# 6.2we - 6.2 SOLUTION OF INITIAL VALUE PROBLEMS KIAM HEONG...

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KIAM HEONG KWA In order to solve diﬀerential equations using Laplace transformation, it is necessary to be able to calculate the Laplace transform L f 0 of the derivative f 0 of a function f in terms of L f . This is the content of the following theorem. Theorem 1 (The derivative theorem) . Suppose that f has exponen- tial order α and is continuous on the interval [0 , ) except at a ﬁnite number of jump discontinuities 0 = t 0 < t 1 < t 2 < ··· < t n . Suppose also that f 0 is piecewise continuous on the interval [0 , ) . Then (1) L f 0 ( s ) = s L f ( s ) - f (0+) - n X i =1 e - st i [ f ( t i +) - f ( t i - )] . for s > α . Remark 1. If f is complex-valued and the parameter s is complex, then the condition s > α can be replaced by Re( s ) > α . Remark 2. To be precise, f (0+) is really given by f (0+) = lim δ 0+ ± f ( τ ) - Z τ δ f 0 ( t ) dt ² for any suﬃciently small τ . Date

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6.2we - 6.2 SOLUTION OF INITIAL VALUE PROBLEMS KIAM HEONG...

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