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The Algebra of Convolution Quotients
Applications to Differential and Integral Equations
6 other sections not shown
absolutely integrable algebraic operations be~xs boundary conditions clearly complex numbers continuous function convergent series converges uniformly convolution quotients corresponding defined delta function denote differentiable with respect differential equation z"(x diffusion equation e~xs equivalence classes Example exists exponential functions finite interval fixed follows func function of slow given ha(x Heaviside's unit function impulse functions infinite series infinitely differentiable initial conditions integral equation Laplace transform lemma locally integrable derivative locally integrable function logarithm mathematical Mikusinski Mikusiriski 1959 nontrivial solution numerical function obtain operational calculus operational form pa(x partial differential equation possesses a locally proof of theorem prove result Riemann integral right-hand side section 4.4 semi-infinite interval sense of definition sequence sifting property slow growth solve theorem 18 tion twice continuously differentiable uniformly convergent unit element vanishes identically wave equation zero zt(x ztt(x zxx(x