MA360Handout 4a

MA360Handout 4a - MA360 Handout 4a (week of Jan 30 ) Sp2006...

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1 MA360 Handout 4a (week of Jan 30 ) Sp2006 Kallfelz Note : New (absolute) due date for Assignment I (See posted assignment sheet for details) Assignment: pp. 52-55 Sheng: 2a), b), e), 3b), c), 4a), d) Summary of Theorems (Ch 1, Sheng) Theorems Description 1. L { af ( t ) + bg ( t )} = aL { f ( t )} + bL {g( t )} Corrolat 1a.) For any piecewise continuous f : lim s →∞ L { f ( t )} = 0 Linearity and asymptotic property of LT 2 .: L { D t f ( t )} = sL { f ( t )} – f (0) 2a. Corrolary to 2 (not found in Sheng): L { D t n f ( t )} = s n L { f ( t )} – n j= 1 s n-j f ( j -1) (0) 4 : L { D t -1 f ( t )} = 1 / sL { f ( t )} 2. & 4. give formulae in terms of the LT of f ( i.e., L { f ( t )} = F ( s ) ) when you’re faced with computing the LT for the derivative of f (or in the case of Thm 2a, the n -th derivative) or the antiderivative of f ( D t -1 { f ( t )} t 0 f ( ω ) d 3. L { f ( t )} = F ( s ) L { f ( at )} = 1 / a F ( s / a ) An obviously useful simplification formula, but it also tells you that the LT is not linear with respect to the variable arguments of f. LT is linear with respect to its functional arguments 1 ! 5. L { t n f ( t )} = (-1) n D n s L { f ( t )} for n = 0,1,2,… Again, another obviously useful simplification formula 5a. Corrolary (Example 1-2-4) { } ( ) 1 1 + + Γ = p p s p t L , for any p > -1 (From Handout 1c, Formula 1c.1) The formula comes especially in handy of integer and half- integer exponents, given Properties 1-4 as of the Gamma function as summarized in Handout 1c 6. ( ) { } ( ) d F t f t L s - = 1 where: L { f ( t )} = F ( s ) Discussed here (Handout 2). Calculus analogy: Recall the power-rule for integration did not hold for t p when p = -1? (You had to introduce a rigorous defn of logarithm) 7. L { f ( t )} = F ( s ) L { e at f ( t )} = F
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MA360Handout 4a - MA360 Handout 4a (week of Jan 30 ) Sp2006...

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