Sivkumr Exmple Sheet 6 Infinite nd Improper Integrls MATH 5H Mteril presented here is extrcted from Stewrt s text s well s from R. G. Brtle s The elements of rel nlysis. Infinite Integrls: These integrls re lso clled improper integrls of the first kind ; we shll, however, use the term infinite integrls, which is due to the gret British nlyst G. H. Hrdy (877 947). Suppose is fixed rel number, nd tht f is Riemnn integrble on the intervl [, c] for every c >. The infinite integrl of f over [, ) is defined s follows: f(x) := provided the it exists (s finite number). We shll sy tht the infinite integrl does not exist if the foresid it does not exist.. Find ll vlues of the rel number α for which the infinite integrl exist.. Find ll vlues of the rel number α for which the infinite integrl exist. 3. Find ll vlues of the rel number α for which the infinite integrl not exist. e f(x) x α exists/ does not e αx exists/ does not (ln x) α exists/ does x If is fixed rel number nd f is Riemnn integrble on the intervl [b, ] for every b <, then the infinite integrl of f over (, ] is defined s follows: f(x) = b b provided the it exists (s finite number). We sy tht the infinite integrl not exist if the foresid it does not exist. f(x) does We shll sy tht f is loclly Riemnn integrble throughout R if the definite integrl µ λ f(x) exists for every pir of rel numbers λ nd µ. 4. Suppose tht f is loclly Riemnn integrble throughout R. Prove the following sttements: (i) If the infinite integrl rel number. f(x) exists for some rel number, then it exists for every
(ii) If the infinite integrl f(x) exists for some rel number, then it exists for every rel number. Suppose f if loclly Riemnn integrble throughout R, nd ssume tht ech of the infinite integrls f(x) nd f(x) exists for some vlue of. We define the infinite integrl of f over (, ) s their sum, nmely: f(x) := f(x) + f(x) = b b f(x) + f(x). (6.) The existence of both its in the definition bove implies the existence of the it [ ] f(x) + f(x) = (6.) c c nd the equlity of the its in (6.) nd (6.). On the other hnd, it is possible to consider only the specil kind of symmetric it given in (6.). Indeed, when this it in (6.) exists (regrdless of whether the it(s) in (6.) exist), we cll it the Cuchy Principl Vlue of the infinite integrl of f over (, ), nd denote it by (CPV) f(x). As noted erlier, if the infinite integrl of f over (, ) exists (tht is, (6.) exists s finite number), then tht number is lso the CPV of the infinite integrl of f over (, ). However, the converse is flse in generl: 5. Give n exmple of function f which stisfies ech of the following properties: (i) f is loclly Riemnn integrble throughout R, (ii) (CPV) f(x) does not exist. integrl is, in generl, weker requirement. f(x) exists, nd (iii) the infinite integrl This shows tht the existence of the Cuchy Principl Vlue Theorem (Comprison Test). Suppose is fixed rel number. Assume tht f nd g re continuous functions with f(x) g(x) for every x. The following hold: (i) If the infinite integrl g(x). (ii) If the infinite integrl f(x) does not exist, then neither does the infinite integrl g(x) exists, then so does the infinite integrl f(x). Theorem (Limit Comprison Test). Suppose is fixed rel number. Assume tht f nd g re continuous functions with f(x) nd g(x) > for every x. If f(x) x g(x) = L >,
then both or neither of the infinite integrls 6. Determine whether the following infinite integrls exist: (i) (ii) (iii) (iv) x 3 + x x + 3x 3 + x + 5x + x 5 + x + (x ) x 5/ + x + 3 g(x) exist. Improper Integrls: These re sometimes clled improper integrls of the second kind. Suppose [, b] is closed bounded intervl. Let f be function which is defined t lest for x stisfying < x b, nd ssume tht f is Riemnn integrble on [c, b] for every < c < b. The improper integrl of f over [, b] is defined s follows: f(x) := c + c provided the it exists. Entirely nlogously, if f is defined for every x < b nd is Riemnn integrble on [, c] for every < c < b, then the improper integrl of f over [, b] is defined by provided the it exists. f(x) := c b 7. Find the vlues of the rel number α such tht the integrl 8. Study (the importnt) Exmple 8 on Pge 55 in Stewrt s text. 9. Suppose n is positive integer. (i) Show tht (ln x) n = n (ln x) n. x α exists/does not exist. (ii) Deduce tht (ln x) n = ( ) n n!. Suppose now tht p is n interior point of the intervl [, b], tht is, < p < b. Assume tht f is defined t every point of [, b] except perhps t p. If both the improper integrls 3 p f(x) nd
p f(x) exist, then we define the improper integrl of f over [, b] s their sum. In terms of its, we hve: f(x) := ɛ + p ɛ f(x) + f(x). (6.3) δ + p+δ As in the cse of infinite integrls, the existence of both its in (6.3) implies the existence of the symmetric it [ p ɛ ] b f(x) + f(x). (6.4) ɛ + p+ɛ On the other hnd, if we only require the symmetric it in (6.4) to exist (regrdless of (6.3)), we obtin the Cuchy Principl Vlue of the improper integrl of f over [, b], nd denote it by (CPV) f(x). Once gin, existence of the CPV of the improper integrl is weker requirement (vide infr):. Give n exmple of function f stisfying ech of the following conditions: (i) f is defined nd continuous on [, ) (, ], (ii) (CPV) f(x) does not exist. f(x) exists, nd (iii) the improper integrl Lplce Trnsforms: The next pir of exmples feture infinite integrls which re known s Lplce Trnsforms. A more detiled tretment of this mteril wits you in course on Differentil Equtions.. Suppose s > nd ω re fixed rel numbers. (i) Show tht the infinite integrl (ii) Evlute the infinite integrl in (i). e st cos(ωt) dt exists.. Suppose s > is fixed, nd let n be positive integer. (i) Show tht the infinite integrl (ii) Show tht (iii) Deduce tht e st t n dt exists. e st t n dt = n s e st t n dt = n! s n+. e st t n dt. A two for one del: The following exmple presents n integrl which is both infinite nd improper. 4
3. Consider the integrl x x 4. which is infinite s well s improper. Evlute it by doing the following: (i) Express the integrl s the sum of two integrls: x x 4 + x x 4. (ii) Compute ech of the two integrls. Note tht the first of the two is n improper integrl, whilst the second is n infinite one. 5