# §20.1 Special Notation

(For other notation see Notation for the Special Functions.)

 , integers. the argument. the lattice parameter, . the nome, , . Since is not a single-valued function of , it is assumed that is known, even when is specified. Most applications concern the rectangular case , , so that and and are uniquely related. for (resolving issues of choice of branch). set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).)

The main functions treated in this chapter are the theta functions where and . When is fixed the notation is often abbreviated in the literature as , or even as simply , it being then understood that the argument is the primary variable. Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21.

Primes on the symbols indicate derivatives with respect to the argument of the function.

## ¶ Other Notations

Jacobi’s original notation: , , , , respectively, for , , , , where . Here the symbol denotes capital eta. See, for example, Whittaker and Watson (1927, p. 479) and Copson (1935, pp. 405, 411).

Neville’s notation: , , , , respectively, for , , , , where again . This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951).

McKean and Moll’s notation: , . See McKean and Moll (1999, p. 125).

Additional notations that have been used in the literature are summarized in Whittaker and Watson (1927, p. 487).