Theta function

For other θ functions, see Theta function (disambiguation).

Jacobi's theta function θ₁ with nome q = e^iπτ = 0.1e^0.1iπ: ${\begin{aligned}\theta _{1}(z,q)&=2q^{\frac {1}{4}}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin(2n+1)z\\&=\sum _{n=-\infty }^{\infty }(-1)^{n-{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}e^{(2n+1)iz}.\end{aligned}}$

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.

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 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 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".

Throughout this article, $(e^{\pi i\tau })^{\alpha }$ should be interpreted as $e^{\alpha \pi i\tau }$ (in order to resolve issues of choice of branch).

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 z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula

${\begin{aligned}\vartheta (z;\tau )&=\sum _{n=-\infty }^{\infty }\exp \left(\pi in^{2}\tau +2\pi inz\right)\\&=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2\pi nz)\\&=\sum _{n=-\infty }^{\infty }q^{n^{2}}\eta ^{n}\end{aligned}}$

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

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

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

$\vartheta (z+\tau ;\tau )=\exp {\bigl (}-\pi i(\tau +2z){\bigr )}\vartheta (z;\tau ).$

Thus, in general,

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

for any integers a and b.

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

$$\left|{\frac {\vartheta (z+a+b\tau ;\tau )}{\vartheta (z;\tau )}}\right|=\exp \left(\pi (b^{2}\Im (\tau )+2b\Im (z))\right)$$

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:

Theorem — If $f:\mathbb {C} \to \mathbb {C}$ is entire and nonconstant, and satisfies the functional equations ${\begin{cases}f(z+1)=f(z)\\f(z+\tau )=e^{az+2\pi ib}f(z)\end{cases}}$ for some constant $a,b\in \mathbb {C}$.

If $a=0$, then $b=\tau$ and $f(z)=e^{2\pi iz}$. If $a=-2\pi i$, then $f(z)=C\vartheta (z+{\frac {1}{2}}\tau +b,\tau )$ for some nonzero $C\in \mathbb {C}$.

Theta function θ₁ with different nome q = e^iπτ. The black dot in the right-hand picture indicates how q changes with τ.

Theta function θ₁ with different nome q = e^iπτ. The black dot in the right-hand picture indicates how q changes with τ.

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

${\begin{aligned}\vartheta _{01}(z;\tau )&=\vartheta \left(z+{\tfrac {1}{2}};\tau \right)\\[3pt]\vartheta _{10}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi iz\right)\vartheta \left(z+{\tfrac {1}{2}}\tau ;\tau \right)\\[3pt]\vartheta _{11}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi i\left(z+{\tfrac {1}{2}}\right)\right)\vartheta \left(z+{\tfrac {1}{2}}\tau +{\tfrac {1}{2}};\tau \right).\end{aligned}}$

This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = e^iπτ rather than τ. In Jacobi's notation the θ-functions are written:

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

Jacobi theta 1

Jacobi theta 2

Jacobi theta 3

Jacobi theta 4

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 z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane. Alternatively, we obtain four functions of q only, defined on the unit disk $|q|<1$. They are sometimes called theta constants:

${\begin{aligned}\vartheta _{11}(0;\tau )&=-\theta _{1}(q)=-\sum _{n=-\infty }^{\infty }(-1)^{n-1/2}q^{(n+1/2)^{2}}\\\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\\\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\\\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\end{aligned}}$

with the nome q = e^iπτ. 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

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

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.

Elliptic nome

Definition and identities to the theta functions

Since the Jacobi functions are defined in terms of the elliptic modulus $k(\tau )$, we need to invert this and find $\tau$ in terms of $k$. We start from $k'={\sqrt {1-k^{2}}}$, the complementary modulus. As a function of $\tau$ it is

$k'(\tau )={\sqrt {1-k^{2}}}={\biggl \{}{\vartheta _{01}[q(k)] \over \vartheta _{00}[q(k)]}{\biggr \}}^{2}$

Let us define the elliptic nome and the complete elliptic integral of the first kind:

$q(k)=\exp {\biggl [}-\pi \,{\frac {K({\sqrt {1-k^{2}}})}{K(k)}}{\biggr ]}=\exp {\biggl [}-\pi \,{\frac {K'(k)}{K(k)}}{\biggr ]}$

These identites exist between the elliptic integral K, the modulus k, the nome q function and the theta functions:

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

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

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

$\theta _{4}[q(k)]=\theta _{4}[q(k)^{2}]{\sqrt[{8}]{1-k^{2}}}$

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

$\theta _{3}[q(k)^{2}]=\theta _{3}[q(k)]\cos[{\tfrac {1}{2}}\arcsin(k)]$

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

$K(k)=\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-k^{2}\sin(\varphi )^{2}}}}\partial \varphi$

$K(k)={\frac {\pi }{2}}\sum _{a=0}^{\infty }{\frac {[(2a)!]^{2}}{16^{a}(a!)^{4}}}k^{2a}$

An identical definition of the nome function can me produced by using a series. Following function has this identity:

${\frac {1-{\sqrt[{4}]{1-k^{2}}}}{1+{\sqrt[{4}]{1-k^{2}}}}}={\frac {\vartheta _{00}[q(k)]-\vartheta _{01}[q(k)]}{\vartheta _{00}[q(k)]+\vartheta _{01}[q(k)]}}={\biggl [}\sum _{n=1}^{\infty }2\,q(k)^{(2n-1)^{2}}{\biggr ]}{\biggl [}1+\sum _{n=1}^{\infty }2\,q(k)^{4n^{2}}{\biggr ]}^{-1}$

Since we may reduce to the case where the imaginary part of $\tau$ is greater than or equal to ${\sqrt {3}}/2$ (see Modular group), we can assume the absolute value of $q$ is less than or equal to $\exp(-\pi {\sqrt {3}}/2)\approx 0.0658$; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for $q$. By solving this function after q we get this result:

$q(k)=\sum _{n=1}^{\infty }{\frac {{\text{Sw}}(n)}{2^{4n-3}}}{\biggl (}{\frac {1-{\sqrt[{4}]{1-k^{2}}}}{1+{\sqrt[{4}]{1-k^{2}}}}}{\biggr )}^{4n-3}={\biggl [}\sum _{n=1}^{\infty }{\frac {{\text{Sw}}(n)}{2^{4n-3}}}{\biggl (}{\frac {k}{1+{\sqrt {1-k^{2}}}}}{\biggr )}^{4n-3}{\biggr ]}^{2}=k^{2}{\biggl \{}{\frac {1}{2}}+{\biggl [}\sum _{n=1}^{\infty }{\frac {{\text{Sw}}(n+1)}{2^{4n+1}}}k^{2n}{\biggr ]}{\biggr \}}^{4}$

$q(k)=k^{2}{\bigl (}{\color {limegreen}{\frac {\color {navy}1}{2}}+{\frac {\color {navy}2}{32}}k^{2}+{\frac {\color {navy}15}{512}}k^{4}+{\frac {\color {navy}150}{8192}}k^{6}+{\frac {\color {navy}1707}{131072}}k^{8}+\ldots }{\bigr )}^{4}$

The Schwarz numbers are in the numerators and the doubles of the powers of sixteen are in the denominators.

In relation to this following limits are valid:

$\lim _{n\to \infty }{\sqrt[{n}]{\text{Sw(n)}}}=16$

$\lim _{n\to \infty }{\frac {\text{Sw(n + 1)}}{\text{Sw(n)}}}=16$

This table shows numbers of the Schwarz integer sequence A002103 accurately:

Sw(1)	Sw(2)	Sw(3)	Sw(4)	Sw(5)	Sw(6)	Sw(7)	Sw(8)
1	2	15	150	1707	20910	268616	3567400

Elliptic integer sequences

The Silesian German mathematician Hermann Amandus Schwarz wrote in his work Formeln und Lehrsätze zum Gebrauche der elliptischen Funktionen in the chapter Berechnung der Grösse k on pages 54 to 56 an integer number sequence. This Schwarz number sequence Sw(n) was also analyzed by the mathematicians Karl Theodor Wilhelm Weierstrass and Louis Melville Milne-Thomson in the 20th century. The mathematician Adolf Kneser determined a synthesis method for this sequence based on the following pattern:

${\text{Sw}}(n+1)={\frac {2}{n}}\sum _{m=1}^{n}{\text{Sw}}(m){\text{Kn}}(n+1-m)$

The mathematician Karl Heinrich Schellbach also researched this integer sequence relation and in his work Die Lehre von den elliptischen Integralen und den Thetafunktionen he dealt with it in detail. The Schwarz sequence Sw(n) is entered in the online encyclopedia of number sequences under the number A002103 and the Kneser sequence Kn(n) is entered under the number A227503. The Kneser integer sequence Kn(n) can be constructed by using a special Apéry sequence Ap(n) (OEIS A036917) defined as follows:

${\text{Ap}}(n)=\sum _{a=0}^{n-1}{\binom {2a}{a}}^{2}{\binom {2n-2-2a}{n-1-a}}^{2}$

And in this way the Kneser number sequence can be defined for every natural number n:

${\text{Kn}}(n+1)=2^{4n+1}-{\tfrac {1}{8}}{\text{Ap}}(n+2)-\sum _{b=1}^{n}{\text{Kn}}(b){\text{Ap}}(n+2-b)$

The Kneser sequence appears in the Taylor series of the period ratio (half period ratio):

${\frac {1}{4}}\ln {\bigl (}{\frac {16}{x^{2}}}{\bigr )}-{\frac {\pi \,K'(x)}{4\,K(x)}}=\sum _{n=1}^{\infty }{\frac {{\text{Kn}}(n)}{2^{4n-1}n}}\,x^{2n}$

${\color {limegreen}{\frac {1}{4}}\ln {\bigl (}{\frac {16}{x^{2}}}{\bigr )}-{\frac {\pi \,K'(x)}{4\,K(x)}}={\frac {\color {cornflowerblue}1}{8}}x^{2}+{\frac {\color {cornflowerblue}13}{256}}x^{4}+{\frac {\color {cornflowerblue}184}{6144}}x^{6}+{\frac {\color {cornflowerblue}2701}{131072}}x^{8}+{\frac {\color {cornflowerblue}40456}{2621440}}x^{10}+\ldots }$

The derivative of this equation after $x$ leads to this equation that shows the generating function of the Kneser number sequence:

${\frac {\pi ^{2}}{8x(1-x^{2})K(x)^{2}}}-{\frac {1}{2x}}=\sum _{n=1}^{\infty }{\frac {{\text{Kn}}(n)}{2^{4n-2}}}x^{2n-1}$

${\color {limegreen}{\frac {\pi ^{2}}{8x(1-x^{2})K(x)^{2}}}-{\frac {1}{2x}}={\frac {\color {cornflowerblue}1}{4}}x+{\frac {\color {cornflowerblue}13}{64}}x^{3}+{\frac {\color {cornflowerblue}184}{1024}}x^{5}+{\frac {\color {cornflowerblue}2701}{16384}}x^{7}+{\frac {\color {cornflowerblue}40456}{262144}}x^{9}+\ldots }$

This result appears because of the Legendre's relation $K\,E'+E\,K'-K\,K'={\tfrac {1}{2}}\pi$ in the numerator.

Following table contains the Schwarz numbers and the Kneser numbers and the Apery numbers:

sequence construction method according to Kneser

Index n	$[(2n)!]^{2}/(n!)^{4}$	Ap(n) (A036917)	Kn(n) (A227503)	Sw(n) (A002103)
1	1	1	1	1
2	4	8	13	2
3	36	88	184	15
4	400	1088	2701	150
5	4900	14296	40456	1707
6	853776	195008	613720	20910
7	11778624	2728384	9391936	268616
8	165636900	38879744	144644749	3567400

In the following, it will be shown as an example how the Schwarz numbers are built up successively. For this, the examples with the Schwarz numbers Sw(4) = 150, Sw(5) = 1707 and Sw(6) = 20910 are used:

$\mathrm {Sw} (4)={\frac {2}{3}}\sum _{m=1}^{3}\mathrm {Sw} (m)\,\mathrm {Kn} (4-m)={\frac {2}{3}}{\bigl [}{\color {navy}\mathrm {Sw} (1)}\,{\color {cornflowerblue}\mathrm {Kn} (3)}+{\color {navy}\mathrm {Sw} (2)}\,{\color {cornflowerblue}\mathrm {Kn} (2)}+{\color {navy}\mathrm {Sw} (3)}\,{\color {cornflowerblue}\mathrm {Kn} (1)}{\bigr ]}=$

${\color {navy}\mathrm {Sw} (4)}={\frac {2}{3}}{\bigl (}{\color {navy}1}\times {\color {cornflowerblue}184}+{\color {navy}2}\times {\color {cornflowerblue}13}+{\color {navy}15}\times {\color {cornflowerblue}1}{\bigr )}={\color {navy}150}$

$\mathrm {Sw} (5)={\frac {2}{4}}\sum _{m=1}^{4}\mathrm {Sw} (m)\,\mathrm {Kn} (5-m)={\frac {2}{4}}{\bigl [}{\color {navy}\mathrm {Sw} (1)}\,{\color {cornflowerblue}\mathrm {Kn} (4)}+{\color {navy}\mathrm {Sw} (2)}\,{\color {cornflowerblue}\mathrm {Kn} (3)}+{\color {navy}\mathrm {Sw} (3)}\,{\color {cornflowerblue}\mathrm {Kn} (2)}+{\color {navy}\mathrm {Sw} (4)}\,{\color {cornflowerblue}\mathrm {Kn} (1)}{\bigr ]}=$

${\color {navy}\mathrm {Sw} (5)}={\frac {2}{4}}{\bigl (}{\color {navy}1}\times {\color {cornflowerblue}2701}+{\color {navy}2}\times {\color {cornflowerblue}184}+{\color {navy}15}\times {\color {cornflowerblue}13}+{\color {navy}150}\times {\color {cornflowerblue}1}{\bigr )}={\color {navy}1707}$

$\mathrm {Sw} (6)={\frac {2}{5}}\sum _{m=1}^{5}\mathrm {Sw} (m)\,\mathrm {Kn} (6-m)={\frac {2}{5}}{\bigl [}{\color {navy}\mathrm {Sw} (1)}\,{\color {cornflowerblue}\mathrm {Kn} (5)}+{\color {navy}\mathrm {Sw} (2)}\,{\color {cornflowerblue}\mathrm {Kn} (4)}+{\color {navy}\mathrm {Sw} (3)}\,{\color {cornflowerblue}\mathrm {Kn} (3)}+{\color {navy}\mathrm {Sw} (4)}\,{\color {cornflowerblue}\mathrm {Kn} (2)}+{\color {navy}\mathrm {Sw} (5)}\,{\color {cornflowerblue}\mathrm {Kn} (1)}{\bigr ]}=$

${\color {navy}\mathrm {Sw} (6)}={\frac {2}{5}}{\bigl (}{\color {navy}1}\times {\color {cornflowerblue}40456}+{\color {navy}2}\times {\color {cornflowerblue}2701}+{\color {navy}15}\times {\color {cornflowerblue}184}+{\color {navy}150}\times {\color {cornflowerblue}13}+{\color {navy}1707}\times {\color {cornflowerblue}1}{\bigr )}={\color {navy}20910}$

Jacobi identities

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 ≡ n² mod 2). For the second, let

$\alpha =(-i\tau )^{\frac {1}{2}}\exp \left({\frac {\pi }{\tau }}iz^{2}\right).$

Then

${\begin{aligned}\vartheta _{00}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{00}(z;\tau )\quad &\vartheta _{01}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{10}(z;\tau )\\[3pt]\vartheta _{10}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{01}(z;\tau )\quad &\vartheta _{11}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=-i\alpha \,\vartheta _{11}(z;\tau ).\end{aligned}}$

Theta functions in terms of the nome

Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the nome q, where w = e^πiz and q = e^πiτ. In this form, the functions become

${\begin{aligned}\vartheta _{00}(w,q)&=\sum _{n=-\infty }^{\infty }\left(w^{2}\right)^{n}q^{n^{2}}\quad &\vartheta _{01}(w,q)&=\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n}q^{n^{2}}\\[3pt]\vartheta _{10}(w,q)&=\sum _{n=-\infty }^{\infty }\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w,q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}.\end{aligned}}$

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 fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers.

Product representations

The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have

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

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 q = e^πiτ (noting some authors instead set q = e^2πiτ) and take w = e^πiz then

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

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

$\vartheta (z;\tau )=\prod _{m=1}^{\infty }{\big (}1-\exp(2m\pi i\tau ){\big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau +2\pi iz{\big )}{\Big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau -2\pi iz{\big )}{\Big )}.$

In terms of w and q:

${\begin{aligned}\vartheta (z;\tau )&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+q^{2m-1}w^{2}\right)\left(1+{\frac {q^{2m-1}}{w^{2}}}\right)\\&=\left(q^{2};q^{2}\right)_{\infty }\,\left(-w^{2}q;q^{2}\right)_{\infty }\,\left(-{\frac {q}{w^{2}}};q^{2}\right)_{\infty }\\&=\left(q^{2};q^{2}\right)_{\infty }\,\theta \left(-w^{2}q;q^{2}\right)\end{aligned}}$

where (  ;  )_∞ is the q-Pochhammer symbol and θ(  ;  ) is the q-theta function. Expanding terms out, the Jacobi triple product can also be written

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

which we may also write as

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

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

${\begin{aligned}\vartheta _{01}(z\mid q)&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]\vartheta _{10}(z\mid q)&=2q^{\frac {1}{4}}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right),\\[3pt]\vartheta _{11}(z\mid q)&=-2q^{\frac {1}{4}}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).\end{aligned}}$

In particular,

$$\lim _{q\to 0}{\frac {\vartheta _{10}(z\mid q)}{2q^{\frac {1}{4}}}}=\cos(\pi z),\quad \lim _{q\to 0}{\frac {-\vartheta _{11}(z\mid q)}{2q^{-{\frac {1}{4}}}}}=\sin(\pi z)$$

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:

${\begin{aligned}\vartheta _{00}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{01}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{10}(z;\tau )&=-ie^{i\pi z+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi u+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{11}(z;\tau )&=e^{i\pi z+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u.\end{aligned}}$

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). Define,

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

with the nome $q=e^{\pi i\tau },$ $\tau =n{\sqrt {-1}},$ and Dedekind eta function $\eta (\tau ).$ Then for $n=1,2,3,\dots$

${\begin{aligned}\varphi \left(e^{-\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}={\sqrt {2}}\,\eta \left({\sqrt {-1}}\right)\\\varphi \left(e^{-2\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {2+{\sqrt {2}}}}{2}}\\\varphi \left(e^{-3\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {1+{\sqrt {3}}}}{\sqrt[{8}]{108}}}\\\varphi \left(e^{-4\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {2+{\sqrt[{4}]{8}}}{4}}\\\varphi \left(e^{-5\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {2+{\sqrt {5}}}{5}}}\\\varphi \left(e^{-6\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}}}{\sqrt[{8}]{12^{3}}}}\\\varphi \left(e^{-7\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}}}{{\sqrt[{8}]{14^{3}}}\cdot {\sqrt[{16}]{7}}}}\\\varphi \left(e^{-8\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {2+{\sqrt {2}}}}+{\sqrt[{8}]{128}}}{4}}\\\varphi \left(e^{-9\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}\\\varphi \left(e^{-10\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{64}}+{\sqrt[{4}]{80}}+{\sqrt[{4}]{81}}+{\sqrt[{4}]{100}}}}{\sqrt[{4}]{200}}}\\\varphi \left(e^{-11\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {4+{\sqrt {11}}-3{\sqrt {3}}\tanh \left({\tfrac {1}{4}}\operatorname {arcosh} \left({\tfrac {7}{4}}\right)+{\tfrac {1}{6}}\operatorname {artanh} \left({\tfrac {27}{47}}{\sqrt {3}}\right)\right)}{{\sqrt[{8}]{44}}\cdot {\sqrt {-66+22{\sqrt {11}}}}}}\\\varphi \left(e^{-12\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{2}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{24}}}}{2{\sqrt[{8}]{108}}}}\\\varphi \left(e^{-13\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {-{\tfrac {1}{13}}+{\tfrac {2}{13}}{\sqrt {3}}\coth \left({\tfrac {1}{6}}\operatorname {artanh} \left({\tfrac {6}{11}}{\sqrt {3}}\right)\right)}}\\\varphi \left(e^{-14\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}+{\sqrt {10+2{\sqrt {7}}}}+{\sqrt[{8}]{28}}{\sqrt {4+{\sqrt {7}}}}}}{\sqrt[{16}]{28^{7}}}}\\\varphi \left(e^{-15\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {7+3{\sqrt {3}}+{\sqrt {5}}+{\sqrt {15}}+{\sqrt[{4}]{60}}+{\sqrt[{4}]{1500}}}}{{\sqrt[{8}]{12^{3}}}\cdot {\sqrt {5}}}}\\2\varphi \left(e^{-16\pi }\right)&=\varphi \left(e^{-4\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{1+{\sqrt {2}}}}{\sqrt[{16}]{128}}}\\\varphi \left(e^{-17\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {2}}(1+{\sqrt[{4}]{17}})+{\sqrt[{8}]{17}}{\sqrt {5+{\sqrt {17}}}}}{\sqrt {17+17{\sqrt {17}}}}}\\2\varphi \left(e^{-20\pi }\right)&=\varphi \left(e^{-5\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {3+2{\sqrt[{4}]{5}}}{5{\sqrt {2}}}}}\\6\varphi \left(e^{-36\pi }\right)&=3\varphi \left(e^{-9\pi }\right)+2\varphi \left(e^{-4\pi }\right)-\varphi \left(e^{-\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt[{3}]{{\sqrt[{4}]{2}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{216}}}}\end{aligned}}$

Note that the following modular identities hold:

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

where $s(q)=s\left(e^{\pi i\tau }\right)=-R\left(-e^{-\pi i/(5\tau )}\right)$ is the Rogers–Ramanujan continued fraction:

${\begin{aligned}s(q)&={\sqrt[{5}]{\tan \left({\frac {1}{2}}\arctan \left({\frac {5}{2}}{\frac {\varphi ^{2}\left(q^{5}\right)}{\varphi ^{2}(q)}}-{\frac {1}{2}}\right)\right)\cot ^{2}\left({\frac {1}{2}}\operatorname {arccot} \left({\frac {5}{2}}{\frac {\varphi ^{2}\left(q^{5}\right)}{\varphi ^{2}(q)}}-{\frac {1}{2}}\right)\right)}}\\&={\cfrac {e^{-\pi i/(25\tau )}}{1-{\cfrac {e^{-\pi i/(5\tau )}}{1+{\cfrac {e^{-2\pi i/(5\tau )}}{1-\ddots }}}}}}\end{aligned}}$

Equianharmonic values

The mathematician Bruce Berndt found out further values of the theta function:

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

Further values

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

${\begin{array}{lll}\varphi \left(\exp(-{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{7/8}\\\varphi \left(\exp(-2{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{1/8}{\Bigl (}1+{\sqrt {{\sqrt {2}}-1}}{\Bigr )}\\\varphi \left(\exp(-3{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{3/8}3^{-1/2}({\sqrt {3}}+1){\sqrt {\tan({\tfrac {5}{24}}\pi )}}\\\varphi \left(\exp(-4{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{-1/8}{\Bigl (}1+{\sqrt[{4}]{2{\sqrt {2}}-2}}{\Bigr )}\\\varphi \left(\exp(-5{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{15/8}\\&&\cdot {\biggl (}{\tfrac {1}{5}}{\sqrt {3}}\cos({\tfrac {1}{10}}\pi ){\dfrac {{(9+{\sqrt {6}})}^{1/3}(1+{\sqrt {3}})+{\sqrt {2}}~3^{1/6}5^{1/3}}{({9+{\sqrt {6}}})^{1/3}(1+{\sqrt {3}})-{\sqrt {2}}~{3^{1/6}5^{1/3}}}}-{\tfrac {1}{5}}({\sqrt {5}}+{\sqrt {2}})\sin({\tfrac {1}{5}}\pi ){\biggr )}\\\varphi \left(\exp(-{\sqrt {6}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {5}{24}}\right){\Gamma \left({\tfrac {5}{12}}\right)}^{-1/2}2^{-13/24}3^{-1/8}{\sqrt {\sin({\tfrac {5}{12}}\pi )}}\\\varphi \left(\exp(-{\tfrac {1}{2}}{\sqrt {6}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {5}{24}}\right){\Gamma \left({\tfrac {5}{12}}\right)}^{-1/2}2^{5/24}3^{-1/8}\sin({\tfrac {5}{24}}\pi )\end{array}}$

Some series identities

The next two series identities were proved by István Mező:

${\begin{aligned}\theta _{4}^{2}(q)&=iq^{\frac {1}{4}}\sum _{k=-\infty }^{\infty }q^{2k^{2}-k}\theta _{1}\left({\frac {2k-1}{2i}}\ln q,q\right),\\[6pt]\theta _{4}^{2}(q)&=\sum _{k=-\infty }^{\infty }q^{2k^{2}}\theta _{4}\left({\frac {k\ln q}{i}},q\right).\end{aligned}}$

These relations hold for all 0 < q < 1. Specializing the values of q, we have the next parameter free sums

${\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}=i\sum _{k=-\infty }^{\infty }e^{\pi \left(k-2k^{2}\right)}\theta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right)$

${\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}$

Zeros of the Jacobi theta functions

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

${\begin{aligned}\vartheta (z;\tau )=\vartheta _{00}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}+{\frac {\tau }{2}}\\[3pt]\vartheta _{11}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau \\[3pt]\vartheta _{10}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}\\[3pt]\vartheta _{01}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {\tau }{2}}\end{aligned}}$

where m, n are arbitrary integers.

Relation to the Riemann zeta function

The relation

$\vartheta \left(0;-{\frac {1}{\tau }}\right)=\left(-i\tau \right)^{\frac {1}{2}}\vartheta (0;\tau )$

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

$\Gamma \left({\frac {s}{2}}\right)\pi ^{-{\frac {s}{2}}}\zeta (s)={\frac {1}{2}}\int _{0}^{\infty }{\bigl (}\vartheta (0;it)-1{\bigr )}t^{\frac {s}{2}}{\frac {\mathrm {d} t}{t}}$

which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z ≠ 0 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

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

where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of ℘(z) at z = 0 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 q-gamma function via the relation

$\left(\Gamma _{q^{2}}(x)\Gamma _{q^{2}}(1-x)\right)^{-1}={\frac {q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)_{\infty }^{3}\left(q^{2}-1\right)}}\theta _{4}\left({\frac {1}{2i}}(1-2x)\log q,{\frac {1}{q}}\right).$

Relations to Dedekind eta function

Let η(τ) be the Dedekind eta function, and the argument of the theta function as the nome q = e^πiτ. Then,

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

and,

$\theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q)=2\eta ^{3}(\tau ).$

See also the Weber modular functions.

Elliptic modulus

The elliptic modulus is

$k(\tau )={\frac {\vartheta _{10}(0;\tau )^{2}}{\vartheta _{00}(0;\tau )^{2}}}$

and the complementary elliptic modulus is

$k'(\tau )={\frac {\vartheta _{01}(0;\tau )^{2}}{\vartheta _{00}(0;\tau )^{2}}}$

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. Taking z = x to be real and τ = it with t real and positive, we can write

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

which solves the heat equation

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

This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic delta function, or Dirac comb, in the sense of distributions

$\lim _{t\to 0}\vartheta (x;it)=\sum _{n=-\infty }^{\infty }\delta (x-n)$.

General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 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 F is a quadratic form in n variables, then the theta function associated with F is

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

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

${\hat {\theta }}_{F}(z)=\sum _{k=0}^{\infty }R_{F}(k)e^{2\pi ikz},$

the numbers R_F(k) are called the representation numbers of the form.

Theta series of a Dirichlet character

For χ a primitive Dirichlet character modulo q and ν = 1 − χ(−1)/2 then

$\theta _{\chi }(z)={\frac {1}{2}}\sum _{n=-\infty }^{\infty }\chi (n)n^{\nu }e^{2i\pi n^{2}z}$

is a weight 1/2 + ν modular form of level 4q² and character

$\chi (d)\left({\frac {-1}{d}}\right)^{\nu },$

which means

$\theta _{\chi }\left({\frac {az+b}{cz+d}}\right)=\chi (d)\left({\frac {-1}{d}}\right)^{\nu }\left({\frac {\theta _{1}\left({\frac {az+b}{cz+d}}\right)}{\theta _{1}(z)}}\right)^{1+2\nu }\theta _{\chi }(z)$

whenever

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

Ramanujan theta function

Further information: Ramanujan theta function and mock theta function

Riemann theta function

Let

$\mathbb {H} _{n}=\left\{F\in M(n,\mathbb {C} )\,{\big |}\,F=F^{\mathsf {T}}\,,\,\operatorname {Im} F>0\right\}$

be the set of symmetric square matrices whose imaginary part is positive definite. $\mathbb {H} _{n}$ 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,$\mathbb {Z}$); for n = 1, Sp(2,$\mathbb {Z}$) = SL(2,$\mathbb {Z}$). The n-dimensional analogue of the congruence subgroups is played by

$\ker {\big \{}\operatorname {Sp} (2n,\mathbb {Z} )\to \operatorname {Sp} (2n,\mathbb {Z} /k\mathbb {Z} ){\big \}}.$

Then, given τ ∈ $\mathbb {H} _{n}$, the Riemann theta function is defined as

$\theta (z,\tau )=\sum _{m\in \mathbb {Z} ^{n}}\exp \left(2\pi i\left({\tfrac {1}{2}}m^{\mathsf {T}}\tau m+m^{\mathsf {T}}z\right)\right).$

Here, z ∈ $\mathbb {C} ^{n}$ 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 τ ∈ $\mathbb {H}$ where $\mathbb {H}$ 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 τ 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 $\mathbb {C} ^{n}\times \mathbb {H} _{n}$.

The functional equation is

$\theta (z+a+\tau b,\tau )=\exp \left(2\pi i\left(-b^{\mathsf {T}}z-{\tfrac {1}{2}}b^{\mathsf {T}}\tau b\right)\right)\theta (z,\tau )$

which holds for all vectors a, b ∈ $\mathbb {Z} ^{n}$, and for all z ∈ $\mathbb {C} ^{n}$ and τ ∈ $\mathbb {H} _{n}$.

Poincaré series

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

Theta function coefficients

If a and b are positive integers, χ(n) any arithmetical function and |q| < 1, then

$\sum _{n=1}^{\infty }\chi (n)q^{an^{2}+bn}=\sum _{n=1}^{\infty }q^{n}\sum _{\stackrel {d|n}{ad^{2}+bd=n}}\chi (d).$

The general case, where f(n) and χ(n) are any arithmetical functions, and f(n) : $\mathbb {N}$ → $\mathbb {N}$ is strictly increasing with f(0) = 0, is

$\sum _{n=1}^{\infty }\chi (n)q^{f(n)}=\sum _{n=1}^{\infty }q^{n}\sum _{d|n}\sum _{f(\delta )|d}\chi (\delta )\mu \left({\frac {d}{f(\delta )}}\right).$

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:

$\beta (x)={\frac {\Gamma (x)^{2}}{\Gamma (2x)}}$

In general, for all natural numbers n ∈ ℕ this formula of the Euler beta function is valid:

${\frac {4^{-1/(n+2)}}{n+2}}\csc {\bigl (}{\frac {\pi }{n+2}}{\bigr )}\beta {\biggl [}{\frac {n}{2(n+2)}}{\biggr ]}=\int _{0}^{\infty }{\frac {1}{\sqrt {x^{n+2}+1}}}\,\mathrm {d} x$

Exemplary elliptic integrals

In the following some Elliptic Integral Singular Values are derived:

The ensuing function has the following lemniscatically elliptic antiderivative: ${\frac {1}{\sqrt {x^{4}+1}}}={\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {1}{2}}F{\biggl [}2\arctan(x);{\frac {1}{2}}{\sqrt {2}}\,{\biggr ]}$ For the value $n=2$ this identity appears: ${\frac {1}{4{\sqrt {2}}}}\csc {\bigl (}{\frac {\pi }{4}}{\bigr )}\beta {\bigl (}{\frac {1}{4}}{\bigr )}=\int _{0}^{\infty }{\frac {1}{\sqrt {x^{4}+1}}}\,\mathrm {d} x={\biggl \{}{\color {blue}{\frac {1}{2}}F{\biggl [}2\arctan(x);{\frac {1}{2}}{\sqrt {2}}\,{\biggr ]}}{\biggr \}}_{x=0}^{x=\infty }=$ $={\frac {1}{2}}F{\bigl (}\pi ;{\frac {1}{2}}{\sqrt {2}}{\bigr )}=K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}$ This result follows from that equation chain: ${\color {ForestGreen}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}={\frac {1}{4}}\beta {\bigl (}{\frac {1}{4}}{\bigr )}}$

The following function has the following equianharmonic elliptic antiderivative: ${\frac {1}{\sqrt {x^{6}+1}}}={\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {1}{6}}{\sqrt[{4}]{27}}F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {x^{2}+1}}}{\biggr )};{\frac {1}{4}}({\sqrt {6}}+{\sqrt {2}}){\biggr ]}$ For the value $n=4$ that identity appears: ${\frac {1}{6{\sqrt[{3}]{2}}}}\csc {\bigl (}{\frac {\pi }{6}}{\bigr )}\beta {\bigl (}{\frac {1}{3}}{\bigr )}=\int _{0}^{\infty }{\frac {1}{\sqrt {x^{6}+1}}}\,\mathrm {d} x={\biggl \{}{\color {blue}{\frac {1}{6}}{\sqrt[{4}]{27}}F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}\,x}{\sqrt {x^{2}+1}}}{\biggr )};{\frac {1}{4}}({\sqrt {6}}+{\sqrt {2}}){\biggr ]}}{\biggr \}}_{x=0}^{x=\infty }=$ $={\frac {1}{6}}{\sqrt[{4}]{27}}F{\bigl [}2\arctan({\sqrt[{4}]{3}});{\frac {1}{4}}({\sqrt {6}}+{\sqrt {2}}){\bigr ]}={\frac {2}{9}}{\sqrt[{4}]{27}}K{\bigl [}{\frac {1}{4}}({\sqrt {6}}+{\sqrt {2}}){\bigr ]}={\frac {2}{3}}{\sqrt[{4}]{3}}K{\bigl [}{\frac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}$ This result follows from that equation chain: ${\color {ForestGreen}K{\bigl [}{\frac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}={\frac {1}{2{\sqrt[{3}]{2}}{\sqrt[{4}]{3}}}}\beta {\bigl (}{\frac {1}{3}}{\bigr )}}$

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

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:

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

$\theta _{3}[\exp(-\pi )]=\theta _{3}[q({\tfrac {1}{2}}{\sqrt {2}})]={\sqrt {2\pi ^{-1}K({\tfrac {1}{2}}{\sqrt {2}})}}=2^{-1/2}\pi ^{-1/2}\beta ({\tfrac {1}{4}})^{1/2}=2^{-1/4}{\sqrt[{4}]{\pi }}\,{\Gamma {\bigl (}{\tfrac {3}{4}}{\bigr )}}^{-1}$

$\theta _{3}[\exp(-{\sqrt {3}}\,\pi )]=\theta _{3}{\bigl \{}q{\bigl [}{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}{\bigr \}}={\sqrt {2\pi ^{-1}K{\bigl [}{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}}}=2^{-1/6}3^{-1/8}\pi ^{-1/2}\beta ({\tfrac {1}{3}})^{1/2}$

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

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

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

Quintic equations

Solution of the Bring-Jerrard-Form

According to the Abel-Ruffini theorem, the general quintic equation cannot be solved with elementary roots. But a general solution is very well possible by using the elliptic functions. The Jacobi theta function can also be used to solve the general case of the equation of the fifth degree depending on the elliptic nome of an elliptic modulus that is always elementarily dependent on the coefficients. Here, the fifth power and the fifth root of the affected elliptic nome is inserted into the theta function. For the following quintic equation in Bring-Jerrard form, the general solution can be represented in simplified form especialls with the theta function $\theta _{3}$ but also with the two functions $\theta _{4}$ and $\theta _{2}$:

$x^{5}+5\,x=4\,c$

For all real values $c$, the shown sum of 5th power function and Identity function for $x$ in dependence of $c$ has exact a real solution. And this real solution $x$ can be evoked exactly correctly for all real values $c$ with the following algorithm:

Solution algorithm of the quintic equation using the theta function

Bring-Jerrard-Equation: $x^{5}+5\,x=4\,c$

Elliptic nome function value: $Q=q{\bigl [}{\bigl (}2c^{2}+2+2{\sqrt {c^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {c^{4}+1}}+1}}+c{\bigr )}{\bigr ]}=q{\bigl \{}\mathrm {ctlh} {\bigl [}{\tfrac {1}{2}}\mathrm {aclh} (c){\bigr ]}^{2}{\bigr \}}$

Real solution x: $x={\frac {{\bigl [}\theta _{3}(Q^{1/5})^{2}-5\,\theta _{3}(Q^{5})^{2}{\bigr ]}{\sqrt {\theta _{3}(Q^{1/5})^{2}+5\,\theta _{3}(Q^{5})^{2}-4\,\theta _{3}(Q)^{2}-2\,\theta _{3}(Q^{1/5})\,\theta _{3}(Q^{5})}}}{4\,\theta _{2}(Q)\,\theta _{4}(Q)\,\theta _{3}(Q)}}$

Alternative equivalent Rogers-Ramanujan continued fraction expression for the same solution x: $x={\biggl [}1-{\frac {R(Q^{2})}{S(Q)^{2}}}{\biggr ]}\times {\frac {1-R(Q^{2})S(Q)}{R(Q^{2})^{2}}}\times {\frac {\theta _{3}(Q^{5})\theta _{3}(Q^{1/5})^{2}-5\,\theta _{3}(Q^{5})^{3}}{4\,\theta _{2}(Q)\,\theta _{4}(Q)\,\theta _{3}(Q)}}$

The abbreviation ctlh stands for the Hyperbolic Lemniscate Cotangent and the abbreviation aclh stnds for the Hyperbolic Lemniscate Areacosine!

$\operatorname {sl} (\varphi )=\tan {\biggl \langle }2\arctan {\biggl \{}{\frac {4}{G}}\sin \left({\frac {\varphi }{G}}\right)\sum _{n=1}^{\infty }{\frac {\cosh[(2n-1)\pi ]}{\cosh[(2n-1)\pi ]^{2}-\cos(\varphi /G)^{2}}}{\biggr \}}{\biggr \rangle }$ $\operatorname {cl} (\varphi )=\tan {\biggl \langle }2\arctan {\biggl \{}{\frac {4}{G}}\cos \left({\frac {\varphi }{G}}\right)\sum _{n=1}^{\infty }{\frac {\cosh[(2n-1)\pi ]}{\cosh[(2n-1)\pi ]^{2}-\sin(\varphi /G)^{2}}}{\biggr \}}{\biggr \rangle }$ $\arctan[\operatorname {sl} (\varphi )^{2}]+\arctan[\operatorname {cl} (\varphi )^{2}]={\tfrac {1}{4}}\pi$ $[\operatorname {sl} (\varphi )^{2}+1][\operatorname {cl} (\varphi )^{2}+1]=2$

$\operatorname {ctlh} (\varrho )=\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}\varrho ){\biggl [}{\frac {\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}\varrho )^{2}+1}{\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}\varrho )^{2}+\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}\varrho )^{2}}}{\biggr ]}^{1/2}$ $\operatorname {ctlh} (\varrho )={\frac {\operatorname {cd} (\varrho ;{\tfrac {1}{2}}{\sqrt {2}})}{\sqrt[{4}]{\operatorname {cd} (\varrho ;{\tfrac {1}{2}}{\sqrt {2}})^{4}+\operatorname {sn} (\varrho ;{\tfrac {1}{2}}{\sqrt {2}})^{4}}}}$

$\operatorname {aclh} (s)={\tfrac {1}{2}}F[2\operatorname {arccot}(s);{\tfrac {1}{2}}{\sqrt {2}}]$ $\operatorname {aclh} (s)={\frac {1}{2}}{\sqrt {2}}\,\pi \,G-\int _{0}^{1}{\frac {s}{\sqrt {s^{4}t^{4}+1}}}\operatorname {d} t$

The letter $G$ represents the Gauss constant and the letter $\varpi =\pi \,G$ represents the lemniscate constant.

The expressions $\mathrm {sn}$ and $\mathrm {cd}$ stand for the Jacobi elliptic amplitude functions!

Important informations about the Rogers-Ramanujan continued fractions:

$R(y)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(y)^{2}}{2\theta _{4}(y^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(y)^{2}}{2\theta _{4}(y^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}$

$S(y)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\theta _{3}(y)^{2}}{2\theta _{3}(y^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\theta _{3}(y)^{2}}{2\theta _{3}(y^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}$

$R(y)=y^{1/5}{\frac {(y;y^{5})_{\infty }(y^{4};y^{5})_{\infty }}{(y^{2};y^{5})_{\infty }(y^{3};y^{5})_{\infty }}}$

$S(y)={\frac {R(y^{4})}{R(y^{2})R(y)}}$

Calculation example of a quintic equation

In the following, an algebraic example value that cannot be represented with elementary root operators is treated, which can be produced with the algorithm shown in the table above:

$x^{5}+5\,x=4$

$Q=q{\bigl [}{\bigl (}2c^{2}+2+2{\sqrt {c^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {c^{4}+1}}+1}}+c{\bigr )}{\bigr ]}{\bigl (}c=1{\bigr )}$

$=q{\bigl \{}{\text{ctlh}}{\bigl [}{\tfrac {1}{2}}\operatorname {aclh} (1){\bigr ]}^{2}{\bigr \}}=q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}=q{\bigl [}{\frac {\sqrt[{4}]{2}}{2}}+\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\bigr ]}$

$Q\approx 0.18520287008030014142515182307361246060360377625$

$x={\frac {\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}^{1/5}\}^{2}-5\,\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}^{5}\}^{2}}{4\,\theta _{2}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}\}\,\theta _{4}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}\}\,\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}\}}}\times$

$\times {\sqrt {\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}^{1/5}\}^{2}+5\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}^{5}\}^{2}-4\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}\}^{2}-2\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}^{1/5}\}\,\theta _{3}\{q{\bigl [}{\text{ctlh}}{\bigl (}{\tfrac {\varpi }{4{\sqrt {2}}}}{\bigr )}^{2}{\bigr ]}^{5}\}}}$

$x\approx 0.75192639869405948026865366345020738740978383913$

Discovery of the modular formula by Hermite

In 1799 the mathematician Paolo Ruffini published an incomplete proof of the insolvability of the general quintic equation over elementary root expressions. He was the first to deal with the proof of this thesis and he used the laws of group theory in his proof. In 1824 Niels Henrik Abel gave a complete proof of this elementary unsolvability of the fifth-degree generalized equations. Around 1830, the French mathematician Évariste Galois presented the theory named after him on the solvability criteria of quintic equations and higher-degree equations using elementary root expressions. This theory is called Galois theory. Later, the mathematician Charles Hermite, also from France, researched an algorithm for determining the elliptic module or the numerical eccentricity for the expression of the solution using the Bring-Jerrard form of the quintic equation with the same sign in front of the quintic and linear term elliptic modular functions. He recognized the fact that for the representation of Bring's radical via module functions, the module corresponds exactly to the shown cotangent-lemniscatus-hyperbolicus square. Hermite wrote down this connection in his work Sur la résolution de l'Équation du cinquiéme degré Comptes rendus. The Italian version of his work Sulla risoluzione delle equazioni del quinto grado contains on page 258 the formula from which the module mentioned here emerges. Likewise, the mathematicians John Stuart Glashan, George Paxton Young and Carl Runge in 1885 the solution of the Bring-Jerrard form. They set up a parameterized formula of the Bring-Jerrard form that describes exactly whether a given quintic equation is solvable with elementary root expressions or not. Based on their parametric formula, they were able to construct a fifth-root expression in terms of the parameters of the absolute term of the normalized Bring-Jerrard form. In this way they determined a quintic radical solution expression depending on an elliptic key. For them the same real solution with the elliptic key and depending on quintic radical expressions looks exactly like this:

$x^{5}+5\,x=4\,c$

$x={\frac {2}{5}}y^{-1/4}{\sqrt[{4}]{50+75y-50y^{2}}}\cosh {\biggl \{}{\frac {1}{5}}\operatorname {arcosh} {\biggl [}{\frac {5{\sqrt {5+5y^{2}}}}{(1+2y){\sqrt {4+6y-4y^{2}}}}}{\biggr ]}{\biggr \}}-$

$-{\frac {2}{5}}y^{-1/4}{\sqrt[{4}]{50+75y-50y^{2}}}\sinh {\biggl \{}{\frac {1}{5}}\operatorname {arsinh} {\biggl [}{\frac {5y{\sqrt {5+5y^{2}}}}{(2-y){\sqrt {4+6y-4y^{2}}}}}{\biggr ]}{\biggr \}}$

Associated elliptic key:

$y={\frac {5\,\theta _{3}{\bigl \langle }q{\bigl \{}\mathrm {ctlh} {\bigl [}{\tfrac {1}{2}}\mathrm {aclh} (c){\bigr ]}^{2}{\bigr \}}^{5}{\bigr \rangle }^{2}}{2\,\theta _{3}{\bigl \langle }q{\bigl \{}\mathrm {ctlh} {\bigl [}{\tfrac {1}{2}}\mathrm {aclh} (c){\bigr ]}^{2}{\bigr \}}{\bigr \rangle }^{2}}}-{\frac {1}{2}}$

In general, Bring's radical is used to solve the generalized fifth degree equation and was invented by the mathematician Erland Samuel Bring from Swede. The elliptic key shown enabled the solution of the general Bring-Jerrard form following the pattern of the mathematicians Glashan, Young and Runge. The Russian mathematicians Viktor Prasolov and Yuri Solovyev also dealt with the elliptic identity of Bring's radical in their work Elliptic Functions and Elliptic Integrals from 1991. And based on the essays by Soon Yi Kang and Nikolaos Bagis, about the shown solution expression with the elliptic key, a solution expression of the same solution can be derived via the Rogers-Ramanujan continued fractions R and S.

References

• Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of Mathematical Functions. New York: Dover Publications. sec. 16.27ff. ISBN 978-0-486-61272-0.
• Akhiezer, Naum Illyich (1990) [1970]. Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs. Vol. 79. Providence, RI: AMS. ISBN 978-0-8218-4532-5.
• Farkas, Hershel M.; Kra, Irwin (1980). Riemann Surfaces. New York: Springer-Verlag. ch. 6. ISBN 978-0-387-90465-8.. (for treatment of the Riemann theta)
• Hardy, G. H.; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford: Clarendon Press.
• Mumford, David (1983). Tata Lectures on Theta I. Boston: Birkhauser. ISBN 978-3-7643-3109-2.
• Pierpont, James (1959). Functions of a Complex Variable. New York: Dover Publications.
• Rauch, Harry E.; Farkas, Hershel M. (1974). Theta Functions with Applications to Riemann Surfaces. Baltimore: Williams & Wilkins. ISBN 978-0-683-07196-2.
• Reinhardt, William P.; Walker, Peter L. (2010), "Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Whittaker, E. T.; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge: Cambridge University Press. ch. 21. (history of Jacobi's θ functions)

Further reading

• Farkas, Hershel M. (2008). "Theta functions in complex analysis and number theory". In Alladi, Krishnaswami (ed.). Surveys in Number Theory. Developments in Mathematics. Vol. 17. Springer-Verlag. pp. 57–87. ISBN 978-0-387-78509-7. Zbl 1206.11055.
• Schoeneberg, Bruno (1974). "IX. Theta series". Elliptic modular functions. Die Grundlehren der mathematischen Wissenschaften. Vol. 203. Springer-Verlag. pp. 203–226. ISBN 978-3-540-06382-7.
• Ackerman, Michael (1 February 1979). "On the generating functions of certain Eisenstein series". Mathematische Annalen. 244 (1): 75–81. doi:10.1007/BF01420339. S2CID 120045753.

Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, ISBN 0-683-07196-3.

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

External links

• Moiseev Igor. "Elliptic functions for Matlab and Octave".

This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.