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L, BarnesTranrnenta in LimwTSystems,Vol. 1, Wiley, New York, 1942. f(t – O)]at zHere and throughoutthis sectionf(t) a point of discontinuity. ALY81S AND TRA~SIENT RESPONSE k%C. 4 This result is proved by showing that e-c’u(t) is bounded for all t; and since c is any number > k, g(t) is an E-function with normal exponent k. The main result of this section will now be proved. -l-transform can be achieved in a few lines. THEOREM11. Let F(p) be the S-transform of an E-function j(t) with normal exponent k.
THEOREM 1. Let j(t)be an E-function; then ~ ~(r) d. is also an ! E-function of normal exponent k. It might be thought that by confining attention to the E-functions an appreciable 10SS would be suffered in the power and flexibility of the method. Thus all functions possessing poles on the real axis are barred from discussion and in particular the Kirchhoff-Dirac &function defined by a(t) = o, t#o, . / -m 3(tj dt = (7) 1. 1 I See GustaveDoetsch, Thcorieund A rwendung der Laplace-transformation, Dover Publications,New York, 1943, for a discussionof those propertiesrelevant to the Laplacctransformtheory.
1The Laplacetransformis used to solve not only linearintegro-differentialequations in a singleindependentvariablebut also linearequationswith an arbitrarynumberof independentvariables,aswellas partialdifferential,integral,and finitedifference equations. The Laplace transform may also be applied to evaluate definite integrals,to sum series,and to developthe theoryof functionsboth of the realand the complexvariable. 14] THE THEORY OF THE LAPLACE TRANSFORM 23 On the first reading it may be desirable to turn, at this stage, to Sec.