Laplace transforms formula set, Schemes and Mind Maps of Mathematics

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LAPLACE TRANSFORM (L.T ) 1) Laplace transform of f(t) =L[f (t)] = Spe (t) dt= F(s); L-[F(s)]=f(t) is the inverse L.T 2) If f(t) is a periodic function with period Tie f(t+T)=f(t) then LUf(t)] = Par fo e-*f(tyde 3) Convolution of f(t) and gilt) is f(t) + g(t) = ft Fwg(t—wdu or ff g(u)f(t —wdu 4) Convolution theorem: LIf(t) + 9(0)] = LiFe) 5) L[2] = fe up@as = f° F(s) as 8) L [fp fear] = Od _ re 7) LEF'CO) = Lif] - fF) = sF(s) — f(0) or LLy'(t)] = stLy(t)] — (0) 8) LEP") = s* LEF()1 — sf(0) — (0) = s? F(s) — sf(0) - f'(0) or Lly"'()] = s?L[y(€)] — sy(0) — y'(0) 9) bef ey] = — eso) _ _ ween 20) Len (O)] = (ayn SEO = (ayn = (ayes) 11) LIU a} == and LIf(t—au(t—a)] =e LIF) 12) L[5(t —a)] =e" and L[f(t)6(t —a)] = f(ajes 13) Suppose f(t) = be ‘s 2 In terms of unit step function f(t) = f,(t) + Le (@) — A(t — a) fit) forOb fO=AO+hW-AO(t—a) +h O-AWluE-4) 15) Unit step (or Heaviside) function is H(t—a) or U(t—a)) = tsa Lt>a (a is non negative constant) “ -_ 16) Unit impulse (or Dirac Delta) function d(t — a) = £48[6, (t — a)] where z, forast