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The generalized Laplace transformation with applications to problems involving finite differences

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The generalized Laplace transformation is defined by an infinite sum 1 LF tE=f s=n=0infinity Fn sn+1, where F(t) must be a function of exponential order, |F(t)| < Malpha|t|, and defined at all integral points. The series converges uniformly and absolutely for all values of |s| = alpha + epsilon, epsilon > 0. From (1) it is readlily found that 2 LEm Ft E=smfs- k=0m-1s m-1-kFk, an equation which is useful for the solution of linear finite difference equations with constant coefficients, a type which often occurs in the theory of statistics, interpolation, structures, electrical circuits and other branches of physics and engineering. In the case of linear difference equations with rational coefficients the formula 3 Lt mFt E=-d dsmsm fs reduces the problem to a linear differential equation;The complex inversion integral and Faltung integral of the regular Laplace transformation have counterparts in the theory of the generalized Laplace transformation, 4 Ft =12ni Cztf zdz and 5 L-1 fsgs E=F tE *Gt =GtE *Ft =n=0 t-1Ft-n-1 Gn. The convolution sum may also be written as a Faltung integral of the related functions F¯(t) and G¯(t), 6 L-1 fsgs E&harrr;0 tFt-x Gxdx, where F¯(t) is defined by the power series, 7 F&d1; t= n=0infinity cnat nn!, and the cn are the Newton coefficients of a tF(t);Extension to the two dimensional case is immediate with particular emphasis on the M transformation, 8 LF tM=2 n=0infinityF nsn+1- F0s, where M F(t) = Ft+1-F t-12 and Delta F(t) = F(t + 1) - 2F(t) + F(t - 1). The M and Delta operators are useful in many complicated physical systems where they may be used to replace the first and second order differential operators, thereby reducing an unwieldy partial differential equation to a numerical solution.