differential equations of Jacobi functions
It is easy to check that each in the series which define the theta functions this differential equation. Furthermore, by the Weierstrass M-test, the series obtained by differentiating the series which define the theta functions term-by-term converge absolutely, and hence one may compute derivatives of the theta functions by taking derivatives of the series term-by-term.
Students of mathematical physics will recognize this equation as a one-dimensional diffusion equation. Furthermore, as may be seen by examining the series defining the theta functions, the theta functions approach periodic delta distributions in the limit . Hence, the theta functions are the Green’s functions of the one-dimensional diffusion equation with periodic boundary conditions.
|Title||differential equations of Jacobi functions|
|Date of creation||2013-03-22 14:41:19|
|Last modified on||2013-03-22 14:41:19|
|Last modified by||rspuzio (6075)|