Jacobi Polynomials, I. New Proofs of Koornwinder’s Laplace Type Integral Representation and Bateman’s Bilinear Sum

1974 ◽  
Vol 5 (1) ◽  
pp. 119-124 ◽  
Author(s):  
Richard Askey
Keyword(s):  
Integral Representation ◽  
Jacobi Polynomials ◽  
Type Integral ◽  
Laplace Type

An Application to H2+ of Laplace Type Integral Transform and its Inverse

1985 ◽  
Vol 40 (3) ◽  
pp. 246-250 ◽  
Author(s):  
M. Primorac ◽  
K. Kovačević
Keyword(s):  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Integral Transform ◽  
Integral Transformation ◽  
Inverse Transform ◽  
Type Integral ◽  
Analytical Formulae ◽  
Molecular Computations ◽  
Laplace Type

Laplace type integral transformation (LIT) has been applied to wavefunctions. The effect of the inverse transform is also discussed. LIT wavefunctions are tested in the calculation of the ground-state energy of H2+, where the untransformed functions were 1s, 12s, 123s and 1234s- STO. The results presented here show that LIT wavefunctions are applicable in molecular computations. The analytical formulae for two-centre one-electron integrals over LIT wavefunctions are derived by use of a Barnett-Coulson-like expansion of rbN (rb + p)-v.

scholarly journals A complete analytical solution of the Fokker–Planck and balance equations for nucleation and growth of crystals

2018 ◽  
Vol 376 (2113) ◽  
pp. 20170327 ◽  
Author(s):  
Eugenya V. Makoveeva ◽  
Dmitri V. Alexandrov
Keyword(s):  
Distribution Function ◽  
Analytical Description ◽  
Supercooled Liquid ◽  
Intermediate Stage ◽  
Nucleation And Growth ◽  
Time Dependent ◽  
Supersaturated Solution ◽  
Nucleation Kinetics ◽  
Type Integral ◽  
Laplace Type

This article is concerned with a new analytical description of nucleation and growth of crystals in a metastable mushy layer (supercooled liquid or supersaturated solution) at the intermediate stage of phase transition. The model under consideration consisting of the non-stationary integro-differential system of governing equations for the distribution function and metastability level is analytically solved by means of the saddle-point technique for the Laplace-type integral in the case of arbitrary nucleation kinetics and time-dependent heat or mass sources in the balance equation. We demonstrate that the time-dependent distribution function approaches the stationary profile in course of time. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

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