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This book is based on lectures, given at the Applied Mathematics Laboratory of the David Taylor Model Basin. It ill devoted to the Laplace Transformation and its application to linear ordinary differential equations with variable coefficients, to linear partial differential equations, with two independent variables and constant coefficients, and to the determination of asymptotic series. The treatment of the Laplace Transformation is baaed on the Fourier Integral Theorem and the ordinary different equations selected for detailed treatment are those of Laguerre and Bessel. The
partial differential equation governing the motion of a tightly stretched vibrating string and a generalization of the equation are fully treated. Asymptotic series for the integral from p to infinity of exp (-z2) dz and for the modified Bessel function I_n(p), absolute value of arg p (pi2), are obtained by means of the Laplace Transformation and finally, asymptotic series useful in the calculation of the ordinary Bessel functions J_n(t) are treated.
Care has been taken to make the treatment self-contained, and details of the proofs of the basic mathematical theorems are given