Theta Function

The theta functions are the elliptic analogs of the Exponential Function, and may be used to express the Jacobi Elliptic Functions. Let $t$ be a constant Complex Number with $\Im[t]> 0$. Define the Nome

$$q\equiv e^{i\pi t}=e^{\pi K'(k)/K(k)},$$ (1)

where

$$t\equiv -i{K'(k)\over K(k)},$$ (2)

and $K(k)$ is a complete Elliptic Integral of the First Kind, $k$ is the Modulus, and $k'$ is the complementary Modulus. Then the theta functions are, in the Notation of Whittaker and Watson,

$$\vartheta _1(z,q) \equiv 2 \sum_{n=0}^\infty (-1)^n q^{(n+1/2)^2}\sin [(2n+1)z]$$

$$= z q^{1/4} \sum_{n=0}^\infty (-1)^n q^{n(n+1)}\sin[(2n+1)z]$$ (3)

$$\vartheta _2(z,q) \equiv 2\sum_{n=0}^\infty q^{(n+1/2)^2}\cos[(2n+1)z]$$

$$= 2q^{1/4} \sum_{n=0}^\infty q^{n(n+1)}\cos[(2n+1)z]$$ (4)

$$\vartheta _3(z,q) \equiv 1+2\sum_{n=1}^\infty q^{n^2}\cos(2nz)$$ (5)

$$\vartheta _4(z,q) \equiv \sum_{n=-\infty}^\infty (-1)^nq^{n^2}e^{2niz}$$

$$= 1+2\sum_{n=1}^\infty (-1)^nq^{n^2}\cos(2nz).$$ (6)

Written in terms of $t$,

$$\vartheta _2(t,q) = \sum_{n=-\infty}^\infty q^{(n+1/2)^2}e^{\Im[t]}$$ (7)

$$\vartheta _3(t,q) = \sum_{n=-\infty}^\infty q^{n^2} e^{\Im[t]}.$$ (8)

These functions are sometimes denoted $\Theta_i$ or $\theta_i$, and a number of indexing conventions have been used. For a summary of these notations, see Whittaker and Watson (1990). The theta functions are quasidoubly periodic, as illustrated in the following table.

$\vartheta _i$	$\vartheta _i(z+\pi)/\vartheta _i(z)$	$\vartheta _i(z+t\pi)/\vartheta _i(z)$
$\vartheta _1$	$-1$	$-N$
$\vartheta _2$	$-1$	$N$
$\vartheta _3$	1	$N$
$\vartheta _4$	1	$-N$

Here,

$$N\equiv q^{-1}e^{-2iz}.$$ (9)

The quasiperiodicity can be established as follows for the specific case of $\vartheta _4$,

$$\vartheta _4(z+\pi,q) = \sum_{n=-\infty}^\infty (-1)^nq^{n^2}e^{2niz}e^{2ni\pi}$$

$$= \sum_{n=-\infty}^\infty (-1)^nq^{n^2}e^{2niz} = \vartheta _4(z,q)$$ (10)

$$\vartheta _4(z+\pi t,q) = \sum_{n=-\infty}^\infty (-1)^nq^{n^2}e^{2ni\pi t}e^{2niz}$$

$$= \sum_{n=-\infty}^\infty (-1)^nq^{n^2}q^{2n}e^{2niz}$$

$$= -q^{-1}e^{-2iz}\sum_{n=-\infty}^\infty (-1)^{n+1} q^{(n+1)^2}q^{2(n+1)iz}$$

$$= -q^{-1}e^{-2iz}\sum_{n=-\infty}^\infty (-1)^n q^{n^2}q^{2niz}$$

$$= -q^{-1}e^{-2iz}\vartheta _4(z,q).$$ (11)

The theta functions can be written in terms of each other:

$$\vartheta _1(z,q) = -ie^{iz+\pi it/4}\vartheta _4(z+{\textstyle{1\over 4}}\pi t,q)$$ (12)

$$\vartheta _2(z,q) = \vartheta _1(z+{\textstyle{1\over 2}}\pi,q)$$ (13)

$$\vartheta _3(z,q) = \vartheta _4(z+{\textstyle{1\over 2}}\pi,q).$$ (14)

Any theta function of given arguments can be expressed in terms of any other two theta functions with the same arguments.

Define

$$\vartheta _i\equiv \vartheta _i(z=0),$$ (15)

which are plotted above. Then we have the identities

$${\vartheta _1}^2(z){\vartheta _4}^2={\vartheta _3}^2(z){\vartheta _2}^2-{\vartheta _2}^2(z){\vartheta _3}^2$$ (16)

$${\vartheta _2}^2(z){\vartheta _4}^2={\vartheta _4}^2(z){\vartheta _2}^2-{\vartheta _1}^2(z){\vartheta _3}^2$$ (17)

$${\vartheta _3}^2(z){\vartheta _4}^2={\vartheta _4}^2(z){\vartheta _3}^2-{\vartheta _1}^2(z){\vartheta _2}^2$$ (18)

$${\vartheta _4}^2(z){\vartheta _4}^2={\vartheta _3}^2(z){\vartheta _3}^2-{\vartheta _2}^2(z){\vartheta _2}^2.$$ (19)

Taking $z=0$ in the last gives the special case

$${\vartheta _4}^4={\vartheta _3}^4-{\vartheta _2}^4.$$ (20)

In addition,

$$\vartheta _3(x) = \sum_{n=-\infty}^\infty x^{n^2}=1+2x+2x^4+2x^9+\ldots$$ (21)

$${\vartheta _3}^2(x) = 1+4\left({{x\over 1-x}-{x^3\over 1-x^3}+{x^5\over 1-x^5}-{x^7\over 1-x^7}+\ldots}\right)$$ (22) (23)

$${\vartheta _3}^4(x) = 1+8\left({{x\over 1-x}+{2x^2\over 1+x^2}+{3x^3\over 1-x^3}+{4x^4\over 1+x^4}+\ldots}\right).$$ (24)

The theta functions obey addition rules such as

$$\vartheta _3(z+y)\vartheta _3(z-y){\vartheta _3}^2={\varthet... ...(y){\vartheta _3}^2(z)+{\vartheta _1}^2(y){\vartheta _1}^2(z).$$ (25)

Letting $y=z$ gives a duplication Formula

$$\vartheta _3(2z){\vartheta _3}^3={\vartheta _3}^4(z)+{\vartheta _1}^4(z).$$ (26)

For more addition Formulas, see Whittaker and Watson (1990, pp. 487-488). Ratios of theta function derivatives to the functions themselves have the simple forms

$${\vartheta _1'(z)\over\vartheta _1(z)} = \cot z+4 \sum_{n=1}^\infty {q^{2n}\over 1-q^{2n}} \sin(2nz)$$ (27)

$${\vartheta _2'(z)\over\vartheta _2(z)} = -\tan z+4\sum_{n=1}^\infty (-1)^n {q^{2n}\over 1-q^{2n}}\sin(2nz)$$ (28)

$${\vartheta _3'(z)\over\vartheta _3(z)} = 4\sum_{n=1}^\infty (-1)^n{q^n\over 1-q^{2n}} \sin(2nz)$$ (29)

$${\vartheta _4'(z)\over\vartheta _4(z)} = \sum_{n=1}^\infty {q^{2n-1}\sin(2z)\over 1-2q^{2n-1}\cos(2z)+q^{4n-2}}$$

$$= \sum_{n=1}^\infty {4q^n\sin(2nz)\over 1-q^{2n}}.$$ (30)

The theta functions can be expressed as products instead of sums by

$$\vartheta _1(z) = 2Gq^{1/4}\sin z \prod_{n=1}^\infty [1-2q^{2n} \cos(2z)+q^{4n}]$$ (31)

$$\vartheta _2(z) = 2Gq^{1/4}\cos z \prod_{n=1}^\infty [1+2q^{2n} \cos(2z)+q^{4n}]$$ (32)

$$\vartheta _3(z) = G \prod_{n=1}^\infty [1+2q^{2n-1}\cos(2z)+q^{4n-2}]$$ (33)

$$\vartheta _4(z) = G \prod_{n=1}^\infty [1-2q^{2n-1}\cos(2z)+q^{4n-2}],$$ (34)

where

$$G\equiv \prod_{n=1}^\infty (1-q^{2n})$$ (35)

(Whittaker and Watson 1990, pp. 469-470).

The theta functions satisfy the Partial Differential Equation

$${\textstyle{1\over 4}}\pi i{\partial^2y\over\partial z^2}+{\partial y\over\partial t}=0,$$ (36)

where $y\equiv \vartheta _j(z\vert t)$. Ratios of the theta functions with $\vartheta _4$ in the Denominator also satisfy differential equations

$${d\over dz}\left[{\vartheta _1(z)\over \vartheta _4(z)}\righ... ... _4}^2{\vartheta _2(z)\vartheta _3(z)\over{\vartheta _4}^2(z)}$$ (37)

$${d\over dz}\left[{\vartheta _2(z)\over \vartheta _4(z)}\righ... ... _3}^2{\vartheta _1(z)\vartheta _3(z)\over{\vartheta _4}^2(z)}$$ (38)

$${d\over dz}\left[{\vartheta _3(z)\over \vartheta _4(z)}\righ... ..._2}^2{\vartheta _1(z)\vartheta _2(z)\over{\vartheta _4}^2(z)}.$$ (39)

Some additional remarkable identities are

$$\vartheta _1'=\vartheta _2\vartheta _3\vartheta _4$$ (40)

$$\vartheta _3(z,t)=-(it)^{1/2}e^{z^2/\pi it}\vartheta _3\left({{2\over t},-{1\over t}}\right),$$ (41)

which were discovered by Poisson in 1827 and are equivalent to

$$\sum_{n=-\infty}^\infty e^{-t(x+n)^2} = \sqrt{\pi\over t} \sum_{k=-\infty}^\infty 2^{2\pi ikx-(\pi^2k^2/t)}.$$ (42)

Another amazing identity is

$2\vartheta _1[{\textstyle{1\over 2}}(-b+c+d+e)]\vartheta _2[{\textstyle{1\over 2}}(b-c+d+e)]\vartheta _3[{\textstyle{1\over 2}}(b+c-d+e)]$
$\quad \times\vartheta _4[{\textstyle{1\over 2}}(b+c+d-e)] =\vartheta _3(b)\vartheta _4(c)\vartheta _1(d)\vartheta _2(e)$
$\quad +\vartheta _2(b)\vartheta _1(c)\vartheta _4(d)\vartheta _3(e)-\vartheta _1(b)\vartheta _2(c)\vartheta _3(d)\vartheta _4(e)$
$\quad +\vartheta _4(b)\vartheta _3(c)\vartheta _2(d)\vartheta _1(e)$	(43)

(Whittaker and Watson 1990, p. 469).

The complete Elliptic Integrals of the First and Second Kinds can be expressed using theta functions. Let

$$\xi\equiv {\vartheta _1(z)\over\vartheta _4(z)},$$ (44)

and plug into (37)

$$\left({d\xi\over dz}\right)^2=({\vartheta _2}^2-\xi^2{\vartheta _3}^2)({\vartheta _3}^2-\xi^2{\vartheta _2}^2).$$ (45)

Now write

$$\xi{\vartheta _3\over\vartheta _2}\equiv y$$ (46)

and

$$z{\vartheta _3}^2\equiv u.$$ (47)

Then

$$\left({dy\over du}\right)^2=(1-y^2)(1-k^2y^2),$$ (48)

where the Modulus is defined by

$$k=k(q)={{\vartheta _2}^2(q)\over {\vartheta _3}^2(q)}.$$ (49)

Define also the complementary Modulus

$$k'=k'(q)={{\vartheta _4}^2(-q)\over {\vartheta _3}^2(q)}.$$ (50)

Now, since

$${\vartheta _2}^4+{\vartheta _4}^4={\vartheta _3}^4,$$ (51)

we have shown

$$k^2+k'^2=1.$$ (52)

The solution to the equation is

$$y={\vartheta _3\over \vartheta _2} {\vartheta _1(u{\vartheta... ...rtheta _3}^{-2}\vert t)} \equiv\mathop{\rm sn}\nolimits (u,k),$$ (53)

which is a Jacobi Elliptic Function with periods

$$4K(k)=2\pi{\vartheta _3}^2(q)$$ (54)

and

$$2iK'(k)=\pi t{\vartheta _3}^2(q).$$ (55)

Here, $K$ is the complete Elliptic Integral of the First Kind,

$$K(k)={\textstyle{1\over 2}}\pi{\vartheta _3}^2(q).$$ (56)

See also Blecksmith-Brillhart-Gerst Theorem, Elliptic Function, Eta Function, Euler's Pentagonal Number Theorem, Jacobi Elliptic Functions, Jacobi Triple Product, Landen's Formula, Mock Theta Function, Modular Equation, Modular Transformation, Mordell Integral, Neville Theta Function, Nome, Poincaré-Fuchs-Klein Automorphic Function, Prime Theta Function, Quintuple Product Identity, Ramanujan Theta Functions, Schröter's Formula, Weber Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 577, 1972.

Bellman, R. E. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961.

Berndt, B. C. ``Theta-Functions and Modular Equations.'' Ch. 25 in Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 138-244, 1994.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 430-432, 1953.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.