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The Hermite Equation and the Generalised Laplace Transform
Author(s) -
Deakin M. A. B.
Publication year - 1997
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19970770316
Subject(s) - laplace transform , laplace transform applied to differential equations , mathematics , two sided laplace transform , green's function for the three variable laplace equation , inverse laplace transform , hermite polynomials , mellin transform , laplace–stieltjes transform , mathematical analysis , laplace's equation , integral transform , partial differential equation , fourier transform , fractional fourier transform , fourier analysis
The second order Laplace equation (i.e. the linear o.d.e. with linear coefficients) may be reduced to one of a number of standard forms. One of these is the Hermite equation which has hitherto been thought not to be treatable by the Laplace transform. This is true because the defining integral for that transform diverges in this case. However this problem does not arise with a generalised Laplace transform. This paper gives the detailed working out of the theory from this point of view, and in the process derives results for the parabolic cylinder functions; these derivations are simpler than the standard ones or else produce these (known) results in a somewhat different form.