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Generalized functions, e.g., Dirac delta functions and derivatives of the same, are currently in vogue. In this paper there is developed a practical approach to generalized functions and the many uses thereof in Fourier and Laplace transform theory. A novel concept, the " line of convergence" concept, is central to the approach. By use of it, generalized steady-state analysis (e. g., sinusoidal and exponentially weighted sinusoidal analysis) of linear time-invariant systems is tied with transient analysis to a common methodology in transform theory.