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A generalized jacobi theta function
Author(s) -
Lehnigk S. H.,
Roach G. F.
Publication year - 1984
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670060120
Subject(s) - mathematics , function (biology) , mathematical analysis , heat equation , theta function , sequence (biology) , hamilton–jacobi equation , functional equation , pure mathematics , variable (mathematics) , partial differential equation , genetics , evolutionary biology , biology
The delta function initial condition solution v * ( x,t;y ) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function \documentclass{article}\pagestyle{empty}\begin{document}$ \Theta (x,t) = \upsilon *(x,t;0) + 2\sum\limits_{n = 1}^\infty {v*(x,t;y_n)} $\end{document} for a sufficiently rapidly increasing and unbounded positive sequence { y y }. It is shown that Θ( x,t ) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ( x,t ) reduces to the third Jacobi Theta function.