SUBSTITUTION RELATIONS FOR LAPLACE TRANSFORMATIONS

Additional results are obtained which center around expressions for the Laplace transforms of functions of the form k(t)F(g(t)). The finite Laplace transformation is involved in a number of the formulas. Examples involving several special cases of g and k are included.

paper involve both the Laplace transformation and the finite Laplace transforma- tion (that is, the integral is over a finite interval).They do not seem to have appeared in the literature and further, although they are not difficult to obtain, do not seem to be otherwise well known.
Throughout this work we assume the form f(s) L{F(t)} foo e_St F(t) dt (i.i) 0 for the Laplace transformation and we use the notations f(s;(B,y)) L{F(t) [U(t-B) U(t-y)]} f e -st F(t) dt.
(1.2) for the finite Laplace transformation.(We also allow y .)We refer to the tables of Roberts and Kaufman [3] throughout and we use the notation [II. 3.2   (4)], for example, to refer to Part II, Section 3.2, Formula 4. The Heaviside (unit step) function is denoted by U.
Since our results are centered around modifications of it, we restate Theorem i of [i].
For our first result we relax the conditions on g from those stated in Theorem i.In connection with this we introduce the finite Laplace transformation (1.2).Now if g is strictly monotone on some subinterval of (0, ) we have the following result.THEOREM 2. Under the hypotheses of Theorem i, except that now let g be monotone on (b,c) with g(b) B < T g(c) (or with > y), then /_{k(t) F[g(t)] [U(t-b) U(t-c)]} foo (s u) f(u;(B,y)) du 0 (2.4) For our second result we modify Theorem i by the introduction of an adjustment function as follows.THEOREM 3.Under the hypotheses of Theorem i, except that we assume the relations t{F(p)A(p)} f(s;A), I{(s,u;A)} (s,p)/A(p), The proofs of Theorems 2 and 3 follow directly the lines of proof given for Theorem i in [I] and hence they are omitted.Two special cases of Theorem 2 are noted.COROLLARY  From Corollary L{(t2-a2 10(a(u2-s2) I/2) f(u; (a,)) (4 3) 0 For the three corresponding cases a > b > 0; a > b, a > 0; and 0 > a > b; must be replaced by I v, a and b interchanged in the U-functions, (b-a) replaced by (a-b), and the f's replaced by f(u;(a/b,)), f(u;(0,1)), and f(u;(a/b,l)) in (4.1), (4.2), and (4.3) respectively.
If a>b>0, e-bS F e-u (u(a-b)/s)V/2 jvi2(s(a_b)u )I/2) f(u) du; (4.5) For the corresponding cases b > a > 0 and b > 0, a < 0 we again change J to I v, interchange a and b in the U-function and whenever a-b appears, and in (4.5) replace f(u;(0,-a/b)) by f(u;(-a/b,)).