Computing the inverse Laplace transform for rational functions vanishing at infinity

Transcription

1 CUBO A Mathematical Journal Vol.6, N ō 3, (97 7. October 24 Computing the inverse Laplace transform for rational functions vanishing at infinity Takahiro Sudo Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Senbaru, Nishihara, Okinawa 93-23, Japan. sudo@math.u-ryukyu.ac.jp ABSTRACT We compute explicitly the inverse Laplace transform for rational functions vanishing at infinity in the general case. We also compute explicitly convolution product for continuous elementary functions involved in the general case. We then consider algebraic structure about the Laplace transform via convolution product. RESUMEN Calculamos explícitamente la transformada de Laplace inversa para funciones racionales que se anulan en infinito en el caso general. Además calculamos explícitamente el producto de convolución para funciones elementales continuas que participan en el caso general. Luego, consideramos estructuras algebraicas de la transformada de Laplace por medio del producto de convolución. Keywords and Phrases: Laplace transform, rational function, convolution. 2 AMS Mathematics Subject Classification: 44A, 44A35, 26C5, 26A6, 26A9.

2 98 Takahiro Sudo CUBO Introduction 6, 3 (24 The Laplace transform L(f of a real-valued, function f on the interval [, is defined by L(f(s for s C in the domain of convergence (cf. [2] or [3]. e st f(tdt The Laplace transform L for continuous functions f on [, is injective. This fact is known as Lerch s theorem as a fundamental theorem in the Laplace transform theory (or deduced from switching the Laplace transform as to be the Fourier transform, so that the inverse Laplace transform is well defined as the inverse image of L(f: L (L(f(t f(t. In this paper we consider real-valued, elementary functions f that are defined on the real line R and are continuous on R, as well as the injectivity of the Laplace transform for these continuous functions is preserved. The reason for this assumption of R just comes from that we do make most of the statements simplified and also that this additional symmetry induces several symmetric results and it makes it to be possible to consider the usual algebraic structure about continuous elementary functions, and is perhaps more natural than cutting down functions to be zero on the negative part of R. But this assumption is not the same as the usual convention that f is assumed to be zero on the interval (,. Indeed, the inverse Laplace trasform defined as f(t 2πi αi α i e st L(f(sds known as Bromwich integral requires that convention, where the complex line integral is taken for some real α and is computed by residue theorem as sums of residue functions. This says that even real-valued functions of one variable as well as the complex case are determined by singularities of their images under the Laplace transform. However, we do not use the complex integral in what follows. In this paper, by elementary calculation we compute explicitly the inverse Laplace transform for rational functions vanishing at infinity in the general case and determine the inverse image. We also compute explicitly convolution product for continuous elementary functions involved in the general case. We then consider algebraic structure about the Laplace transform from our view point, which may not be written in the literature. As a reference, there is another inductive computation known as a recursive formula for multiples of convolution in general (see [], but without using it we compute more explictly multiples of convolution for certain concrete continuous elementary functions. There are 4 sections after this introduction as follows: 2. Inverse Laplace transform for rational functions in a special case; 3. Inverse Laplace transform for rational functions in another special case; 4. Inverse Laplace transform for rational functions; 5. Algebraic structure.

3 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions Our elemetary but explicit computation results obtained in the several general cases of those sections and our consideration and determination on the algebraic structure about the Laplace transform via convolution product would be new as well as useful and helpful as a reference. Notation. We denote by e x the exponential function to the base e for x R and by sinx and cosx the trigonometric functions for x R. We denote by ( n k the combination of k items from n items mutually different. 2 Inverse Laplace transform for rational functions in a special case As a well known fact, we have Lemma 2.. Let s C with the real part Re(s > and λ R a constant with λ. Then ( L (s 2 λ sinλt t cosλt (t R. 2 Proof. By using a fact that the Laplace transform of the convolution product of two functions f(t and g(t is the pointwise multiplication of their Laplace transforms: L(f g(s L(f(s L(g(s with f g(t f(t τg(τdτ, we compute ( L (s 2 λ 2 2 L ( s 2 λ 2 λ 2 sinλ(t τsinλτdτ 2 [ 2 sinλ(t 2τ τcosλt s 2 λ 2 λ 2 sinλt sinλt ] t {cosλ(t 2τ cosλt}dτ τ 3 sinλt t 2 cosλt. Note that L(sinλt λ s 2 λ 2 well known. As the second step in induction, we have Lemma 2.2. Let s C with Re(s > and λ R a constant with λ. Then ( ( L 3 λ 2 t 2 (s 2 λ 2 3 8λ 5 sinλt 3t 8λ 4 cosλt.

4 Takahiro Sudo CUBO 6, 3 (24 Proof. We compute ( L (s 2 λ 2 3 ( L s 2 λ 2 ( λ sinλt L (s 2 λ 2 2 (s 2 λ 2 2 (t. Inserting the result of Lemma 2. above for L ( (s 2 λ 2 2 and computing the convolution product by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. As the third step in induction, we have Lemma 2.3. Let s C with Re(s > and λ R a constant with λ. Then ( ( L 5 2 t 2 ( 3 λ 2 t 3 5t (s 2 λ 2 4 6λ 7 sinλt 6λ 6 cos λt. Proof. We compute ( L (s 2 λ 2 4 ( L s 2 λ 2 ( λ sinλt L (s 2 λ 2 3 (s 2 λ 2 3 (t. Inserting the result of Lemma 2.2 above for L ( (s 2 λ 2 3 and computing the convolution product by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. Theorem 2.4. Let s C with Re(s > and λ R a constant with λ. Then, for an integer n, ( L (s 2 λ 2 2n e 2n 2 (tsinλto 2n (tcosλt where e 2n 2 (t is an even polynomial of t with degree 2n 2 and with real coefficients involving λ, and o 2n (t is an odd polynomial of t with degree 2n and with real coefficients involving λ. Similarly, for an integer n 2, ( L (s 2 λ 2 2n e 2n 2 (tsinλto 2n 3 (tcosλt. Proof. By induction, suppose that the formula for 2n in the statement holds. We then compute ( ( L (s 2 λ 2 2n L s 2 λ 2 (s 2 λ 2 2n ( λ sinλt L (s 2 λ 2 2n (t

5 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... (and inserting the formula assumed for L ( (s 2 λ 2 2n we have: λ sinλt {e 2n 2(tsinλto 2n (tcosλt} λ sinλ(t τ{e 2n 2 (τsinλτo 2n (τcosλτ}dτ (and by addition theorem of trigonometric functions, we have: e 2n 2 (τ{cosλ(t 2τ cosλt}dτ o 2n (τ{sinλtsinλ(t 2τ}dτ [ e 2n 2 (τcosλ(t 2τdτ [ o 2n (τdτ sinλt ] e 2n 2 (τdτ cosλt ] o 2n (τsinλ(t 2τdτ. By using integration by parts repeatedly, we compute the first integral term among four terms as: e 2n 2(τcosλ(t 2τdτ [ e 2n 2 (τ sinλ(t 2τ { e2n 2 (t e 2n 2 ( ] t τ } sinλt e 2n 2(τsinλ(t 2τdτ e 2n 2 (τsinλ(t 2τdτ and note that the first coefficient { } is an even polynomial of t of degree 2n 2, and the integral in the second term is computed as: e 2n 2 (τsinλ(t 2τdτ [ e 2n 2 (τ cosλ(t 2τ { e 2n 2 (t e 2n 2 ( ] t τ } cosλt e 2n 2 (τcosλ(t 2τdτ e 2n 2 (τcosλ(t 2τdτ with the differential e 2n 2 ( and the coefficient { } an odd polynomial of t of degree 2n, and moreover, the last integral e 2n 2 (τcosλ(t 2τdτ is computed inductively and finitely by integration by parts to obtain the similar coefficients of sinλt and cosλt summed as even and odd polynomials of t with degrees less than 2n 2 and 2n, respectively. Thesameconsiderationasforthefirstintegralisappliedforthefourthintegral: o 2n (τsinλ(t 2τdτ to be computed as the sum of cosλt and sinλt with coefficients odd and even polynomials of t of degree 2n and 2n, respectively. As for e 2n 2(τdτ cosλt and o 2n (τdτ sinλt, the second and third integrals among four terms are computed to be odd and even polynomials of t with degrees 2n and 2n 2, respectively.

6 2 Takahiro Sudo CUBO 6, 3 (24 Summing up the computations above, we obtain ( L e (s 2 λ 2 2(n 2(n 2 (tsinλto 2(n 3 (tcosλt for some even e 2n (t and odd o 2n (t. This is the case where n is replaced with n in the second formula in the statement. Similarly, the case of 2n 2 is deduced from the case of 2n that now we have proved above. Remark. Perhaps, the real coefficients involving λ of even and odd polynomials in general 2n or 2n could be determined explicitly as given in Lemmas 2. to 2.3. Corollary 2.5. For any integer n, ( L (s 2 λ 2 n (t is an odd function as for t R. 3 Inverse Laplace transform for rational functions in another special case As a well known fact, we have Lemma 3.. Let s C with Re(s > and λ R a constant with λ. Then ( L s (s 2 λ 2 2 tsinλt (t R. Proof. We compute as in the previous section, ( ( L s (s 2 λ 2 2 L s 2 λ 2 s s 2 λ 2 λ sinλt cosλt λ sinλ(t τcosλτdτ [ cosλ(t 2ττsinλt Note that L(cosλt s s 2 λ 2 well known. As the second step in induction, we have ] t τ {sinλ(t 2τsinλt}dτ tsinλt. Lemma 3.2. Let s C with Re(s > and λ R a constant with λ. Then ( ( L s λ 2 t 2 (s 2 λ 2 3 4λ 3 sinλt t cosλt.

7 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 3 Proof. We compute ( L s (s 2 λ 2 3 ( L s 2 λ 2 ( λ sinλt L s (s 2 λ 2 2 s (s 2 λ 2 2 (t. Inserting the result of Lemma 3. above for L ( and computing the convolution product (s 2 λ 2 2 by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. One can use the following decomposition and Lemma 2. in the previous section: ( ( L s s 2 λ 2 (s 2 λ 2 2 cosλt L (s 2 λ 2 2 (t. s As the third step in induction, we have Lemma 3.3. Let s C with Re(s > and λ R a constant with λ. Then ( ( L s λ 3 t 2 (s 2 λ 2 4 6λ 5 sinλt ( 3λt 3 3 t 3 6λ 4 cos λt. Proof. We compute ( L s (s 2 λ 2 4 ( L s 2 λ 2 ( λ sinλt L s (s 2 λ 2 3 s (s 2 λ 2 3 (t. Inserting the result of Lemma 3.2 above for L ((s 2 λ 2 and computing the convolution product 3 by using integration by parts and addition theorem of trigonometric functions we obtain the formula in the statement. Theorem 3.4. Let s C with Re(s > and λ R a constant with λ. Then, for an integer n 2, ( L s (s 2 λ 2 2n e 2n 2 (tsinλto 2n (tcosλt where e 2n 2 (t is an even polynomial of t with degree 2n 2 and with real coefficients involving λ, and o 2n (t is an odd polynomial of t with degree 2n and with real coefficients involving λ. Similarly, for an integer n 2, ( L s (s 2 λ 2 2n e 2n 2 (tsinλto 2n 3 (tcosλt. s

8 4 Takahiro Sudo CUBO 6, 3 (24 Proof. By induction, suppose that two formula in the statement hold for 2n. We then compute ( ( L s (s 2 λ 2 2n L s 2 λ 2 s (s 2 λ 2 2n ( s λ sinλt L (s 2 λ 2 2n (t and inserting the formula assumed for L ((s 2 λ 2 we have 2n λ sinλt {e 2n 2(tsinλto 2n (tcosλt} λ s sinλ(t τ{e 2n 2 (τsinλτo 2n (τcosλτ}dτ. ( Note that this integral is exactly the same as the integral in the case of L proof of Theorem 2.4. Thus, we omit the rest of the proof. (s 2 λ 2 2n Similarly, the case of 2n2 is deduced from the case of 2n that now we have proved. Corollary 3.5. For any integer n 2, ( L s (s 2 λ 2 n (t is an odd function as for t R. in the Remark. Note that the polynomials obtained in Theorem 3.4 are not the same as those in Theorem 2.4, but we use the same symbols for both of the polynomials. Anyhow, combining both of Theorem 2.4 and Theorem 3.4 we get Corollary 3.6. Both L ( (s 2 λ 2 2n for n and L ( s (s 2 λ 2 2n for n 2 are written as the same form: e 2n 2 (tsinλto 2n (tcosλt; and both L ( (s 2 λ 2 2n for n and L ( s (s 2 λ 2 2n for n 2 are written as the same form: e 2n 2 (tsinλto 2n 3 (tcosλt. 4 Inverse Laplace transform for rational functions It iswell knownthat arationalfunction fofs Csuchthat f(s p(s q(s with p(s,q(s polynomials of s C with real coefficients and with degp(s < degq(s is decomposed into partial fractions as: f(s m j c j s j l 2 n k k j l m k k j c kj (s a k j l2 d kj s ((s b k 2 c 2 k j n k k j e kj ((s b k 2 c 2 k j, where q(s q s m Π l k (s a k m k Π l 2 k ((s b k 2 c 2 k n k

9 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 5 the factorization ofq(s in real R with a k real roots ofmultiplicity m k and with b k ±ic k imaginary roots of multiplicity n k, for some q in R, m,m k,n k, and a k,b k or, c k in R, and l,l 2 N, and c j,c kj R but c m or c kmk corresponding to the highest terms if nonzero; and d kj,e kj R but either d knk or e knk in R corresponding to the highest terms if nonzero. Therefore, using the basic facts of Laplace transform for polynomials and translation and our results in the previous sections we obtain Theorem 4.. Let f(s be a rational function of s C vanishing at infinity with the factorization as above. Assume that the real part of s satisfies the following inequality: Re(s > max{,a k,b j k l, j l 2 } if m, l, and l 2, and otherwise, some elements of the set may be dropped. Then, in general, L (f(s(t l 2 k l 2 k n2 l 2 m j d k c k e b kt sinc k t n k 2 k n2 l 2 k l 2 c j t j l (j! l 2 k m k k j c kj e a kt t j (j! d k2 2c k e b kt tsinc k t d k(2n e b kt [ e k(2n 2 (tsinc k to k(2n 3 (tcosc k t ] n k 2 d k(2n e b kt [ e k(2n 2 (tsinc k to k(2n (tcosc k t ] e k e b kt cosc k t n k 2 k n2 l 2 n k 2 k n [ ] e k(2n e b kt e k(2n 2 (tsinc kto k(2n 3 (tcosc kt [ ] e k(2n e b kt e k(2n 2 (tsinc kto k(2n (tcosc kt, where x means the minimum integer y such that y x, and x means the maximum integer y such that y x, and e kj (t, o kj (t, e kj (t, and o kj (t are polynomials of t with degree j and with real coefficients involving c k. Remark. The restriction on s C comes from the existence of the Laplace transform L(g(t(s f(s for some g(t. Namely, s C should belong to a domain of convergence of L(g(t(s. Note that the maximum in the statement is just that of real parts of poles of the rational function f(s given.

10 6 Takahiro Sudo CUBO 6, 3 (24 Remark. Note that L(δ(t for δ(t the Dirac function, viewed as a distribution, i.e. a functional, so that L ( δ(t, where the constant unit function on C is also viewed as a distribution. 5 Algebraic structure In this section we consider algebraic structure about the Laplace transform via convolution product. We denote by R (C the set of rational functions on C with real coefficients, vanishing at infinity. Under pointwise addition and multiplication, R (C becomes a non-unital algebra over R. Indeed, Lemma 5.. The set R (C is an algebra over R under point-wise operations. Proof. Let f,g R (C such that f(s p (s q (s and g(s p 2(s q 2 (s for some polynomials p j(s,q j (s with degp j < degq j (j,2. Then f(sg(s p (sq 2 (sp 2 (sq (s q (sq 2 (s and f(sg(s p (sp 2 (s q (sq 2 (s R (C R (C and other axioms can be also easily checked. We denote by A(R the algebra over R generated by the sets of elementary continuous functions on R: {t n n N {,,2, }} of monomials and {e µt µ R} and {sinλt,cosλt λ R} under point-wise addition and point-wise multiplication. Under (extended convolution product defined as: (f g(t f(t τg(τdτ, (t R for f,g A(R, which is a commutative and associative operation as well known as the usual case on [,, A(R becomes an algebra over R. Indeed, check it in details as follows: Proposition 5.2. The real algebra A(R under point-wise operations is viewed as an algebra over R under convolution product, as given in the following: [ n ( ] n ( ( (t n t m k (t t nm k km for n,m N; and for µ,λ R with µ λ, k (2 (e µt e λt (t { e λt e µt λ µ if µ λ, te µt if µ λ;

11 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 7 and for n in N and µ R with µ, (3 (t n e µt (t with the integral τk e µτ dτ equal to and for λ,µ R, and and n k ( n t n k ( k k k e µt ( l k! µ l (k l! tk l (4 (sinλt cosµt(t (cosλt cosµt(t (sinλt sinµt(t and moreover, for µ,λ R with µ, and and furthermore, { λ ( ( k k! µ k τ k e µτ dτ ; cosµt λ cosλt if λ ±µ, λ 2 µ 2 λ 2 µ 2 tsinλt if λ ±µ, 2 { µ sinµt λ sinλt if λ ±µ, λ 2 µ 2 λ 2 µ 2 2 tcosλt sinλt if λ ±µ, { λ (5 (e µt sinλt(t (e µt cosλt(t (6 (t n sinλt(t sinµt µ sinλt if λ ±µ, λ 2 µ 2 λ 2 µ 2 2 tcosλt sinλt if λ ±µ; λ 2 µ 2{λe µt µsinλt λcosλt}. λ 2 µ 2{µe µt µcosλtλsinλt}; n k ( n t t n k ( k τ k sinλτdτ k with I k τk sinλτdτ given by, for m N with m, if k 2m, and if k 2m, I 2m m m I 2m ( l k! λ 2l (k 2l2! tk 2l2 cosλt ( (l k! λ 2l (k 2l! tk 2l sinλt ( m k! λ 2m m ( l k! λ 2l (k 2l2! tk 2l2 cosλt m ( (l k! λ 2l (k 2l! tk 2l sinλt

12 8 Takahiro Sudo CUBO 6, 3 (24 so that (t n sinλt(t n 2 m ( n { 2m m n 2 m m ( 2ml k! λ 2l (k 2l2! tn 2l2 cosλt ( (2ml k! λ 2l (k 2l! tn 2l sinλt ( 3m k! λ 2m tn 2m } ( n { 2m m ( 2ml k! λ 2l (k 2l2! tn 2l2 cosλt m ( (2ml k! λ 2l (k 2l! tn 2l sinλt} and also, (t n cosλt(t n k ( n t t n k ( k τ k cosλτdτ k with J k τk cosλτdτ given by, for m N with m, if k 2m, and if k 2m, J 2m m m ( (l k! λ 2l (k 2l2! tk 2l2 sinλt ( (l k! λ 2l (k 2l! tk 2l cosλt so that J 2m (t n cosλt(t m ( (l k! λ 2l (k 2l2! tk 2l2 sinλt m ( (l k! λ 2l (k 2l! tk 2l cosλt ( m k! λ 2m2 n 2 m ( n { 2m m n 2 m m ( (2ml k! λ 2l (k 2l2! tn 2l2 sinλt ( (2ml k! λ 2l (k 2l! tn 2l cosλt} ( n { 2m m ( (2ml k! λ 2l (k 2l2! tn 2l2 sinλt m ( (2ml k! λ 2l (k 2l! tn 2l cosλt ( 3m2 k! λ 2m2 t n 2m }.

13 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 9 Finally, the convolution of general monomials is given by, as an example, (7 (t n e µ t sinλ t (t m e µ 2t cosλ 2 t(t n ( e µ t n t t n k ( k τ km e (µ 2 µ τ sinλ (t τcosλ 2 τdτ k k with sinλ (t τcosλ 2 τ 2 {sin(λ t (λ λ 2 τsin(λ t (λ λ 2 τ}, and then the following integral is computed inductively as I km,sin τ km e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ µ 2 µ (µ 2 µ 2 (λ ±λ 2 2tkm e (µ 2 µ t sin( λ 2 t λ ±λ 2 (µ 2 µ 2 (λ ±λ 2 2tkm e (µ 2 µ t cos( λ 2 t (km(µ 2 µ (µ 2 µ 2 (λ ±λ 2 2 (km(λ ±λ 2 (µ 2 µ 2 (λ ±λ 2 2 τ km e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ τ km e (µ 2 µ τ cos(λ t (λ ±λ 2 τdτ, where the last two integrals are denoted by I km,sin and I km,cos respectively, and the integrals can be inductively reduced to the cases of I j,sin and I j,cos for j km 2 and finally to the case of I,sin and I,cos that are given by and I,sin e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ µ 2 µ 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ sin( λ 2 t sin(λ t} λ ±λ 2 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ cos( λ 2 t cos(λ t} I,cos e (µ 2 µ τ cos(λ t (λ ±λ 2 τdτ µ 2 µ 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ cos( λ 2 t cos(λ t} λ ±λ 2 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ sin( λ 2 t sin(λ t} Other cases of convolution products of general monomials with sin and cos changed are also computed similarly, but omitted.

14 Takahiro Sudo CUBO 6, 3 (24 Proof. For (, check first that for n,m in N, (t n t m (t (t τ n τ m dτ n ( n n ( n t n k ( k τ km dτ t n k ( k tkm k k km k k [ n ( ] n ( k t nm A(R. k km k Also, for (2, for µ,λ R with µ λ, If µ λ, then (e µt e λt (t e µ(t τ e λτ dτ e µt e (λ µτ dτ e µt [ e (λ µτ λ µ (e µt e λt (t ]t τ eλt e µt λ µ e µt dτ te µt A(R. Moreover, for (3, for n in N and µ R with µ, (t n e µt (t n k (t τ n e µτ dτ ( n t n k ( kτ k e µτ dτ k n k ( n t n k ( k k We then compute the following integral by integration by parts: I k τ k e µτ dτ µ tk e µt k µ I k µ tk e µt k µ 2tk e µt k(k µ 2 I k 2 µ tk e µt k µ 2tk e µt ( k k(k 2 µ k k e µt ( l ( k! ( k k! µ l (k l! tk l µ k with I eµτ dτ µ (eµt. A(R. τ k e µτ dτ. ( t µ eµt µ I

15 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... Next, for (4, for λ,µ R with λ ±µ, (sinλt cosµt(t sinλ(t τcosµτdτ {sin(λt (λ µτsin(λt (λµτ}dτ 2 [ cos(λt (λ µτ cos(λt (λµτ ] t 2 λ µ λµ τ λ λ 2 µ 2 cosµt If λ ±µ, then (sinλt cosµt(t 2tsinλt A(R. Moreover, for λ,µ R with λ ±µ, (cosλt cosµt(t cosλ(t τcosµτdτ λ λ 2 cosλt A(R. µ 2 {cos(λt (λ µτcos(λt (λµτ}dτ 2 [ sin(λt (λ µτ sin(λt (λµτ ] t 2 (λ µ (λµ τ µ λ 2 µ 2 sinµt λ λ 2 sinλt A(R. µ 2 If λ ±µ, then (cosλt cosµt(t 2 tcosλt sinλt A(R. Furthermore, for λ,µ R with λ ±µ, (sinλt sinµt(t 2 2 sinλ(t τsinµτdτ {cos(λt (λ µτ cos(λt (λµτ}dτ [ sin(λt (λ µτ (λ µ sin(λt (λµτ ] t (λµ τ λ λ 2 µ 2 sinµt µ λ 2 sinλt A(R. µ 2 If λ ±µ, then (sinλt sinµt(t 2 tcosλt sinλt A(R. Next, for (5, for µ,λ R with µ, (e µt sinλt(t e µt e µτ sinλτdτ

16 2 Takahiro Sudo CUBO 6, 3 (24 with the integral I s e µτ sinλτdτ computed as ] t [ e µτ I s µ sinλτ λ τ µ e µτ cosλτdτ e µt µ sinλt λ µ 2( eµt cosλt λ2 µ 2I s so that Similarly, I s λ 2 µ 2{λ e µt (µsinλtλcosλt}. (e µt cosλt(t e µt e µτ cosλτdτ with the integral I c e µτ cosλτdτ computed as [ ] e µτ t I c µ cosλτ λ τ µ e µτ sinλτdτ µ ( e µt cosλt λ µ 2e µt sinλt λ2 µ 2I c so that I c λ 2 µ 2{µ e µt (µcosλt λsinλt}. On the other hand, for (6, for n in N and λ R with λ, (t n sinλt(t n k (t τ n sinλτdτ ( n t n k ( k τ k sinλτdτ k n k We then compute the following integral by integration by parts: ( n t t n k ( k τ k sinλτdτ. k I k τ k sinλτdτ λ tk cosλt k λ Inductively, if k 2m with m N and m, then I 2m m τ k cosλτdτ λ tk cosλt k λ 2tk sinλt k(k λ 2 I k 2. ( l k! λ 2l (k 2l2! tk 2l2 cosλt m ( (l k! λ 2l (k 2l! tk 2l sinλt ( m k! λ 2m I

17 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 3 with I sinλτdτ λ λ cosλt, and hence, I 2m m m If k 2m with m N and m, then I 2m m ( l k! λ 2l (k 2l2! tk 2l2 cosλt ( (l k! λ 2l (k 2l! tk 2l sinλt ( m k! λ 2m. ( l k! λ 2l (k 2l2! tk 2l2 cosλt m ( (l k! λ 2l (k 2l! tk 2l sinλt ( m k! λ 2m I with I τsinλτdτ λ tcosλt λ sinλt, and hence, 2 I 2m m ( l k! λ 2l (k 2l2! tk 2l2 cosλt m ( (l k! λ 2l (k 2l! tk 2l sinλt. Similarly, for n in N and λ R with λ, (t n cosλt(t n k (t τ n cosλτdτ ( n t n k ( k τ k cosλτdτ k n k We then compute the following integral by integration by parts: ( n t t n k ( k τ k cosλτdτ. k J k τ k cosλτdτ λ tk sinλt k λ Inductively, if k 2m with m N and m, then J 2m m τ k sinλτdτ λ tk sinλt k λ 2tk cosλt k(k λ 2 J k 2. ( (l k! λ 2l (k 2l2! tk 2l2 sinλt m ( (l k! λ 2l (k 2l! tk 2l cosλt ( m k! λ 2m J

18 4 Takahiro Sudo CUBO 6, 3 (24 with J cosλτdτ λ sinλt, and hence, J 2m m m If k 2m with m N and m, then J 2m m ( (l k! λ 2l (k 2l2! tk 2l2 sinλt ( (l k! λ 2l (k 2l! tk 2l cosλt. ( (l k! λ 2l (k 2l2! tk 2l2 sinλt m ( (l k! λ 2l (k 2l! tk 2l cosλt ( m k! λ 2m J with J τcosλτdτ λ tsinλt λ 2 (cosλt, and hence, J 2m m ( (l k! λ 2l (k 2l2! tk 2l2 sinλt m ( (l k! λ 2l (k 2l! tk 2l cosλt ( m k! λ 2m2. Finally, for (7, the convolution of general monomials is given by, as an example, (t n e µ t sinλ t (t m e µ 2t cosλ 2 t(t e µ t (t τ n e µ (t τ sinλ (t ττ m e µ 2τ cosλ 2 τdτ n k ( n t t n k ( k τ km e (µ 2 µ τ sinλ (t τcosλ 2 τdτ k with sinλ (t τcosλ 2 τ 2 {sin(λ t (λ λ 2 τsin(λ t (λ λ 2 τ}, and then the following integral is computed inductively as I km,sin τ km e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ [τ km e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ (km ] t τ ( τ km e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ dτ

19 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 5 where the indefinite integral is computed by using integration by parts twice as: e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ and hence, we obtain µ 2 µ 2 µ τ (µ 2 µ 2 (λ ±λ 2 2e(µ sin(λ t (λ ±λ 2 τ λ ±λ 2 2 µ τ (µ 2 µ 2 (λ ±λ 2 2e(µ cos(λ t (λ ±λ 2 τ µ 2 µ I km,sin (µ 2 µ 2 (λ ±λ 2 2tkm e (µ 2 µ t sin( λ 2 t λ ±λ 2 (µ 2 µ 2 (λ ±λ 2 2tkm e (µ 2 µ t cos( λ 2 t (km(µ 2 µ (µ 2 µ 2 (λ ±λ 2 2 (km(λ ±λ 2 (µ 2 µ 2 (λ ±λ 2 2 τ km e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ τ km e (µ 2 µ τ cos(λ t (λ ±λ 2 τdτ, wherethelasttwointegralsaredenotedbyi km,sin andi km,cos respectively,andtheintegrals can be inductively reduced to the case of I,sin and I,cos that are given by and I,sin e (µ 2 µ τ sin(λ t (λ ±λ 2 τdτ µ 2 µ 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ sin( λ 2 t sin(λ t} λ ±λ 2 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ cos( λ 2 t cos(λ t} I,cos e (µ 2 µ τ cos(λ t (λ ±λ 2 τdτ µ 2 µ 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ cos( λ 2 t cos(λ t} λ ±λ 2 2 µ t (µ 2 µ 2 (λ ±λ 2 2{e(µ sin( λ 2 t sin(λ t}. Other cases of convolution products of general monomials with sin and cos changed are also computed similarly. Theorem 5.3. The Laplace transform L is an algebra homomorphism from A(R with convolution product to R (C with point-wise multiplication. Also, the inverse Laplace transform L is an algebra homomorphism from R (C to A(R. Then L L id A(R and L L id R (C the identity maps on A(R and R (C.

20 6 Takahiro Sudo CUBO 6, 3 (24 Remark. Note that we identify a rational function g in R (C with the image of f A(R under L with domain of convergence such that L(f g. Indeed, the infimum of the real part of the domain of convergence for L(f g is defined to be the maximum of the real parts of poles of g, so that the domain of convergence for L(f is determined by g uniquely. Proof. It is well known that L(f g(s L(f(s L(g(s for f,g A(R. It follows by the convolution products checked explicitly in Proposition 5.2, in particular, that any element of A(R, which is a linear combination of multiples of elementary continuous functions, is a continuous function on R, so that, as also a well known fact, the Laplace transform is injective on A(R but restricted to [,, and hence the inverselaplace transforml is alsoinjective on L(A(R. Note that real coefficients in such linear combinations are determined uniquely by the injectivity, so that we may extend the definition domains from [, to R preserving the injectivity. It is clear that L(A(R is contained in R (C by using basic formulae in Laplace transform and by Proposition 5.2. For instance, check that L(e µt t n sinλt(s L(t n sinλt(s µ ( ( n L(( t n sinλt(s µ ( n dn λ ds n (s µ 2 λ 2 R (C. Thelastbelongingisprovedbyinduction. Indeed,if p(s q(s R (C, then d ds (p(s q(s p (sq(s p(sq (s q(s 2 R (C. It is also checked explicitly in Theorem 4. that any element of R (C is mapped to an element of A(R under L. Corollary 5.4. It follows that the algebra A(R with convolution product is isomorphic to R (C, as an algebra. It also follows that the algebra A(R with point-wise multiplication is isomorphic to R (C, as a real vector space. Remark. The Laplace transform L (as well as the inverse L is linear but dose not preserve point-wise multiplication. For instance, L(t 2 2 s 3 L(t L(t s 2 s 2 s 4. Received: October 23. Accepted: August 24. References [] Hiroshi Fukawa, Laplace transformation and ordinary differential equations (in Japanese, Sho-Ko-dou (995.

21 CUBO 6, 3 (24 Computing the inverse Laplace transform for rational functions... 7 [2] M. S. J., Mathematics Dictionary, (Sugaku Jiten, in Japanese, Math. Soc. Japan, 4th edition, Iwanami (27. [3] A. Vretblad, Fourier Analysis and Its Applications, GTM 223, Springer (23.