## Theta Function

The theta functions are the elliptic analogs of the Exponential Function, and may be used to express the Jacobi Elliptic Functions. Let be a constant Complex Number with . Define the Nome

 (1)

where
 (2)

and is a complete Elliptic Integral of the First Kind, is the Modulus, and is the complementary Modulus. Then the theta functions are, in the Notation of Whittaker and Watson,
 (3) (4) (5) (6)

Written in terms of ,
 (7) (8)

These functions are sometimes denoted or , 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.

 1 1

Here,

 (9)

The quasiperiodicity can be established as follows for the specific case of ,
 (10) (11)

The theta functions can be written in terms of each other:
 (12) (13) (14)

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

Define

 (15)

which are plotted above. Then we have the identities
 (16)

 (17)

 (18)

 (19)

Taking in the last gives the special case
 (20)

 (21) (22) (23) (24)

The theta functions obey addition rules such as

 (25)

Letting gives a duplication Formula
 (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
 (27) (28) (29) (30)

The theta functions can be expressed as products instead of sums by
 (31) (32) (33) (34)

where
 (35)

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

The theta functions satisfy the Partial Differential Equation

 (36)

where . Ratios of the theta functions with in the Denominator also satisfy differential equations
 (37)

 (38)

 (39)

 (40)

 (41)

which were discovered by Poisson in 1827 and are equivalent to
 (42)

Another amazing identity is
 (43)
(Whittaker and Watson 1990, p. 469).

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

 (44)

and plug into (37)
 (45)

Now write
 (46)

and
 (47)

Then
 (48)

where the Modulus is defined by
 (49)

Define also the complementary Modulus
 (50)

Now, since
 (51)

we have shown
 (52)

The solution to the equation is
 (53)

which is a Jacobi Elliptic Function with periods
 (54)

and
 (55)

Here, is the complete Elliptic Integral of the First Kind,
 (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.