Formula1

Formula1 - 1 f.t sF.s ± f.0 f.bt 1 b F ±.s=b ² e at f.t...

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Math 421:01 and 02 — Spring 2008 Formula Sheet Trigonometry The formula e it D cos t C i sin t (with i 2 D 1 ) connects exponential and trigonometric functions. Consequences include the basic identities cos C ˇ/ D cos ˛ cos ˇ sin ˛ sin ˇ sin C ˇ/ D sin ˛ cos ˇ C cos ˛ sin ˇ Laplace transforms The Laplace transform of a function of t is a function of a new variable s defined by ˇ f f .t/ g D Z 1 0 f .t/e st dt This definition shows that the Laplace transform is a linear operator . In describing properties of the Laplace transform, it is conventional to write F.s/ for ˇ f f .t/ g . Some useful properties (in addition to linearity) are f .t/ F.s/ 1 1 s t n 1 s n C 1
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Unformatted text preview: 1 f .t/ sF.s/ ± f.0/ f.bt/ 1 b F ± .s=b/ ² e at f.t/ F.s ± a/ tf.t/ ± F .s/ f.t ± a/ U .t ± a/ e ± as F.s/ ı.t ± a/ e ± as .f ² g/.t/ F.s/G.s/ In the last formula, .f ² g/.t/ denotes the convolution of f.t/ and g.t/ which is deﬁned by .f ² g/.t/ D Z t f.±/g.t ± ±/d± Calculation of Laplace transforms or inverse transforms typically requires several steps taken from this table. To apply these rules, functions must be found so that the given expression has the form shown in a line of this table....
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