See also Carlson (2011, §2).
Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices . The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012).
See also Carlson (2011, §§1 and 4).
Addendum: For a companion equation see (20.7.34).
See Lawden (1989, pp. 19–20). This reference also gives the eleven additional identities for the permutations of the four theta functions.
See also Carlson (2011, §3).
In the following equations , and all square roots assume their principal values.
These are examples of modular transformations; see §23.15.