Universidd Crlos III de Mdrid Clulus II Mrin Delgdo Téllez de Ceped Unit 3 Integrls Depending on Prmeter Definition 3.1. Let f : [,b] [,d] R, if for eh fixed t [,d] the funtion f(x,t) is integrble over [, b] on the x vrible, we define the following funtion F : [, b] R s F(t) = f(x,t)dx. We ll F(t) n integrl depending on prmeter. Theorem 3.2. f ontinuous on [, b] [, d] F ontinuous on [, d]. Theorem 3.3. f nd f t = f ontinuous on [,b] [,d] F differentible on [,d] t nd F (t) = f t (x,t)dx. Note. The ide is tht we n interhnge the proesses of integrtion nd differentition. Theorem 3.4 (Leibniz s Theorem). Let f nd f t = f be ontinuous funtions on t [,b] [,d] nd α,β differentible funtions on [,d] with imge on [,b], tht is, α(t),β(t): [,d] [,b], x [α(t),β(t)] [,b]. We define G(t) = then G is differentible on [, d] nd β(t) α(t) f(x,t)dx, G (t) = f(β(t),t) β (t) f(α(t),t) α (t) + β(t) α(t) f t (x,t)dx
Universidd Crlos III de Mdrid Clulus II 2 Mrin Delgdo Téllez de Ceped Integrls Depending on Prmeter The previows theorems nnot be used when the integrls re improper. So we hve the following results tht re vlid if the integrls re improper or not. Theorem 3.5. Let f(x,t) be ontinuous funtion of t on [,d] for lmost every x [, b], if there exists funtion g(x) integrble on [, b] suh tht f(x, t) g(x), t [, d] nd for lmost every x [, b], then F is ontinuous on [, d]. Theorem 3.6. Let f t (x,t) be ontinuous funtion of t on [,d] for lmost every x [,b], if there exists funtion g(x) integrble on [,b] suh tht f t (x,t) g(x), t [, d] nd for lmost every x [, b], then F is differentible on [, d] nd F (t) = f t (x,t)dx. Theorem 3.7. Let f(x, t) be integrble over [, b] [, d], then F(t) is integrble over [,d] nd d F(t)dt = d f(x,t)dxdt = d f(x, t)dtdx. Note. The integrls tht depend on prmeter n be omputed by tking the derivtive with respet to the prmeter, F (t) = f t(x,t)dx, if we re ble to ompute this integrl, then we just hve to integrte with respet to the prmeter, nd finlly to find the vlue of the onstnt of integrtion tht ppers, fixing the vlue of the prmeter. We n lso use this ide to ompute n integrl I tht does not depend on prmeter, by mking it depend on prmeter, I(t), then we ompute I(t) by differentiting nd integrting with respet to t, nd finlly fixing the prmeter to t tht gives us bk to the vlue of I = I(t ).
Universidd Crlos III de Mdrid Clulus II Integrls Depending on Prmeter Mrin Delgdo Téllez de Ceped 3 The Gmm Funtion Γ(x) = t x 1 e t dt, x >. It is generliztion of the ftoril funtion to rel nd omplex numbers. Properties of the Gmm funtion 1. Γ(x) is ontinuous nd differentible. 2. Γ(x) C nd dn dx nγ(x) = 3. Γ(1) = Γ(2) = 1, Γ(1/2) = π. (log t) n t x 1 e t dt. 4. Γ(x + 1) = xγ(x). Γ(n + 1) = n!, n N. 5. lim Γ(x) =. x + The B(p,q) = Bet funtion 1 x p 1 (1 x) q 1 dx, p,q > It is n integrl depending on two prmeters, ll the theory is vlid with the nturl modifitions. 1. B(p,q) = B(q,p). Properties of the Bet funtion 2. B(p,q) is ontinuous nd differentible on eh vrible. 3. B(p,q) C, nd n+m p n q mb(p,q) = 1 x p 1 (log x) n (1 x) q 1 (log (1 x)) m dx, p,q >. 4. B(p,q) = q 1 B(p,q 1), q > 1. p + q 1 ( ) 1 m + n 1 5. B(m + 1,n + 1) =, m,n N. m + n + 1 n 6. B(p,q) = 2 π/2 7. B(p,q) = Γ(p)Γ(q) Γ(p + q). 8. B(1/2,1/2) = π. (os t) 2p 1 (sin t) 2q 1 dt = t p 1 dt. (1 + t) p+q
Universidd Crlos III de Mdrid Clulus II 4 Mrin Delgdo Téllez de Ceped Integrls Depending on Prmeter The Trnsform of Lple It is useful to solve some integrls, solve systems of ordinry differentil equtions with initil vlue nd to study systems whose equtions ontin derivtives nd integrls. Definition 3.8. Let f : [, ) R be integrble, with exponentil growth, tht is, f(x) Ce αx, x > T, where C,α,T re onstnts depending of f, we define the Lple Trnsform of f s L[f(x)](s) F(s) = e sx f(x)dx. Properties of the Trnsform of Lple 1. L[f(x)](s) onverges for s (α, ) nd is ontinuous on (α, ). 2. L[f(x)](s) C s α, s > α. 3. Linerity: L[αf(x) + βg(x)](s) = αl[f(x)](s) + βl[g(x)](s). 4. L[1](s) = 1 s, s >. L[ex ](s) = 1 s, s >. L[x n ](s) = n! s n+1, s >, n N. Γ(α + 1) L[xα ](s) = s α+1, s >, α > 1. L[sin (x)](s) = s 2 + 2, s >. L[os (x)](s) = s s 2 + 2, s >. 5. Trnsltion: L[e x f(x)](s) = L[f(x)](s + ) = F(s + ), R. { f(x), x 6. Redefining f(x) =, x < L[f(x )](s) = e s L[f(x)](s), >. 7. L[f(x)](s) = 1 L[f(x)] ( s 8. ), >. d n ds nl[f(x)](s) = ( 1)n L[x n f(x)](s), n = 1,2,3... 9. Let f (n 1) be differentible on (, ) nd f,f,...,f (n 1) ontinuous on x, then [ ] d n L dx nf(x) (s) = s n L[f(x)](s) s n 1 f() s n 2 f (). f (n 1) (). In prtiulr L[f (x)](s) = sl[f(x)](s) f(), L[f (x)](s) = s 2 L[f(x)](s) sf() f ().
Universidd Crlos III de Mdrid Clulus II Integrls Depending on Prmeter Mrin Delgdo Téllez de Ceped 5 Definition 3.9. The onvolution of f(x) nd g(x) is the funtion Properties f g(x) = x f(u)g(x u)du, x. f g(x) = g f(x). L[f g(x)](s) = F(s)G(s). Definition 3.1. Given funtion F(s), if there exists f(x), ontinuous on x, suh tht L[f(x)](s) = F(s), we define the Inverse Lple Trnsform of F(s) s L 1 [F(s)](x) = f(x). The Inverse Lple Trnsform is lso liner. Applition of The Trnsform of Lple We n use the Lple trnsform to solve systems of differentil equtions: Differentil eqution for f(x) L Trnsformed eqution for F(s) Solve F(s) L 1 The solution is f(x)