exp( 2 m \pi i \tau) ) ( 1 + \exp ( ( 2 m -1 ) \pi i \tau + 2 \pi i z ) ) ( 1 + \exp ( ( 2 m -1 ) \pi i \tau - 2 \pi i z ) ) .

In terms of and :

\begin{align} \vartheta(z; \tau) & = \prod ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right) \left( 1 + q ^{2m-1} w ^{2\right) \left(} 1 + (q ^{2m-1})/(w ^{2}) \right) \\ & = \left(q ^{2;q} ^{2 q;q} ^{inf ty\left( - w} ^{2 \right)} ^{inf ty\left( - (q)/(w ^{2})} ;q ^{2 \right)} _{inf ty} \\ & = \left(q ^{2;q} ^{2 q;q} ^{inf ty\theta\left( - w} ^{2\right)} \end{align}

where is the -Pochhammer symbol and is the -theta function. Expanding terms out, the Jacobi triple product can also be written

\prod ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right) ( 1 + \left(w ^{2+w} ^{-2} \right)q ^{2m-1} + q ^{4m-2} ) ,

which we may also write as

\vartheta(z\mid q) = \prod ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right) \left( 1 + 2 \cos( 2 \pi z)q ^{2m-1} + q ^{4m-2} \right).

This form is valid in general but clearly is of particular interest when is real. Similar product formulas for the auxiliary theta functions are

\begin{align} \vartheta _{01} (z\mid q) & = \prod ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right) \left( 1 - 2 \cos( 2 \pi z)q ^{2m-1} + q ^{4m-2} \right),\\[ 3 pt] \vartheta _{10} (z\mid q) & = 2 q ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right) \left( 1 + 2 \cos( 2 \pi z)q ^{2m} + q ^{4m} \right),\\[ 3 pt] \vartheta _{11} (z\mid q) & = -2 q ^{inf ty} _{m =1} \left( 1 - q ^{2m} \right)\left( 1 - 2 \cos( 2 \pi z)q ^{2m} + q ^{4m} \right). \end{align} In particular, $$\backslash lim\_\backslash frac\; =\; \backslash cos(\backslash pi\; z),\backslash quad\; \backslash lim\_\backslash frac\; =\; \backslash sin(\backslash pi\; z)$$so we may interpret them as one-parameter deformations of the periodic functions \sin, \cos , again validating the interpretation of the theta function as the most general 2 quasi-period function.

Integral representations

The Jacobi theta functions have the following integral representations:

\begin{align} \vartheta _{00} (z; \tau) & = - i\int ^{i +inf ty} _{i -inf ty} e ^{i \pi \tau u ^{2}} (\cos( 2 \pi uz + \pi u))/(\sin(\pi u)) d u; \\[ 6 pt] \vartheta _{01} (z; \tau) & = - i\int ^{i +inf ty} _{i -inf ty} e ^{i \pi \tau u ^{2}} (\cos ( 2 \pi uz))/(\sin (\pi u)) d u; \\[ 6 pt] \vartheta _{10} (z; \tau) & = - ie ^{i \pi z + (14)/(i) \pi\tau} \int ^{i +inf ty} _{i -inf ty} e ^{i\pi\tau u ^{2}} (\cos( 2 \pi uz + \pi u + \pi \tau u))/(\sin (\pi u)) d u; \\[ 6 pt] \vartheta _{11} (z; \tau) & = e ^{i \pi z + (14)/(i) \pi \tau} \int ^{i +inf ty} _{i -inf ty} e ^{i\pi\tau u ^{2}} (\cos( 2 \pi uz + \pi \tau u))/(\sin(\pi u)) d u. \end{align}

The Theta Nullwert function

\theta _{3} (q) as this integral identity:\theta _{3} (q) = 1 + (4 q\sqrt{ ln ( 1 /q))/() } \int_^ \frac \,\mathrmx

This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.

Based on this formula following three eminent examples are given:

l [ (2)/(K l ( \pi) (1)/(2) \sqrt{ 2 } r ) r ] ^{1/2} = \theta _{3} l [\exp( - \pi) r ] = 1 + 4 \exp( - \pi) \int ^{inf ty} _{0} (\exp( - \pi x ^{2)[1} - \exp( -2 \pi)\cos( 2 \pi x)])/(1-2 \exp( -2 \pi)\cos( 2 \pi x) + \exp( -4 \pi)) d x l [ (2)/(\pi) K(\sqrt{ 2 } - 1 ) r ] ^{1/2} = \theta _{3} l [\exp( - \sqrt{ 2 }\pi) r ] = 1 + 4 \sqrt[ 4 ]{ 2 }\exp( - \sqrt{ 2 }\pi) \int ^{inf ty} _{0} (\exp( - \sqrt{ 2)/(\pi) x ^{2)[1} - \exp( -2 \sqrt{ 2 }\pi)\cos( 2 \sqrt{ 2 }\pi x)]}{ 1 - 2 \exp( -2 \sqrt{ 2 }\pi)\cos( 2 \sqrt{ 2 }\pi x) + \exp( -4 \sqrt{ 2 }\pi)} d x l \{ (2)/(K l [\sin l ( \pi) (\pi)/(12) r ) r ] r \} ^{1/2} = \theta _{3} l [\exp( - \sqrt{ 3 }\pi) r ] = 1 + 4 \sqrt[ 4 ]{ 3 }\exp( - \sqrt{ 3 }\pi) \int ^{inf ty} _{0} (\exp( - \sqrt{ 3)/(\pi) x ^{2)[1} - \exp( -2 \sqrt{ 3 }\pi)\cos( 2 \sqrt{ 3 }\pi x)]}{ 1 - 2 \exp( -2 \sqrt{ 3 }\pi)\cos( 2 \sqrt{ 3 }\pi x) + \exp( -4 \sqrt{ 3 }\pi)} d x

Furthermore, the theta examples

\theta _{3} (\tfrac{ 1 }{ 2 }) and \theta _{3} (\tfrac{ 1 }{ 3 }) shall be displayed:\theta _{3} l ( (1)/(2) r ) = 1+2 \sum ^{inf ty} _{n =1} (1)/(2 ^{n ^{2}}) = 1 + 2 \pi ^{-1/2} \sqrt{ ln ( 2 )} \int ^{inf ty} _{0} (\exp[ -ln ( 2 )x ^{2]\{16} -4 \cos[ 2ln ( 2 )x]\)/() } \,\mathrmx \theta _{3} l ( (1)/(2) r ) = 2 . 128936827211877158669 \ldots \theta _{3} l ( (1)/(3) r ) = 1+2 \sum ^{inf ty} _{n =1} (1)/(3 ^{n ^{2}}) = 1 + (4)/(3) \pi ^{-1/2} \sqrt{ ln ( 3 )} \int ^{inf ty} _{0} (\exp[ -ln ( 3 )x ^{2]\{81} -9 \cos[ 2ln ( 3 )x]\)/() } \,\mathrmx \theta _{3} l ( (1)/(3) r ) = 1 . 691459681681715341348 \ldots

Explicit values

Lemniscatic values

Proper credit for most of these results goes to Ramanujan. See Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004).^[6] Define,

\varphi(q) = \vartheta _{00} ( 0 ;\tau) = \theta _{3(0;q)=\sum} ^{inf ty} ^{n =-inf ty} q ^{n ^{2}}

with the nome

q = e ^{\pi} , \tau = n\sqrt{ -1 }, and Dedekind eta function η (\tau). Then for n = 1 , 2 , 3 , ... \begin{align} \varphi\left(e ^{-\pi} \right) & = (\sqrt[ 4 ]{\pi)/() } = \sqrt2\,\eta\left(\sqrt\right)\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \sqrt\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\2\varphi\left(e^\right) &= \varphi\left(e^\right) + \frac \frac\\\varphi\left(e^\right) &= \frac \frac\\2\varphi\left(e^\right) &= \varphi\left(e^\right) + \frac \sqrt\\6\varphi\left(e^\right) &= 3\varphi\left(e^\right) + 2\varphi\left(e^\right) - \varphi\left(e^\right) + \frac \sqrt[3]\end

If the reciprocal of the Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding

\vartheta _{00} values or \phi values can be represented in a simplified way by using the hyperbolic lemniscatic sine:\varphi l [\exp( - \tfrac{ 1 }{ 5 }\pi) r ] = \sqrt[ 4 ]{\pi}{\Gamma\left(\tfrac{ 3 }{ 4 }\right)} ^{-1} \operatorname{slh} l (\tfrac{ 1 }{ 5 }\sqrt{ 2 }\varpi r )\operatorname{slh} l (\tfrac{ 2 }{ 5 }\sqrt{ 2 }\varpi r ) \varphi l [\exp( - \tfrac{ 1 }{ 7 }\pi) r ] = \sqrt[ 4 ]{\pi}{\Gamma\left(\tfrac{ 3 }{ 4 }\right)} ^{-1} \operatorname{slh} l (\tfrac{ 1 }{ 7 }\sqrt{ 2 }\varpi r )\operatorname{slh} l (\tfrac{ 2 }{ 7 }\sqrt{ 2 }\varpi r ) \operatorname{slh} l (\tfrac{ 3 }{ 7 }\sqrt{ 2 }\varpi r ) \varphi l [\exp( - \tfrac{ 1 }{ 9 }\pi) r ] = \sqrt[ 4 ]{\pi}{\Gamma\left(\tfrac{ 3 }{ 4 }\right)} ^{-1} \operatorname{slh} l (\tfrac{ 1 }{ 9 }\sqrt{ 2 }\varpi r )\operatorname{slh} l (\tfrac{ 2 }{ 9 }\sqrt{ 2 }\varpi r ) \operatorname{slh} l (\tfrac{ 3 }{ 9 }\sqrt{ 2 }\varpi r ) \operatorname{slh} l (\tfrac{ 4 }{ 9 }\sqrt{ 2 }\varpi r ) \varphi l [\exp( - \tfrac{ 1 }{ 11 }\pi) r ] = \sqrt[ 4 ]{\pi}{\Gamma\left(\tfrac{ 3 }{ 4 }\right)} ^{-1} \operatorname{slh} l (\tfrac{ 1 }{ 11 }\sqrt{ 2 }\varpi r )\operatorname{slh} l (\tfrac{ 2 }{ 11 }\sqrt{ 2 }\varpi r ) \operatorname{slh} l (\tfrac{ 3 }{ 11 }\sqrt{ 2 }\varpi r ) \operatorname{slh} l (\tfrac{ 4 }{ 11 }\sqrt{ 2 }\varpi r ) \operatorname{slh} l (\tfrac{ 5 }{ 11 }\sqrt{ 2 }\varpi r )

With the letter

\varpi the Lemniscate constant is represented.

Note that the following modular identities hold:

\begin{align} 2 \varphi\left(q ^{4\right)} & = \varphi(q) + \sqrt{ 2 \varphi ^{2\left(q} ^{2\right)-\varphi} ^{2(q)}\\ 3\varphi\left(q} ^{9\right)} & = \varphi(q) + \sqrt[ 3 ]{ 9 (\varphi ^{4\left(q} ^{3\right)})/(\varphi(q)) - \varphi ^{3(q)}\\ \sqrt{5}\varphi\left(q} ^{25} \right) & = \varphi\left(q ^{5 \right)\cot\left( (1)/(2)} \arctan\left( (2)/(\sqrt{ 5) }\frac\frac\right)\right)\end

where

s(q) = s\left(e ^{\pi} \right) =- R\left( - e ^{-\pi} \right) is the Rogers–Ramanujan continued fraction:\begin{align} s(q) & = \sqrt[ 5 ]{\tan\left( (1)/(\arctan\left( 2) (5)/(2) (\varphi ^{2\left(q} ^{5\right)})/(- \varphi ^{2(q)}) (1)/(2) \right)\right)\cot ^{2 \left( (1)/(2)} \operatorname{arccot}\left( (5)/(2) (\varphi ^{2\left(q} ^{5\right)})/(- \varphi ^{2(q)}) (1)/(2) \right)\right)}\\ & = \cfrac{e ^{-\pi} }\end

Equianharmonic values

The mathematician Bruce Berndt found out further values^[7] of the theta function:

\begin{array}{lll} \varphi\left(\exp( - \sqrt{ 3 }\pi)\right) & = & \pi ^{-1} {\Gamma\left(\tfrac{ 4 }{ 3 }\right)} ^{3/2} 2 ^{-2/3} 3 ^{13/8} \\ \varphi\left(\exp( -2 \sqrt{ 3 }\pi)\right) & = & \pi ^{-1} {\Gamma\left(\tfrac{ 4 }{ 3 }\right)} ^{3/2} 2 ^{-2/3} 3 ^{13/8} \cos(\tfrac{ 1 }{ 24 }\pi) \\ \varphi\left(\exp( -3 \sqrt{ 3 }\pi)\right) & = & \pi ^{-1} {\Gamma\left(\tfrac{ 4 }{ 3 }\right)} ^{3/2} 2 ^{-2/3} 3 ^{7/8} (\sqrt[ 3 ]{ 2 } +1 ) \\ \varphi\left(\exp( -4 \sqrt{ 3 }\pi)\right) & = & \pi ^{-1} {\Gamma\left(\tfrac{ 4 }{ 3 }\right)} ^{3/2} 2 ^{-5/3} 3 ^{13/8} l ( 1+ \sqrt{\cos(\tfrac{ 1 }{ 12 }\pi)} r ) \\ \varphi\left(\exp( -5 \sqrt{ 3 }\pi)\right) & = & \pi ^{-1} {\Gamma\left(\tfrac{ 4 }{ 3 }\right)} ^{3/2} 2 ^{-2/3} 3 ^{5/8} \sin(\tfrac{ 1 }{ 5 }\pi)(\tfrac{ 2 }{ 5 }\sqrt[ 3 ]{ 100 } + \tfrac{ 2 }{ 5 }\sqrt[ 3 ]{ 10 } + \tfrac{ 3 }{ 5 }\sqrt{ 5 } +1 ) \end{array}

Further values

Many values of the theta function^[8] and especially of the shown phi function can be represented in terms of the gamma function:

\begin{array}{lll} \varphi\left(\exp( - \sqrt{ 2 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 9 }{ 8 }\right){\Gamma\left(\tfrac{ 5 }{ 4 }\right)} ^{-1/2} 2 ^{7/8} \\ \varphi\left(\exp( -2 \sqrt{ 2 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 9 }{ 8 }\right){\Gamma\left(\tfrac{ 5 }{ 4 }\right)} ^{-1/2} 2 ^{1/8} l ( 1+ \sqrt{\sqrt{ 2 } -1 } r ) \\ \varphi\left(\exp( -3 \sqrt{ 2 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 9 }{ 8 }\right){\Gamma\left(\tfrac{ 5 }{ 4 }\right)} ^{-1/2} 2 ^{3/8} 3 ^{-1/2} (\sqrt{ 3 } +1 )\sqrt{\tan(\tfrac{ 5 }{ 24 }\pi)} \\ \varphi\left(\exp( -4 \sqrt{ 2 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 9 }{ 8 }\right){\Gamma\left(\tfrac{ 5 }{ 4 }\right)} ^{-1/2} 2 ^{-1/8} l ( 1+ \sqrt[ 4 ]{ 2 \sqrt{ 2 } -2 } r ) \\ \varphi\left(\exp( -5 \sqrt{ 2 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 9 }{ 8 }\right){\Gamma\left(\tfrac{ 5 }{ 4 }\right)} ^{-1/2} (1)/(15) 2 ^{3/8} x \\ && x l [\sqrt[ 3 ]{ 5 }\sqrt{ 10+2 \sqrt{ 5 }} l (\sqrt[ 3 ]{ 5+ \sqrt{ 2 } +3 \sqrt{ 3 }} + \sqrt[ 3 ]{ 5+ \sqrt{ 2 } -3 \sqrt{ 3 }} r ) - l ( 2- \sqrt{ 2 } r )\sqrt{ 25-10 \sqrt{ 5 }} r ] \\ \varphi\left(\exp( - \sqrt{ 6 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 5 }{ 24 }\right){\Gamma\left(\tfrac{ 5 }{ 12 }\right)} ^{-1/2} 2 ^{-13/24} 3 ^{-1/8} \sqrt{\sin(\tfrac{ 5 }{ 12 }\pi)} \\ \varphi\left(\exp( - \tfrac{ 1 }{ 2 }\sqrt{ 6 }\pi)\right) & = & \pi ^{-1/2} \Gamma\left(\tfrac{ 5 }{ 24 }\right){\Gamma\left(\tfrac{ 5 }{ 12 }\right)} ^{-1/2} 2 ^{5/24} 3 ^{-1/8} \sin(\tfrac{ 5 }{ 24 }\pi) \end{array}

Nome power theorems

Direct power theorems

For the transformation of the nome^[9] in the theta functions these formulas can be used:

\theta _{2} (q ^{2)} = \tfrac{ 1 }{ 2 }\sqrt{ 2 [\theta _{3} (q) ^{2} - \theta _{4} (q) ^{2]}} \theta _{3} (q ^{2)} = \tfrac{ 1 }{ 2 }\sqrt{ 2 [\theta _{3} (q) ^{2} + \theta _{4} (q) ^{2]}} \theta _{4} (q ^{2)} = \sqrt{\theta _{4} (q)\theta _{3} (q)}

The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the Pythagorean triples according to the Jacobi Identity. Furthermore, those transformations are valid:

\theta _{3} (q ^{4)} = \tfrac{ 1 }{ 2 }\theta _{3} (q) + \tfrac{ 1 }{ 2 }\theta _{4} (q)

These formulas can be used to compute the theta values of the cube of the nome:

27 \theta _{3} (q ^{3)} ^{8} - 18 \theta _{3} (q ^{3)} ^{4 \theta} ^{3} (q) ^{4} - \theta _{3} (q) ^{8} = 8 \theta _{3} (q ^{3)} ^{2 \theta} ^{3} (q) ^{2 [ 2 \theta} _{4} (q) ^{4} - \theta _{3} (q) ^{4]} 27 \theta _{4} (q ^{3)} ^{8} - 18 \theta _{4} (q ^{3)} ^{4 \theta} ^{4} (q) ^{4} - \theta _{4} (q) ^{8} = 8 \theta _{4} (q ^{3)} ^{2 \theta} ^{4} (q) ^{2 [ 2 \theta} _{3} (q) ^{4} - \theta _{4} (q) ^{4]}

And the following formulas can be used to compute the theta values of the fifth power of the nome:

[\theta _{3} (q) ^{2} - \theta _{3} (q ^{5)} ^{2 ][ 5 \theta} ^{3} (q ^{5)} ^{2} - \theta _{3} (q) ^{2]} ^{5} = 256 \theta _{3} (q ^{5)} ^{2 \theta} ^{3} (q) ^{2 \theta} _{4} (q) ^{4} [\theta _{3} (q) ^{4} - \theta _{4} (q) ^{4]} [\theta _{4} (q ^{5)} ^{2} - \theta _{4} (q) ^{2 ][ 5 \theta} _{4} (q ^{5)} ^{2} - \theta _{4} (q) ^{2]} ^{5} = 256 \theta _{4} (q ^{5)} ^{2 \theta} ^{4} (q) ^{2 \theta} _{3} (q) ^{4} [\theta _{3} (q) ^{4} - \theta _{4} (q) ^{4]}

Transformation at the cube root of the nome

The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:

l [ (\theta _{3} (q ^{1/3} ) ^{2})/(\theta _{3} (q) ^{2}) - (3 \theta _{3} (q ^{3} ) ^{2})/(\theta _{3} (q) ^{2}) r ] ^{2} = 4 -4 l [ (2 \theta _{2} (q) ^{2} \theta _{4} (q) ^{2})/(\theta _{3} (q) ^{4}) r ] ^{2/3} l [ (3 \theta _{4} (q ^{3} ) ^{2})/(\theta _{4} (q) ^{2}) - (\theta _{4} (q ^{1/3} ) ^{2})/(\theta _{4} (q) ^{2}) r ] ^{2} = 4 +4 l [ (2 \theta _{2} (q) ^{2} \theta _{3} (q) ^{2})/(\theta _{4} (q) ^{4}) r ] ^{2/3}

Transformation at the fifth root of the nome

The Rogers-Ramanujan continued fraction can be defined in terms of the Jacobi theta function in the following way:

R(q) = \tan l \{ (1)/(\arctan l [ 2) (1)/(2) - (\theta _{4} (q) ^{2})/(2 \theta _{4} (q ^{5)} ^{2}) r ] r \} ^{1/5} \tan l \{ (1)/(\arccot l [ 2) (1)/(2) - (\theta _{4} (q) ^{2})/(2 \theta _{4} (q ^{5)} ^{2}) r ] r \} ^{2/5} R(q ^{2)} = \tan l \{ (1)/(\arctan l [ 2) (1)/(2) - (\theta _{4} (q) ^{2})/(2 \theta _{4} (q ^{5)} ^{2}) r ] r \} ^{2/5} \cot l \{ (1)/(\arccot l [ 2) (1)/(2) - (\theta _{4} (q) ^{2})/(2 \theta _{4} (q ^{5)} ^{2}) r ] r \} ^{1/5} R(q ^{2)} = \tan l \{ (1)/(\arctan l [ 2) (\theta _{3} (q) ^{2})/(2 \theta _{3} (q ^{5)} ^{2}) - (1)/(2) r ] r \} ^{2/5} \tan l \{ (1)/(\arccot l [ 2) (\theta _{3} (q) ^{2})/(2 \theta _{3} (q ^{5)} ^{2}) - (1)/(2) r ] r \} ^{1/5}

The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:

S(q) = (R(q ^{4)})/(R(q ^{2)R(q)}) = \tan l \{ (1)/(2) \arctan l [ (\theta _{3} (q) ^{2})/(2 \theta _{3} (q ^{5)} ^{2}) - (1)/(2) r ] r \} ^{1/5} \cot l \{ (1)/(\arccot l [ 2) (\theta _{3} (q) ^{2})/(2 \theta _{3} (q ^{5)} ^{2}) - (1)/(2) r ] r \} ^{2/5}

The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:

(\theta _{3} (q ^{1/5} ))/(\theta _{3} (q ^{5)}) - 1 = (1)/(S(q)) l [S(q) ^{2} + R(q ^{2)r]l[1} + R(q ^{2)S(q)r]} 1 - (\theta _{4} (q ^{1/5} ))/(\theta _{4} (q ^{5)}) = (1)/(R(q)) l [R(q ^{2)} + R(q) ^{2r]l[1} - R(q ^{2)R(q)r]} \theta _{3} (q ^{1/5} ) ^{2} - \theta _{3} (q) ^{2} = l [\theta _{3} (q) ^{2} - \theta _{3} (q ^{5)} ^{2 r ] l [ 1+ (1)/(R(q ^{2)S(q)})} + R(q ^{2 )S(q) + (1)/(R(q ^{2)} ^{2})} + R(q ^{2)} ^{2+ (1)/(S(q))} - S(q) r ] \theta _{4} (q) ^{2} - \theta _{4} (q ^{1/5} ) ^{2} = l [\theta _{4} (q ^{5)} ^{2} - \theta _{4} (q) ^{2 r ] l [ 1- (1)/(R(q ^{2)R(q)})} - R(q ^{2 )R(q) + (1)/(R(q ^{2)} ^{2})} + R(q ^{2)} ^{2- (1)/(R(q))} + R(q) r ]

Modulus dependent theorems

Im combination with the elliptic modulus, following formulas can be displayed:

These are the formulas for the square of the elliptic nome:

\theta _{4} [q(k)] = \theta _{4} [q(k) ^{2]\sqrt[8]{1} - k ^{2}} \theta _{4} [q(k) ^{2]} = \theta _{3} [q(k)]\sqrt[ 8 ]{ 1 - k ^{2}} \theta _{3} [q(k) ^{2]} = \theta _{3} [q(k)]\cos[\tfrac{ 1 }{ 2 }\arcsin(k)]

And this is an efficient formula for the cube of the nome:

\theta _{4} l \langle q l \{\tan l [\tfrac{ 1 }{ 2 }\arctan(t ^{3)r]r\}} ^{3} r \rangle = \theta _{4} l \langle q l \{\tan l [\tfrac{ 1 }{ 2 }\arctan(t ^{3)r]r\}} r \rangle 3 ^{-1/2} l (\sqrt{ 2 \sqrt{t ^{4} - t ^{2} + 1 } - t ^{2} + 2 } + \sqrt{t ^{2} + 1 } r ) ^{1/2}

For all real values

t \in \R the now mentioned formula is valid.

And for this formula two examples shall be given:

First calculation example with the value

t = 1 inserted:

\theta _{4} l \langle q l \{\tan l [\tfrac{ 1 }{ 2 }\arctan( 1 ) r ] r \} ^{3} r \rangle = \theta _{4} l \langle q l \{\tan l [\tfrac{ 1 }{ 2 }\arctan( 1 ) r ] r \} r \rangle 3 ^{-1/2} l (\sqrt{ 3 } + \sqrt{ 2 } r ) ^{1/2}
\theta _{4} l [\exp( -3 \sqrt{ 2 }\pi) r ] = \theta _{4} l [\exp( - \sqrt{ 2 }\pi) r ] 3 ^{-1/2} l (\sqrt{ 3 } + \sqrt{ 2 } r ) ^{1/2}

Second calculation example with the value

t = \Phi ^{-2} inserted:

\theta _{4} l \langle q l \{\tan l [\tfrac{ 1 }{ 2 }\arctan(\Phi ^{-6} ) r ] r \} ^{3} r \rangle = \theta _{4} l \langle q l \{\tan l [\tfrac{ 1 }{ 2 }\arctan(\Phi ^{-6} ) r ] r \} r \rangle 3 ^{-1/2} l (\sqrt{ 2 \sqrt{\Phi ^{-8} - \Phi ^{-4} + 1 } - \Phi ^{-4} + 2 } + \sqrt{\Phi ^{-4} + 1 } r ) ^{1/2}
\theta _{4} l [\exp( -3 \sqrt{ 10 }\pi) r ] = \theta _{4} l [\exp( - \sqrt{ 10 }\pi) r ] 3 ^{-1/2} l (\sqrt{ 2 \sqrt{\Phi ^{-8} - \Phi ^{-4} + 1 } - \Phi ^{-4} + 2 } + \sqrt{\Phi ^{-4} + 1 } r ) ^{1/2}

The constant

\Phi represents the Golden ratio number \Phi = \tfrac{ 1 }{ 2 }(\sqrt{ 5 } + 1 ) exactly.

Some series identities

Sums with theta function in the result

The infinite sum^{[10] [11]} of the reciprocals of Fibonacci numbers with odd indices has this identity:

\sum ^{inf ty} _{n =1} (1)/(F _{2n-1}) = (\sqrt{ 5)/() }\,\sum_^\infty \frac = \frac \sum_^\infty \frac == (\sqrt{ 5)/() }\,\theta_(\Phi^)^2 = \frac\bigl[\theta_{3}(\Phi^{-1})^2 - \theta_{4}(\Phi^{-1})^2\bigr]

By not using the theta function expression, following identity between two sums can be formulated:

\sum ^{inf ty} _{n =1} (1)/(F _{2n-1}) = (\sqrt{ 5)/() }\,\biggl[\sum_{n=1}^\infty 2 \,\Phi^{- (2n - 1)^2 /2} \biggr]^2 \sum ^{inf ty} _{n =1} (1)/(F _{2n-1}) = 1 . 82451515740692456814215840626732817332 \ldots

Also in this case

\Phi = \tfrac{ 1 }{ 2 }(\sqrt{ 5 } + 1 ) is Golden ratio number again.

Infinite sum of the reciprocals of the Fibonacci number squares:

\sum ^{inf ty} _{n =1} (1)/(F ^{2} _{n}) = (5)/(24) l [ 2 \theta _{2} (\Phi ^{-2} ) ^{4} - \theta _{3} (\Phi ^{-2} ) ^{4} + 1 r ] = (5)/(24) l [\theta _{3} (\Phi ^{-2} ) ^{4} - 2 \theta _{4} (\Phi ^{-2} ) ^{4} + 1 r ]

Infinite sum of the reciprocals of the Pell numbers with odd indices:

\sum ^{inf ty} _{n =1} (1)/(P _{2n-1}) = (1)/(\sqrt{ 2) }\,\theta_\bigl[(\sqrt{2}-1)^2\bigr]^2 = \frac\bigl[\theta_{3}(\sqrt{2}-1)^2 - \theta_{4}(\sqrt{2}-1)^2\bigr]

Sums with theta function in the summand

The next two series identities were proved by István Mező:

\begin{align} \theta ^{2 (q)& = iq} _{4} ^{(14)/()} \sum ^{inf ty} _{k =-inf ty} q ^{2 k ^{2-k}} \theta _{1 \left( (2 k -1)/(2 i)} ln q,q\right),\\[ 6 pt] \theta ^{inf ty} _{k =-inf ty} q ^{2 k ^{2}} \theta _{4 \left( (k ln q)/(i)} ,q\right). \end{align}

These relations hold for all . Specializing the values of, we have the next parameter free sums

\sqrt{ (\pi\sqrt{e ^{\pi})/() }}\cdot\frac =i\sum_^\infty e^ \theta_1 \left(\frac(2k-1),e^\right)\sqrt{ (\pi)/(2) }\cdot\frac =\sum_^\infty\frac

Zeros of the Jacobi theta functions

All zeros of the Jacobi theta functions are simple zeros and are given by the following:

\begin{align} \vartheta(z;\tau) = \vartheta _{00} (z;\tau) & = 0 &\Longleftrightarrow&& z & = m + n \tau + (1)/(2) + (\tau)/(2) \\[ 3 pt] \vartheta _{11} (z;\tau) & = 0 &\Longleftrightarrow&& z & = m + n \tau \\[ 3 pt] \vartheta _{10} (z;\tau) & = 0 &\Longleftrightarrow&& z & = m + n \tau + (1)/(2) \\[ 3 pt] \vartheta _{01} (z;\tau) & = 0 &\Longleftrightarrow&& z & = m + n \tau + (\tau)/(2) \end{align} where, are arbitrary integers.

Relation to the Riemann zeta function

The relation

\vartheta\left( 0 ; - (1)/(\tau) \right) = \left( - i\tau\right) ^{(12)/(\vartheta( 0 ;\tau))}

was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform

\Gamma\left( (s)/(2) \right) \pi ^{- (s)/(2)} \zeta(s) = (1)/(2) \int ^{inf ty l (\vartheta( 0 ;it) -1 r )t} _{0} ^{(s)/(2)} (d t)/(t)

which can be shown to be invariant under substitution of by . The corresponding integral for is given in the article on the Hurwitz zeta function.

Relation to the Weierstrass elliptic function

The theta function was used by Jacobi to construct (in a form adapted to easy calculation) 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

\wp(z;\tau) = - ( log \vartheta _{11} (z;\tau) ) '' + c

where the second derivative is with respect to and the constant is defined so that the Laurent expansion of at has zero constant term.

Relation to the q-gamma function

The fourth theta function – and thus the others too – is intimately connected to the Jackson -gamma function via the relation^[12]

\left(\Gamma _{q ^{2}} (x)\Gamma _{q ^{2}} ( 1- x)\right) ^{-1} = (q ^{2x(1-x)})/(\left(q ^{-2} ;q ^{-2} \right) ^{2-1 \right)} _{inf ty\left(q}) \theta _{4 \left( (1)/(2 i)} ( 1-2 x) log q, (1)/(q) \right).

Relations to Dedekind eta function

Let be the Dedekind eta function, and the argument of the theta function as the nome . Then,

\begin{align} \theta _{2(q)} = \vartheta _{10} ( 0 ;\tau) & = (2η ^{2(2\tau)})/(η (\tau)) ,\\[ 3 pt] \theta _{3(q)} = \vartheta _{00} ( 0 ;\tau) & = (η ^{5(\tau)})/(η ^{2 \left( (1)/(2)} \tau\right) η ^{2(2\tau)}) = (η ^{2 \left( (1)/(2)} (\tau +1 )\right))/(η (\tau +1 )) , \\[ 3 pt] \theta _{4(q)} = \vartheta _{01} ( 0 ;\tau) & = (η ^{2 \left( (1)/(2)} \tau\right))/(η (\tau)) , \end{align}

and,

\theta _{2(q)\theta} _{3(q)\theta} _{4(q)} = 2η ^{3(\tau).}

See also the Weber modular functions.

Elliptic modulus

The elliptic modulus is

k(\tau) = (\vartheta _{10} ( 0 ;\tau) ^{2})/(\vartheta _{00} ( 0 ;\tau) ^{2}) and the complementary elliptic modulus isk'(\tau) = (\vartheta _{01} ( 0 ;\tau) ^{2})/(\vartheta _{00} ( 0 ;\tau) ^{2})

Derivatives of theta functions

These are two identical definitions of the complete elliptic integral of the second kind:

E(k) = \int ^{\pi/ 2} _{0} \sqrt{ 1 - k ^{2\sin(\varphi)} ^{2}} \partial\varphi E(k) = (\pi)/(2) \sum ^{inf ty} _{a =0} ([( 2 a)!] ^{2})/(( 1-2 a) 16 ^{a} (a!) ^{4}) k ^{2a}

The derivatives of the Theta Nullwert functions have these MacLaurin series:

\theta _{2} '(x) = (d)/(d x) \theta _{2} (x) = (1)/(2) x ^{-3/4} + \sum ^{inf ty} _{n =1} (1)/(2) ( 2 n + 1 ) ^{2} x ^{(2n-1)(2n+3)/4} \theta _{3} '(x) = (d)/(d x) \theta _{3} (x) = 2+ \sum ^{inf ty} _{n =1} 2 (n + 1 ) ^{2} x ^{n(n+2)} \theta _{4} '(x) = (d)/(d x) \theta _{4} (x) = -2+ \sum ^{inf ty} _{n =1} 2 (n + 1 ) ^{2} ( -1 ) ^{n+1} x ^{n(n+2)}

The derivatives of theta zero-value functions are as follows:

\theta _{2} '(x) = (d)/(d x) \theta _{2} (x) = (1)/(2 \pi x) \theta _{2} (x)\theta _{3} (x) ^{2} E l [ (\theta _{2} (x) ^{2})/(\theta _{3} (x) ^{2}) r ] \theta _{3} '(x) = (d)/(d x) \theta _{3} (x) = \theta _{3} (x) l [\theta _{3} (x) ^{2} + \theta _{4} (x) ^{2 r ] l \{ (1)/(2 \pi x)} E l [ (\theta _{3} (x) ^{2} - \theta _{4} (x) ^{2})/(\theta _{3} (x) ^{2} + \theta _{4} (x) ^{2}) r ] - (\theta _{4} (x) ^{2})/(4 x) r \} \theta _{4} '(x) = (d)/(d x) \theta _{4} (x) = \theta _{4} (x) l [\theta _{3} (x) ^{2} + \theta _{4} (x) ^{2 r ] l \{ (1)/(2 \pi x)} E l [ (\theta _{3} (x) ^{2} - \theta _{4} (x) ^{2})/(\theta _{3} (x) ^{2+ \theta} _{4} (x) ^{2}) r ] - (\theta _{3} (x) ^{2})/(4 x) r \}

The two last mentioned formulas are valid for all real numbers of the real definition interval:

-1 < x < 1 \cap x \in \R

And these two last named theta derivative functions are related to each other in this way:

\vartheta _{4} (x) l [ (d)/(d x) \vartheta _{3} (x) r ] - \vartheta _{3} (x) l [ (d)/(d x) \theta _{4} (x) r ] = (1)/(4 x) \theta _{3} (x)\theta _{4} (x) l [\theta _{3} (x) ^{4} - \theta _{4} (x) ^{4r]}

The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:

(d)/(d x) (\theta _{2} (x))/(\theta _{3} (x)) = (\theta _{2} (x)\theta _{4} (x) ^{4})/(4 x\theta _{3} (x)) (d)/(d x) (\theta _{2} (x))/(\theta _{4} (x)) = (\theta _{2} (x)\theta _{3} (x) ^{4})/(4 x\theta _{4} (x)) (d)/(d x) (\theta _{3} (x))/(\theta _{4} (x)) = (\theta _{3} (x) ^{5} - \theta _{3} (x)\theta _{4} (x) ^{4})/(4 x\theta _{4} (x))

For the derivation of these derivation formulas see the articles Nome (mathematics) and Modular lambda function!

Integrals of theta functions

For the theta functions these integrals^[13] are valid:

\int ^{1} _{0} \theta _{2} (x) d x = \sum ^{inf ty} ^{k =-inf ty} (4)/(( 2 k +1 ) ^{2+4}) = \pi\tanh(\pi) ≈ 3 . 129881 \int ^{1} _{0} \theta _{3} (x) d x = \sum ^{inf ty} ^{k =-inf ty} (1)/(k ^{2+1}) = \pi\coth(\pi) ≈ 3 . 153348 \int ^{1} _{0} \theta _{4} (x) d x = \sum ^{inf ty} ^{k =-inf ty} (( -1 ) ^{k})/(k ^{2+1}) = \pi\operatorname{csch}(\pi) ≈ 0 . 272029

The final results now shown are based on the general Cauchy sum formulas.

A solution to the heat equation

The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions.^[14] Taking to be real and with real and positive, we can write

\vartheta (x;it) =1+2 \sum ^{inf ty} _{n =1} \exp\left( - \pi n ^{2} t\right) \cos( 2 \pi nx)

which solves the heat equation

(\partial)/(\vartheta(x;it) = \partial t) (1)/(4 \pi) (\partial ^{2})/(\partial x ^{2}) \vartheta(x;it).

This theta-function solution is 1-periodic in, and as it approaches the periodic delta function, or Dirac comb, in the sense of distributions

\lim _{t\to} \vartheta(x;it) = \sum ^{inf ty} _{n =-inf ty} \delta(x - n) .

General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at with the theta function.

Relation to the Heisenberg group

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.

Generalizations

If is a quadratic form in variables, then the theta function associated with is

\theta _{F} (z) = \sum _{m\in \Z ^{n}} e ^{2\pi}

with the sum extending over the lattice of integers

Z ^{n} . This theta function is a modular form of weight (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,\hat{\theta} _{F} (z) = \sum ^{inf ty} _{k =0} R _{F(k)} e ^{2\pi} ,

the numbers are called the representation numbers of the form.

Theta series of a Dirichlet character

For a primitive Dirichlet character modulo and then

\theta _{\chi(z)} = (12 \sum)/(_{n=-infty}) ^{infty} \chi(n) n ^{\nu} e ^{2 i \pi n ^{2} z}

is a weight modular form of level and character

\chi(d) \left( (-1)/(d) \right) ^{\nu,} which means^[15] \theta _{\chi\left( (az + b)/(cz + d)} \right) = \chi(d) \left( (-1)/(d) \right) ^{\nu} \left( (\theta _{1 \left( (az + b)/(cz + d)} \right))/(\theta _{1(z)}) \right) ^{1+2\nu} \theta _{\chi(z)}

whenever

a,b,c,d\in \Z ^{4,} ad - bc =1 ,c \equiv 0 \bmod 4 q ^{2.}

Riemann theta function

Let

H _{n=\left\{F\in} M(n,\Complex) | F = F ^{T} , \operatorname{Im} F > 0 \right\}

be the set of symmetric square matrices whose imaginary part is positive definite.

H _{n} is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The -dimensional analogue of the modular group is the symplectic group ; for, . The -dimensional analogue of the congruence subgroups is played by\ker \ {\operatorname{Sp}( 2 n,\Z)\to \operatorname{Sp}( 2 n,\Z/k\Z) \ }.

Then, given, the Riemann theta function is defined as

\theta (z,\tau) = \sum _{m\in \Z ^{n}} \exp\left( 2 \pi i \left(\tfrac 12 m ^{T} \tau m + m ^{T} z \right)\right).

Here, is an -dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with and where 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 surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking to be the period matrix with respect to a canonical basis for its first homology group.

The Riemann theta converges absolutely and uniformly on compact subsets of

C ^{n} x H _{n} .

The functional equation is

\theta (z + a + \tau b, \tau) = \exp\left( 2 \pi i \left( - b ^{Tz-\tfrac12} b ^{T\tau} b\right)\right) \theta (z,\tau)

which holds for all vectors, and for all and .

Poincaré series

The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.

Derivation of the theta values

Identity of the Euler beta function

In the following, three important theta function values are to be derived as examples:

This is how the Euler beta function is defined in its reduced form:

\beta(x) = (\Gamma(x) ^{2})/(\Gamma( 2 x))

In general, for all natural numbers

n \isin N this formula of the Euler beta function is valid:(4 ^{-1/(n})/(\csc l ( n +2) (\pi)/(r )\beta l [ n +2) (n)/(2 (n +2 )) r ] = \int ^{inf ty} _{0} (1)/(\sqrt{x ^{n+2} +1) } \,\mathrm x

Exemplary elliptic integrals

In the following some Elliptic Integral Singular Values^[16] are derived:

The ensuing function has the following lemniscatically elliptic antiderivative:(1)/(\sqrt{x ^{4} +1) } = \frac\,\frac F\biggl[2\arctan(x); \frac{1}{2}\sqrt{2}\,\biggr] For the value n = 2 this identity appears:(1)/(4 \sqrt{ 2) }\csc\bigl(\frac\bigr)\beta\bigl(\frac\bigr) = \int_^ \frac \,\mathrmx = \biggl\_^ == (1)/(2) F l (\pi; (1)/(2) \sqrt{ 2 } r ) = K l ( (1)/(2) \sqrt{ 2 } r ) This result follows from that equation chain: {\color{ForestGreen}K l ( (1)/(2) \sqrt{ 2 } r ) = (1)/(\beta l ( 4) (1)/(4) r )}

The following function has the following equianharmonic elliptic antiderivative:(1)/(\sqrt{x ^{6} +1) } = \frac\,\frac\sqrt [4]F\biggl[2\arctan\biggl(\frac{\sqrt[4]\,x}\biggr);\frac(\sqrt + \sqrt)\biggr] For the value n = 4 that identity appears:(1)/(6 \sqrt[ 3 ]{ 2) }\csc\bigl(\frac\bigr)\beta\bigl(\frac \bigr) = \int_^ \frac \,\mathrmx = \biggl\_^ == (1)/(6) \sqrt[ 4 ]{ 27 } F l [ 2 \arctan(\sqrt[ 4 ]{ 3 }); (1)/(4) (\sqrt { 6 } + \sqrt{ 2 }) r ] = (2)/(9) \sqrt[ 4 ]{ 27 } K l [ (1)/(4) (\sqrt{ 6 } + \sqrt{ 2 }) r ] = (2)/(3) \sqrt[ 4 ]{ 3 } K l [ (1)/(4) (\sqrt{ 6 } - \sqrt{ 2 }) r ] This result follows from that equation chain: {\color{ForestGreen}K l [ (1)/(4) (\sqrt{ 6 } - \sqrt{ 2 }) r ] = (1)/(\sqrt[ 4 ]{ 3 }}\beta l ( 2 \sqrt[ 3 ]{ 2) (1)/(3) r )}

And the following function has the following elliptic antiderivative:(1)/(\sqrt{x ^{8} +1) } == (d)/(d x) (1)/(4) \sec l ( (\pi)/(r )F l \{ 2 \arctan l [ 8) (2 \cos(\pi/ 8 )x)/(\sqrt{x ^{4} + \sqrt{ 2) x ^{2} + 1 } - x ^{2} +1 } r ]; 2 \sqrt[ 4 ]{ 2 }\sin l ( (\pi)/(8) r ) r \} + (1)/(\sec l ( 4) (\pi)/(r )F l \{\arcsin l [ 8) (2 \cos(\pi/ 8 )x)/(r ];\tan l ( x ^{2} +1) (\pi)/(8) r ) r \} For the value n = 6 the following identity appears:(1)/(8 \sqrt[ 4 ]{ 2) }\csc\bigl(\frac\bigr)\beta\bigl(\frac \bigr) = \int_^ \frac \,\mathrmx == l \langle{\color{blue} (1)/(\sec l ( 4) (\pi)/(r )F l \{ 2 \arctan l [ 8) (2 \cos(\pi/ 8 )x)/(\sqrt{x ^{4} + \sqrt{ 2) x ^{2} + 1 } - x ^{2} + 1 } r ]; 2 \sqrt[ 4 ]{ 2 }\sin l ( (\pi)/(8) r ) r \} + (1)/(\sec l ( 4) (\pi)/(8) r )F l \{\arcsin l [ (2 \cos(\pi/ 8 )x)/(r ];\tan l ( x ^{2} +1) (\pi)/(8) r ) r \}} r \rangle ^{x =inf ty} _{x =0} = = (1)/(\sec l ( 4) (\pi)/(8) r )F l [\pi; 2 \sqrt[ 4 ]{ 2 }\sin l ( (\pi)/(8) r ) r ] = (1)/(\sec l ( 2) (\pi)/(8) r )K(\sqrt{ 2 \sqrt { 2 } - 2 } r ) =2 \sin l ( (\pi)/(8) r )K(\sqrt{ 2 } - 1 ) This result follows from that equation chain: {\color{ForestGreen}K(\sqrt{ 2 } - 1 ) = (1)/(8) \sqrt[ 4 ]{ 2 }(\sqrt{ 2 } + 1 ) \beta l ( (3)/(8) r )}

Combination of the integral identities with the nome

The elliptic nome function has these important values:

q(\tfrac{ 1 }{ 2 }\sqrt{ 2 }) = \exp( - \pi) q[\tfrac{ 1 }{ 4 }(\sqrt{ 6 } - \sqrt{ 2 })] = \exp( - \sqrt{ 3 }\pi) q(\sqrt{ 2 } - 1 ) = \exp( - \sqrt{ 2 }\pi)

For the proof of the correctness of these nome values, see the article Nome (mathematics)!

On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:

\theta _{3} [q(k)] = \sqrt{ 2 \pi ^{-1} K(k)}

\theta _{3} [\exp( - \pi)] = \theta _{3} [q(\tfrac{ 1 }{ 2 }\sqrt{ 2 })] = \sqrt{ 2 \pi ^{-1} K(\tfrac{ 1 }{ 2 }\sqrt{ 2 })} = 2 ^{-1/2} \pi ^{-1/2} \beta(\tfrac{ 1 }{ 4 } ) ^{1/2} = 2 ^{-1/4} \sqrt[ 4 ]{\pi}{\Gamma l (\tfrac{ 3 }{ 4 } r )} ^{-1} \theta _{3} [\exp( - \sqrt{ 3 }\pi )] = \theta _{3} l \{q l [\tfrac{ 1 }{ 4 }(\sqrt { 6 } - \sqrt{ 2 }) r ] r \} = \sqrt{ 2 \pi ^{-1} K l [\tfrac{ 1 }{ 4 }(\sqrt{ 6 } - \sqrt { 2 }) r ]} = 2 ^{-1/6} 3 ^{-1/8} \pi ^{-1/2} \beta(\tfrac{ 1 }{ 3 }) ^{1/} \theta _{3} [\exp( - \sqrt{ 2 }\pi )] = \theta _{3} [q(\sqrt{ 2 } - 1 )] = \sqrt{ 2 \pi ^{-1} K(\sqrt{ 2 } - 1 )} = 2 ^{-1/8} \cos(\tfrac{ 1 }{ 8 }\pi)\pi ^{-1/2} \beta(\tfrac{ 3 }{ 8 }) ^{1/2}

\theta _{4} [q(k)] = \sqrt[ 4 ]{ 1 - k ^{2}\sqrt{2\pi} ^{-1} K(k)}

\theta _{4} [\exp( - \sqrt{ 2 }\pi)] = \theta _{4} [q(\sqrt{ 2 } - 1 )] = \sqrt[ 4 ]{ 2 \sqrt{ 2 } - 2 }\sqrt{ 2 \pi ^{-1} K(\sqrt{ 2 } - 1 )} = 2 ^{-1/4} \cos(\tfrac{ 1 } { 8 }\pi) ^{1/2} \pi ^{-1/2} \beta(\tfrac{ 3 }{ 8 }) ^{1/2}

Partition sequences and Pochhammer products

Regular partition number sequence

The regular partition sequence

P(n) itself indicates the number of ways in which a positive integer number n can be splitted into positive integer summands. For the numbers n = 1 to n = 5 , the associated partition numbers P with all associated number partitions are listed in the following table:

Example values of P(n) and associated number partitions!n!P(n)!paying partitions

0	1	empty partition/empty sum
1	1	(1)
2	2	(1+1), (2)
3	3	(1+1+1), (1+2), (3)
4	5	(1+1+1+1), (1+1+2), (2+2), (1+3), (4)
5	7	(1+1+1+1+1), (1+1+1+2), (1+2+2), (1+1+3), (2+3), (1+4), (5)

The generating function of the regular partition number sequence can be represented via Pochhammer product in the following way:

\sum ^{inf ty} ^{k =0} P(k)x ^{k} = (1)/((x;x) _{infty}) = \theta _{3} (x) ^{-1/6} \theta _{4} (x) ^{-2/3} l [ (\theta _{3} (x) ^{4} - \theta _{4} (x) ^{4})/(16 x) r ] ^{-1/24}

The summandization of the now mentioned Pochhammer product is described by the Pentagonal number theorem in this way:

(x;x) _{infty} = 1 + \sum ^{inf ty} _{n =1} l [ - x ^{Fn(2n-1)} - x ^{Kr(2n-1)} + x ^{Fn(2n)} + x ^{Kr(2n)} r ]

The following basic definitions apply to the pentagonal numbers and the card house numbers:

Fn (z) = \tfrac{ 1 }{ 2 }z( 3 z -1 ) Kr (z) = \tfrac{ 1 }{ 2 }z( 3 z +1 )

As a further application^[17] one obtains a formula for the third power of the Euler product:

(x;x) ^{3} = \prod ^{inf ty} _{n =1} ( 1- x ^{n)} ^{3} = \sum ^{inf ty} ^{m =0} ( -1 ) ^{m(2m} +1 )x ^{m(m+1)/2}

Strict partition number sequence

And the strict partition sequence

Q(n) indicates the number of ways in which such a positive integer number n can be splitted into positive integer summands such that each summand appears at most once^[18] and no summand value occurs repeatedly. Exactly the same sequence^[19] is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:

Example values of Q(n) and associated number partitions!n!Q(n)!Number partitions without repeated summands!Number partitions with only odd addends

0	1	empty partition/empty sum	empty partition/empty sum
1	1	(1)	(1)
2	1	(2)	(1+1)
3	2	(1+2), (3)	(1+1+1), (3)
4	2	(1+3), (4)	(1+1+1+1), (1+3)
5	3	(2+3), (1+4), (5)	(1+1+1+1+1), (1+1+3), (5)
6	4	(1+2+3), (2+4), (1+5), (6)	(1+1+1+1+1+1), (1+1+1+3), (3+3), (1+5)
7	5	(1+2+4), (3+4), (2+5), (1+6), (7)	(1+1+1+1+1+1+1), (1+1+1+1+3), (1+3+3), (1+1+5), (7)
8	6	(1+3+4), (1+2+5), (3+5), (2+6), (1+7), (8)	(1+1+1+1+1+1+1+1), (1+1+1+1+1+3), (1+1+3+3), (1+1+1+ 5), (3+5), (1+7)

The generating function of the strict partition number sequence can be represented using Pochhammer's product:

\sum ^{inf ty} ^{k =0} Q(k)x ^{k} = (1)/((x;x ^{2 )} _{inf ty}) = \theta _{3} (x) ^{1/6} \theta _{4} (x) ^{-1/3} l [ (\theta _{3} (x) ^{4} - \theta _{4} (x) ^{4})/(16 x) r ] ^{1/24}

Overpartition number sequence

The Maclaurin series for the reciprocal of the function has the numbers of over partition sequence as coefficients with a positive sign:^[20]

(1)/(\theta _{4} (x)) = \prod ^{inf ty} _{n =1} (1+ x ^{n})/(1- x ^{n}) = \sum ^{inf ty} _{k =0} \overline{P}(k)x ^{k} (1)/(\theta _{4} (x)) = 1+2 x +4 x ^{2+8x} ^{3+14x} ^{4+24x} ^{5+40x} ^{6+64x} ^{7+100x} ^{8+154x} ^{9+232x} ^{10} + ...

If, for a given number

k , all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number^[21] of the marked partitions depending on k by the overpartition function \overline{P}(k) .

First example:

\overline{P}( 4 ) = 14

These 14 possibilities of partition markings exist for the sum 4:

(4), (4), (3+1), (3+1), (3+1), (3+1), (2+2), (2+2), (2+1+1), (2+1+1), (2+1+1), (2+1+1), (1+1+1+1), (1+1+1+1)

Second example:

\overline{P}( 5 ) = 24

These 24 possibilities of partition markings exist for the sum 5:

(5), (5), (4+1), (4+1), (4+1), (4+1), (3+2), (3+2), (3+2), (3+2), (3+1+1), (3+1+1), (3+1+1), (3+1+1), (2+2+1), (2+2+1), (2+2+1), (2+2+1),(2+1+1+1), (2+1+1+1), (2+1+1+1), (2+1+1+1), (1+1+1+1+1), (1+1+1+1+1)

Relations of the partition number sequences to each other

In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers

P(n) is under the code A000041, the sequence of strict partitions is Q(n) under the code A000009 and the sequence of superpartitions \overline{P}(n) under the code A015128. All parent partitions from index n = 1 are even.

The sequence of superpartitions

\overline{P}(n) can be written with the regular partition sequence P^[22] and the strict partition sequence Q^[23] can be generated like this:\overline{P}(n) = \sum ^{n} _{k =0} P(n - k)Q(k) In the following table of sequences of numbers, this formula should be used as an example:

n	P(n)	Q(n)	\overline{P}(n)
0	1	1	1 = 1*1
1	1	1	2 = 1 * 1 + 1 * 1
2	2	1	4 = 2 * 1 + 1 * 1 + 1 * 1
3	3	2	8 = 3 * 1 + 2 * 1 + 1 * 1 + 1 * 2
4	5	2	14 = 5 * 1 + 3 * 1 + 2 * 1 + 1 * 2 + 1 * 2
5	7	3	24 = 7 * 1 + 5 * 1 + 3 * 1 + 2 * 2 + 1 * 2 + 1 * 3

Related to this property, the following combination of two series of sums can also be set up via the function :\theta _{4} (x) = l [\sum ^{inf ty} _{k =0} P(k) x ^{k} r ] ^{-1} l [\sum ^{inf ty} _{k =0} Q(k) x ^{k} r ] ^{-1}

References

• Book: Milton . Abramowitz . Milton Abramowitz . Irene A. . Stegun . Irene Stegun . Handbook of Mathematical Functions . 1964 . . New York . 978-0-486-61272-0 . sec. 16.27ff. . Abramowitz and Stegun .
• Book: Akhiezer, Naum Illyich . Naum Akhiezer . Elements of the Theory of Elliptic Functions . 1970 . AMS Translations of Mathematical Monographs . 79 . 1990 . . Providence, RI . 978-0-8218-4532-5 .
• Book: Hershel M. . Farkas . Hershel M. Farkas . Irwin . Kra . Irwin Kra . Riemann Surfaces . 1980 . . New York . 978-0-387-90465-8 . ch. 6. . (for treatment of the Riemann theta)
• Book: G. H. . Hardy . G. H. Hardy . E. M. . Wright . E. M. Wright . An Introduction to the Theory of Numbers . 4th . 1959 . . Oxford .
• Book: Mumford, David . David Mumford . Tata Lectures on Theta I . 1983 . . Boston . 978-3-7643-3109-2 .
• Book: Pierpont, James . James Pierpont (mathematician) . Functions of a Complex Variable . New York . . 1959 .
• Book: Harry E. . Rauch . Harry Rauch . Hershel M. . Farkas . Theta Functions with Applications to Riemann Surfaces . 1974 . . Baltimore . 978-0-683-07196-2 .
  • Book: E. T. Whittaker . E. T. . Whittaker . G. N. . Watson . G. N. Watson . A Course in Modern Analysis . 4th . . Cambridge . 1927 . ch. 21. (history of Jacobi's functions)

Further reading

• Book: Farkas, Hershel M. . Krishnaswami . Alladi. Krishnaswami Alladi . Surveys in Number Theory . Developments in Mathematics . 17 . . 2008 . 978-0-387-78509-7 . Theta functions in complex analysis and number theory . 57–87 . 1206.11055 .
• Book: Schoeneberg, Bruno . Bruno Schoeneberg . Elliptic modular functions . Die Grundlehren der mathematischen Wissenschaften . 203 . . 1974 . 978-3-540-06382-7 . IX. Theta series . 203–226 .
• Ackerman . Michael . On the generating functions of certain Eisenstein series . Mathematische Annalen . 1 February 1979 . 244 . 1 . 75–81 . 10.1007/BF01420339 . 120045753 .

Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, .

• Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.

External links

• Web site: Elliptic functions for Matlab and Octave. Moiseev Igor.

Notes and References

1. Tyurin. Andrey N.. Quantization, Classical and Quantum Field Theory and Theta-Functions. math/0210466v1. 30 October 2002.
2. Book: Chang, Der-Chen . Heat Kernels for Elliptic and Sub-elliptic Operators . Birkhäuser . 2011 . 7.
3. See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation (e ^{z)} ^{\alpha=e} ^{\alpha} e ^{z}} when z is outside the strip - \pi<\operatorname{Im}z\le\pi . Here, \operatorname{Log} denotes the principal branch of the complex logarithm.
4. Book: Tata Lectures on Theta I . 2007 . Birkhäuser Boston . 978-0-8176-4572-4 . Modern Birkhäuser Classics . Boston, MA . 4 . en . 10.1007/978-0-8176-4577-9 .
5. \theta _{1(q)=0} for all q\in C with |q|< 1 .
6. Jinhee . Yi . Theta-function identities and the explicit formulas for theta-function and their applications . Journal of Mathematical Analysis and Applications . 381–400 . 292 . 2 . 10.1016/j.jmaa.2003.12.009 . 2004. free .
7. Berndt . Bruce C . Rebák . Örs . Explicit Values for Ramanujan's Theta Function (q) . Hardy-Ramanujan Journal . 9 January 2022 . 44 . 8923 . 10.46298/hrj.2022.8923 . 245851672 . free . 2112.11882 .
8. Yi . Jinhee . Theta-function identities and the explicit formulas for theta-function and their applications . Journal of Mathematical Analysis and Applications . 15 April 2004 . 292 . 2 . 381–400 . 10.1016/j.jmaa.2003.12.009 . free .
9. Andreas Dieckmann: Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta. Physikalisches Institut Universität Bonn, Abruf am 1. Oktober 2021.
10. Landau (1899) zitiert nach Borwein, Page 94, Exercise 3.
11. Web site: 2021-07-18. Number-theoretical, combinatorial and integer functions – mpmath 1.1.0 documentation.
12. Mező . István . A -Raabe formula and an integral of the fourth Jacobi theta function . 2012 . Journal of Number Theory . 133 . 2 . 692–704 . 10.1016/j.jnt.2012.08.025 . free . 2437/166217 . free .
13. Web site: integration - Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$ . en . 2022-08-13.
14. Ohyama. Yousuke. 1995. Differential relations of theta functions. Osaka Journal of Mathematics. EN. 32. 2. 431–450. 0030-6126.
15. Shimura, On modular forms of half integral weight
16. Web site: 2023-04-07. Elliptic Integral Singular Value. msu.edu.
17. https://www.researchgate.net/publication/235432739_Ramanujan%27s_theta-function_identities_involving_Lambert_series Ramanujan's theta-function identities involving Lambert series
18. Web site: 2022-03-09. code golf - Strict partitions of a positive integer.
19. Web site: 2022-03-09. A000009 - OEIS.
20. Mahlburg . Karl . The overpartition function modulo small powers of 2 . Discrete Mathematics . 286 . 3 . 2004 . 10.1016/j.disc.2004.03.014 . 263–267.
21. en. Elsevier Enhanced Reader. Discrete Mathematics . 28 April 2009 . 309 . 8 . 2528–2532 . 10.1016/j.disc.2008.05.007 . Kim . Byungchan . free .
22. Web site: 2022-03-11. Eric W. Weisstein. en. Partition Function P.
23. Web site: 2022-03-11. Eric W. Weisstein. en. Partition Function Q.