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The Laplace Transformation
Further Properties of the Transformation
9 other sections not shown
analytic function boundary conditions boundary value problem Cauchy's integral formula coefficients complex number component continuous function convolution corresponding damping defined denote derive the formula differential form displacements Y(x,t eigenfunctions eigenvalue equa erfc exists exponential order factor fixed follows from equation formal Fourier series frequency func function F(t Green's function half plane heat heat equation Hence improper integral initial conditions initial displacement initial temperature integral equation integral formula integrand inverse transform inversion integral Laplace integral Laplace transformation linear mass obtain operational mathematics operational properties parameter partial differential equations polynomials positive number prescribed Prob represents resonance satisfies the conditions shown in Fig simple poles singular points sinh solution Solve string temperature function temperature zero tends to infinity Theorem tion uniform convergence uniformly with respect vanish velocity verify vibration write written