The Laplace Transform: Theory and Applications
by Joel L. Schiff · Springer (Undergraduate Texts in Mathematics)
Springer undergraduate text developing the transform rigorously: existence and basic principles, applications to ODEs, integral and difference equations, the Dirac delta via Riemann-Stieltjes integrals, a complex variable chapter, the complex inversion formula, and PDEs, with many worked examples. Next step after the computational toolkit: proves what the tables assume and covers the Bromwich inversion integral. More accessible and modern than Widder.
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