Appendix Laplace Transforms Involving Fractional and Irrational Operations As the cases of integer-order systems, Laplace transform and its inverse are very important. In this appendix, the definition is given first. Then some of the essential special functions are described. Finally, an inverse Laplace transform table involving fractional and irrational-order operators is given. A.1 Laplace Transforms For a time-domain function f(t), its Laplace transform, in s-domain, is defined as ∞ L [f(t)] = f(t)e−stdt = F (s), (A.53) 0 where L [f(t)] is the notation of Laplace transform. If the Laplace transform of a signal f(t)isF (s), the inverse Laplace transform of F (s) is defined as 1 σ+j∞ f(t)=L −1[F (s)] = F (s)estds, (A.54) j2π σ−j∞ where σ is greater than the real part of all the poles of function F (s). A.2 Special Functions for Laplace Transform Since the evaluation for some fractional-order is difficult, special functions may be needed. Here some of the special functions are introduced and listed in Table A.1. A.3 Laplace Transform Tables An inverse Laplace transform table involving fractional and irrational oper- ators is collected in Table A.2 [86, 300]. 391 392 Appendix Table A.1 Some special functions Special functions Definition ∞ k γ (γ)k z 1 Mittag-Leffler E (z)= , Eα,β (z)=Eα,β (z), Eα(z)=Eα,1(z) α,β Γ(αk + β) k! k=0 t −t2 τ2 Dawson function daw(t)=e e dτ 0 t 2 −τ2 erf function erf(t)= √ e dτ π 0 ∞ 2 −τ2 erfc function erfc(t)= √ e dτ =1− erf(t) π t n t2 d −t2 Hermit polynomial Hn(t)=e e dtn 2 2 2 Bessel function Jν (t) is the solution to t y¨ + ty˙ +(t − ν )y =0 −ν Extended Bessel function Iν (t)=j Iν (jt) Table A.2 Inverse Laplace transforms with fractional and irrational operators F (s) f(t)=L −1[F (s)] F (s) f(t)=L −1[F (s)] αγ−β n n− 1 s β− γ α 1 2 t 2 t 1E −at √ ,n=1, 2, ··· √ (sα + a)γ α,β sn s 1 · 3 · 5 ···(2n − 1) π k πs k 1 coth | sin kt| arctan sin kt s2 +k2 2k s t ) 2 2 √ s − a 2 1 −k s t − 1 k2 k log (1 − cosh at) √ e 2 e 4t − k erfc √ s2 t s s π 2 t √ 2 2 −k s √ s + a 2 e ak a2t k log (1 − cos at) √ √ e e erfc a t + √ s2 t s(a + s) 2 t n √ (1 − s) n! 1 1 −at √ H2n t √ √ e erf (b − a)t n+ 1 − s 2 (2n)! πt s + b(s + a) b a n 1 (1 − s) n! √ √ J0(at) − √ H2n+1 t s2 + a2 n+ 3 (2n + 1)! π s 2 − k − 1 (a b) k − 1 (a+b)t a b √ I0(at) √ √ e 2 Ik t ,k>0 s2 − a2 ( s+a+ s+b)2k t 2 √ √ √ √ s+2a− s 1 −at s +2a − s 1 −at √ √ e I1(at) √ √ e I1(at) t t √s+2a + s s +2a + s ν √ k− 1 ( s2 +a2 −s) 1 π t 2 √ ν J − I a ν (at),ν > 1 k k− 1 (at) 2 2 (s2 − a2) Γ(k) 2a 2 √ s + a ν √ k− 1 ( s2 −a2 +s) 1 π t 2 √ ν I − √ J a ν (at),ν> 1 k− 1 (at) s2 − a2 ( s2 + a2)k Γ(k) 2a 2 ( k k ka s − a 1 bt at ( s2 +a2 −s) Jk(at),k > 0 log e − e t s − b t J − 1 1(at) 1 − 1 (a+b)t a b √ √ √ e 2 I0 t s + s2 + a2 at s + a s + b 2 Appendix 393 Table A.2 (continued) F (s) f(t)=L −1[F (s)] F (s) f(t)=L −1[F (s)] 2 2 √ √ 1 NJN (at) b − a a2t b2t √ , N>0 √ e b−a erf a t −be erfc b t (s+ s2 +a2)N at (s − a2)( s + b) √ √ √ √ 1 bt at s +2a − s −at s−a− s−b √ e − e √ ae I1(at)+I0(at) 2 πt3 s √ 1 −k/s √ 1 −k/s 1 e J0 2 kt √ e √ cos 2 kt s s πt √ √ 1 k/s 1 1 −k/s 1 √ e √ cosh 2 kt √ e √ sin 2 kt s πt s s πk √ 1 (ν−1) √ 1 k/s 1 1 −k/s t 2 √ e √ sinh 2 kt e Jν−1 2 kt ,ν > 0 s s πk sν k √ 1 (ν−1) √ −k s k − 1 k2 1 k/s t 2 k e √ e 4 e Iν−1 2 kt 2 πt3 sν k ) √ √ 1 −k s k 1 − s t − 1 1 e erfc √ √ e 2 e 4t − erfc √ s 2 t s s π 2 t √ √ − s √ 1 −k s 1 − 1 k2 e t+1 1 √ e √ e 4t √ √ e erfc t + √ s πt s( s +1) 2 t α−1 1 t −at 1 α−1 α e t Eα,α −at (s + a)α Γ(α) sα + a α a α s α 1 − Eα −at Eα −at s(sα + a) s(sα + a) α 1 α s α t E1,1+α(at) −t E1,1−α(at), 0 <α<1 sα(s − a) s − a ) 1 1 1 t √ √ √ 2 s πt s s π √ 1 2 √ s 1 2 √ √ √ daw t √ − √ daw t s(s +1) π s +1 πt π √ 1 2 s 1 at √ t E1,3/2 −a t √ √ e (1+2at) s(s + a2) (s − a) s − a πt √ √ s 1 2 1 1 a2t √ E1,1/2 −a t √ √ − ae erfc a t s + a2 t s + a πt √ √ 1 √ s 1 a2t √ erf t √ + ae erf a t s s +1 s − a2 πt √ √ a t 1 1 a2t 1 2 −a2t τ2 √ e erf a t √ √ e e dτ 2 2 s(s − a ) a s(s + a ) a π 0 √ ) √ √ 1 a2t s s t 2 √ √ e erfc a t 2 − √ daw t s( s + a) s +1 π π −t √ 1 e 1 t √ √ √ e erf t s +1 πt s(s − 1) √ √ √ s 1 t k! k− / k √ +e erf t √ t( 1) 2E ( ) ∓λ t , (s) >λ2 s − 1 πt s ± λ 1/2,1/2 α−1 α−1 1 t s α 1/α Eα ∓λt , (s) > |λ| sα Γ(α) sα ± λ 394 Appendix Table A.2 (continued) F (s) f(t)=L −1[F (s)] 1 1 −at/2 a ( √ √ e Iν t ,k > 0 s(s + a)( s + a + s)2ν aν 2 √ k− 1 − Γ(k) t 2 − 1 (a+b)t a b 2 I k k π e k− 1 t (s + a) (s + b) a − b 2 2 1 JN (at) √ √ s2 + a2(s + s2 + a2)N aN 1 J (at) √ √ 1 s2 + a2(s + s2 + a2) a 2 2 √ √ b − a a2t b b2t √ √ e erf a t − 1 +e erfc b t s(s − a2)( s + b) a √ −k s √ ae ak a2t k k √ −e e erfc a t + √ + erfc √ s(a + s) 2 t 2 t − − 1 − 1 (a+b)t a b a b √ √ te 2 I0 t + I1 t s + a(s + b) s + b 2 2 √ − s − 1 √ e e 4k t 1 √ √ − e +1erfc t + √ s +1 πt 2 t √ − s √ e 1 t 1 √ erfc √ − e +1erfc t + √ s( s +1) 2 t 2 t References 1. 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