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

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

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:

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

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

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

Thus, in general,

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

for any integers a and b.

For any fixed ${\displaystyle \tau }$, the function is an entire function on the complex plane, so by Liouville's theorem, it cannot be doubly periodic in ${\displaystyle 1,\tau }$ unless it is constant, and so the best we could do is to make it periodic in ${\displaystyle 1}$ and quasi-periodic in ${\displaystyle \tau }$. Indeed, since

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

and ${\displaystyle \Im (\tau )>0}$, the function ${\displaystyle \vartheta (z,\tau )}$ 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:[3]

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.

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:

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

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

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

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

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 ${\displaystyle |q|<1}$. They are sometimes called theta constants:[note 2]

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

with the nome q = e^iπτ. These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is

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

or equivalently,

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

which is the Fermat curve of degree four.

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

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

Then

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

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

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

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.

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

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

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

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

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

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

In terms of w and q:

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

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

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

which we may also write as

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

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

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

In particular,

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

so we may interpret them as one-parameter deformations of the periodic functions ${\displaystyle \sin ,\cos }$, again validating the interpretation of the theta function as the most general 2 quasi-period function.

The Jacobi theta functions have the following integral representations:

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

The next two series identities were proved by István Mező:[7]

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

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

${\displaystyle {\begin{aligned}{\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=i\sum _{k=-\infty }^{\infty }e^{\pi \left(k-2k^{2}\right)}\theta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right),\\[6pt]{\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}&=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}\end{aligned}}}$

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

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

where m, n are arbitrary integers.

The relation

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

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

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

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.

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

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

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.

The fourth theta function – and thus the others too – is intimately connected to the Jackson q-gamma function via the relation[8]

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

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

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

and,

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

See also the Weber modular functions.

The elliptic modulus is

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

and the complementary elliptic modulus is

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

The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions.[9] Taking z = x to be real and τ = it with t real and positive, we can write

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

which solves the heat equation

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

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

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

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.

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.

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

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

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

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

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

• 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)

• 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.

• 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.