On Fay's tri-secant formula (Q1063653)

The tri-secant formula of \textit{J. D. Fay} [''Theta functions on Riemann surfaces'', Lect. Notes Math. 352 (1973; Zbl 0281.30013)] is a special theta-function relation which occupies a central position in recent exciting work on the Schottky problem, see \textit{R. C. Gunning}'s survey [Bull. Am. Math. Soc., New Ser. 11, 287-316 (1984; Zbl 0552.14009)] for a general appreciation. The present paper gives a proof of the formula which uses only the simplest aspects of function theory on Riemann surfaces and the most elementary facts about theta functions. A derivation of a version of the formula due to Gunning is also included.