Generalized improper integral definition for finite limit
aa r X i v : . [ m a t h . C A ] N ov Generalized improper integral definitionfor finite limit
Michael A. Blischke
Abstract.
A generalization of the definition of a one-dimensional im-proper integral with a finite limit is presented. The new definition ex-tends the range of valid integrals to include integrals which were pre-viously considered to not be integrable. This definition is shown to beequivalent to the infinite limit definition presented in “Generalized im-proper integral definition for infinite limit” via a particular change ofvariable of integration. The definition preserves linearity and unique-ness. Integrals which are valid under the conventional definition havethe same value under the new definition. Criteria for interchanging theorder of integration and differentiation, and for interchanging the orderwith a second integration, are obtained. Examples are provided.
Contents
1. Introduction and preliminaries 12. Generalized definition for finite limit improper integral 23. Conversion to infinite limit integral 54. Equivalence of generalized definitions 95. Examples 126. Conclusion 14References 15
1. Introduction and preliminaries
There are two basic types of improper integrals. Integrals with an infinitelimit are defined as the limit of a series of proper integrals as one of thelimits approaches infinity. Improper integrals with finite limits are neededwhen the integrand does not have a finite limiting value as the variableof integration approaches a particular “critical” value. In this case, theimproper integral is defined as the limit of a series of proper integrals as oneof the limits approaches the critical value. Improper integrals with morethan one critical value, or with interior critical values, can be found as asum of these two basic types.
Mathematics Subject Classification.
Key words and phrases.
Integration, Integral, Improper integral, Leibniz integral rule. MICHAEL A. BLISCHKE
The improper integral with an infinite upper limit defined by Z ∞ a f ( x ) dx ≡ lim b →∞ (cid:26)Z ba f ( x ) dx (cid:27) exists when the limit exists, with a similar definition for an integral with aninfinite lower limit.Similarly, the improper integral with a critical lower limit, defined by Z βα g ( u ) du ≡ lim δ → + (cid:26)Z βα + δ g ( u ) du (cid:27) exists when that limit exists, with a similar definition for an integral with acritical lower limit. For any of the above, when the limit does not exist, theintegrals are said to not exist, or to diverge. We will refer to the above asthe conventional definitions.In “Generalized improper integral definition for infinite limit” [Bli12], thefollowing definition for an integral with an infinite limit was introduced:(1) Z Z ∞ a f ( x ) dx ≡ lim b →∞ (cid:26)Z ba f ( x ) dx + Z b + cb f ( x ) z ( x − b ) dx (cid:27) where f ( x ) is the function to be integrated, and where z ( x ) is a termina-tion function, defined therein. The inclusion of the additional term insidethe limit allows convergence to be rigorously shown for a greater range offunctions, f ( x ). The over-struck Z on integrals using the alternate defini-tion was included there to distinguish them from integrals that exist usingconventional definitions.A useful form that is equivalent to (1) is(2) Z Z ∞ a f ( x ) dx = − F ( a ) − lim b →∞ (cid:26)Z c F ( x + b ) z ′ ( x ) dx (cid:27) where F ( x ) is defined (for some arbitrary lower limit φ ) by(3) F ( x ) ≡ Z xφ f ( x ′ ) dx ′ . When the integrals and the limit in (1) or (2) exist, the integral of f ( x )is said to exist under the alternate definition, and to have the value of thelimit. In [Bli12], it was shown that for all termination functions for whichthe limit exists, the integral will have the same value, so that the definitiongives a unique value. Many other properties of integrals found using thealternative definition were shown there.
2. Generalized definition for finite limit improper integral
It is desirable to introduce a corresponding definition for an improperintegral where the critical limit is finite. We introduce as our definition ofimproper integral for a function with a critical lower limit α and a noncriticalupper limit β : ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 3
Definition 1. (4) Z Z βα g ( u ) du ≡ lim δ → + (cid:26)Z α + δα g ( u ) w (cid:18) ( u − α ) δ (cid:19) du + Z βα + δ g ( u ) du (cid:27) The term w ( v ) is defined below. We will again follow the convention ofusing the overstruck Z in this paper for the integrals using the new definitionsto distinguish them from conventionally defined integrals. An improperintegral with critical finite upper limit and noncritical lower limit is definedsimilarly as Definition 2. (5) Z Z βα g ( u ) du ≡ lim δ → + (cid:26)Z β − δα g ( u ) du + Z ββ − δ g ( u ) w (cid:18) ( β − u ) δ (cid:19) du (cid:27) . For simplicity in the derivations, and without loss of generality, for theremainder of this paper we will take the critical limit to be the lower limit,and to be 0, giving(6) Z Z β g ( u ) du ≡ lim δ → + (cid:26)Z δ g ( u ) w ( u/δ ) du + Z βδ g ( u ) du (cid:27) . The function w ( v ) will be referred to as the initialization function, anal-ogous to the termination functions in [Bli12] for the infinite limit case.As was the case for the termination function, the initialization functionis not arbitrary. Its function is to smooth out the sharp lower bound whentaking the limit. It is required to be finite, to not depend on δ , and to satisfythe following conditions:(7) w ( v ) = (cid:26) v < ǫ v ≥ w ′ ( v ) ≥ ǫ with(9) 0 < ǫ ≤ . Definition 3.
An initialization function is any function satisfying the con-ditions given in (7) through (9).From (7) we have that(10) w ′ ( v ) = (cid:26) v < ǫ v >
1A less restrictive condition on w ( v ) may be possible, for example allowing ǫ = 0 with w ( v ) → v →
0. However, requiring nonzero ǫ will allow us to show equivalence between the finite limit definition andthe infinite limit definition from [Bli12]. Investigation into less restrictive MICHAEL A. BLISCHKE conditions on initialization functions which maintain the desirable proper-ties of the new definition, but which expand the allowable set of integrablefunctions, is a subject for future research.With (7) and (10), we have that(11) Z w ′ ( v ) = Z ǫ w ′ ( v ) = 1 . We can get w ( v ) for 0 < v < w ′ ( v ) as(12) w ( v ) = Z v w ′ ( v ′ ) dv ′ . We now define(13) G ( u ) ≡ Z uφ g ( s ) ds and using integration by parts, (6) becomes Z Z β g ( u ) du = lim δ → + ( G ( u ) w ( u/δ ) (cid:12)(cid:12)(cid:12)(cid:12) δ − δ Z δ G ( u ) w ′ ( u/δ ) du + G ( u ) (cid:12)(cid:12)(cid:12)(cid:12) βδ ) = G ( β ) − lim δ → + (cid:26) δ Z δ G ( u ) w ′ ( u/δ ) du (cid:27) . (14)This form is analogous to (2), and can be an easier form to work withthan (6).We will follow a path analogous to that taken with termination func-tions in [Bli12]. Given two initialization functions w ( v ) and w ( v ), we cancombine them to obtain a third, via their derivatives, as(15) w ′ ( v ) ≡ Z w ′ ( v/v ′ ) w ′ ( v ′ ) v ′ dv ′ . Using (10), (15) can also be written as(16) w ′ ( v ) = Z v w ′ ( v/v ′ ) w ′ ( v ′ ) v ′ dv ′ . From (15), since w ′ ( v ) and w ′ ( v ) are both non-negative, w ′ ( v ) is alsonon-negative, satisfying (8). If w ′ ( v ) satisfies (10), we can see that w ′ ( v )also satisfies (10). We can also show that w ′ ( v ) satisfies (11), and thereforealso (7): ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 5 Z w ′ ( v ) = Z (cid:20)Z w ′ ( v/v ′ ) w ′ ( v ′ ) v ′ dv ′ (cid:21) dv = Z (cid:20)Z w ′ (cid:0) v/v ′ (cid:1) dv (cid:21) w ′ ( v ′ ) v ′ dv ′ = Z "Z v ′ w ′ (cid:0) v/v ′ (cid:1) dv w ′ ( v ′ ) v ′ dv ′ = Z v ′ w ′ ( v ′ ) v ′ dv ′ = 1 . (17)We will denote the combined initialization function using the same nota-tion we used for termination functions,(18) w ( v ) = w ( v ) ⊙ w ( v ) . We also have, substituting v ′′ = v/v ′ , that w ′ ( v ) = Z w ′ ( v/v ′ ) w ′ ( v ′ ) v ′ dv ′ = Z w ′ ( v ′′ ) w ′ ( v/v ′′ ) v ′ vv ′ v ′′ dv ′ = Z w ′ ( v ′′ ) w ′ ( v/v ′′ ) v ′′ dv ′ (19)so the relation (18) satisfies commutivity.
3. Conversion to infinite limit integral
We will now examine what happens when we switch between the gen-eralized finite limit integral and an infinite limit improper integral. Thisdevelopment sets up the next section, where we show equivalence betweenthe generalized finite limit definition and the generalized infinite limit defi-nition.We will use the general change of variable defined by u = ψ ( x )(20) x = ψ − ( u ) . We require for all x > ψ − ( β ) that ψ ( x ) be finite, that it be strictly mono-tonic with(21) ψ ′ ( x ) < , and that(22) lim x →∞ { ψ ( x ) } ≡ . Combining these properties, we obtain that(23) ψ ( x ) > , MICHAEL A. BLISCHKE that(24) 0 ≤ ψ ( x + b ) ψ ( b ) ≤ b ≥ ψ − ( β ) , x ≥ Z ∞ x ψ ′ ( x ′ ) dx ′ = ψ ( x ) . Using (20) in (13) gives(26) G ( u ) = Z ψ − ( u ) ψ − ( φ ) g ( ψ ( t )) ψ ′ ( t ) dt. Defining(27) f ( t ) ≡ − g ( ψ ( t )) ψ ′ ( t )and(28) F ( x ) ≡ Z xψ − ( φ ) f ( t ) dt we get(29) G ( u ) = − Z ψ − ( u ) ψ − ( φ ) f ( t ) dt = − F ( ψ − ( u ))or(30) F ( x ) = − G ( ψ ( x )) . We will now define an infinite limit integral corresponding to our finitelimit integral as(31) Ξ Z ∞ ψ − ( β ) f ( x ) dx ≡ Z Z β g ( u ) du. In (31) we use an over-struck Ξ, instead of an over-struck Z , to allow us todistinguish the resulting integral (41) from the infinite limit integral defi-nition introduced in [Bli12]. We will use the overstruck Ξ throughout thispaper for the infinite limit improper integral that is obtained from our finitelimit improper integral via a change of variable within the definition . Sincethe definition is in terms of conventional integrals, we know we can safelyperform those changes of variable. Note that we haven’t yet shown anyrelation beween this and the infinite limit integral definition from [Bli12]. ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 7
Using (31) in (14) we getΞ Z ∞ ψ − ( β ) f ( x ) dx = G ( β ) − lim δ → + (cid:26) δ Z δ G ( u ) w ′ ( u/δ ) du (cid:27) = G ( β ) − lim δ → + ( δ Z ∞ ψ − ( δ ) G ( ψ ( x )) w ′ (cid:18) ψ ( x ) δ (cid:19) ψ ′ ( x ) dx ) = − F (cid:0) ψ − ( β ) (cid:1) + lim δ → + ( δ Z ∞ ψ − ( δ ) F ( x ) w ′ (cid:18) ψ ( x ) δ (cid:19) ψ ′ ( x ) dx ) = − F (cid:0) ψ − ( β ) (cid:1) + lim b →∞ (cid:26) ψ ( b ) Z ∞ b F ( x ) w ′ (cid:18) ψ ( x ) ψ ( b ) (cid:19) ψ ′ ( x ) dx (cid:27) (32)where b ≡ ψ − ( δ ).We now define ζ ( x, b ) such that(33) ζ (0 , b ) ≡ ζ ′ ( x − b, b ) ≡ ( w ′ (cid:16) ψ ( x ) ψ ( b ) (cid:17) ψ ′ ( x ) ψ ( b ) x ≥ b x < b or(35) ζ ′ ( x, b ) = ( w ′ (cid:16) ψ ( x + b ) ψ ( b ) (cid:17) ψ ′ ( x + b ) ψ ( b ) x ≥ x < . We see from (8), (21), and (23) that(36) ζ ′ ( x, b ) ≤ x .From (10), ζ ′ ( x, b ) = 0 when ψ ( x + b ) < ǫψ ( b ). We will define e ( ǫ, b ) by(37) ψ ( e ( ǫ, b ) + b ) ψ ( b ) ≡ ǫ so we get(38) e ( ǫ, b ) = ψ − ( ǫψ ( b )) − b and then(39) ζ ′ ( x, b ) = 0 for x > e ( ǫ, b ) . Since w ′ ( v ) and ψ ′ ( x + b ) ψ ( b ) are finite, we have that ζ ′ ( x, b ) is finite, and from(35) and (39) it is seen that(40) Z ∞−∞ ζ ′ ( x, b ) dx = Z e ( ǫ,b )0 ζ ′ ( x, b ) dx = − . The integral in (40) is proper and absolutely convergent.
MICHAEL A. BLISCHKE
In a reciprocal fashion, if we have ζ ( x, b ) that can be shown to satisfy (33)through (36) and (39) for some w ′ ( v ), with ζ ′ ( x, b ) finite, then (7) through(11) can be satisfied.Substituting, (32) becomesΞ Z ∞ ψ − ( β ) f ( x ) dx = − F (cid:0) ψ − ( β ) (cid:1) + lim b →∞ (cid:26)Z ∞ b F ( x ) ζ ′ ( x − b, b ) dx (cid:27) = − F (cid:0) ψ − ( β ) (cid:1) + lim b →∞ (cid:26)Z ∞ F ( x + b ) ζ ′ ( x, b ) dx (cid:27) . (41)Because of the similarity of (41) and (2), we will refer to ζ ( x, b ) also asa termination function, even though it is not constant WRT b . When it isnot obvious from context which termination function we are referring to, wewill call the ζ ( x, b ) a termination function of the second type, and will calltermination functions as described in [Bli12] a termination function of thefirst type.Using the relations between w ( v ) and ζ ( x, b ), we can freely switch betweenthe finite limit integral (14) and its corresponding infinite limit integral(41). We’ll denote the relation between an initialization function w ( v ) usedin the finite limit integral and the termination function ζ ( x, b ) used in thecorresponding infinite limit integral and given by (34) or (35), as(42) ζ ( x ) ⇔ w ( v ) . We next show that combining initialization functions and combining ter-mination functions of the second type are equivalent. Beginning with (16)and using(43) v = ψ ( x + b ) ψ ( b )(44) v ′ = ψ ( x ′ + b ) ψ ( b )we get(45) dv ′ = ψ ′ ( x ′ + b ) ψ ( b ) dx ′ and w ′ (cid:18) ψ ( x + b ) ψ ( b ) (cid:19) = − Z x w ′ (cid:16) ψ ( x + b ) ψ ( x ′ + b ) (cid:17) w ′ (cid:16) ψ ( x ′ + b ) ψ ( b ) (cid:17) ( ψ ( x ′ + b ) /ψ ( b )) ψ ′ ( x ′ + b ) ψ ( b ) dx ′ = − Z x h w ′ (cid:16) ψ ( x + b ) ψ ( x ′ + b ) (cid:17) ψ ′ ( x ′ + b ) ψ ( x ′ + b ) i h w ′ (cid:16) ψ ( x ′ + b ) ψ ( b ) (cid:17) ψ ′ ( x ′ + b ) ψ ( b ) i ( ψ ′ ( x + b ) /ψ ( b )) dx ′ = − Z x + bb h w ′ (cid:16) ψ ( x + b ) ψ ( x ′ ) (cid:17) ψ ′ ( x ′ ) ψ ( x ′ ) i h w ′ (cid:16) ψ ( x ′ ) ψ ( b ) (cid:17) ψ ′ ( x ′ ) ψ ( b ) i ( ψ ′ ( x + b ) /ψ ( b )) dx ′ . (46) ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 9
Using (34) and (35) w ′ (cid:18) ψ ( x + b ) ψ ( b ) (cid:19) = − ψ ( b ) ψ ′ ( x + b ) Z x + bb ζ ′ ( x + b − x ′ , b ) ζ ′ ( x ′ − b, b ) dx ′ = − ψ ( b ) ψ ′ ( x + b ) Z x ζ ′ ( x − x ′ , b ) ζ ′ ( x ′ , b ) dx ′ (47)and so ζ ′ ( x, b ) = − Z x ζ ′ ( x − x ′ , b ) ζ ′ ( x ′ , b ) dx ′ (48)Thus we have that the combination of two termination functions of thesecond type is also a termination function of the second type. Note thateach correspondance between termination function and initialization func-tion uses the same change of variable, u = ψ ( x ).We write this more compactly as(49) ζ ′ ( x, b ) = ζ ′ ( x, b ) ⊗ ζ ′ ( x, b )and denote the combined termination function using the notation(50) ζ ( x, b ) = ζ ( x, b ) ⊙ ζ ( x, b ) . Thus, given(51) ζ ( x, b ) ⇔ w ( v ) and ζ ( x, b ) ⇔ w ( v )we find that(52) ζ ( x, b ) ⊙ ζ ( x, b ) ⇔ w ( v ) ⊙ w ( v ) . That is, given two initialization functions, the combination of their cor-responding termination functions is equal to the termination function cor-responding to their combination.
4. Equivalence of generalized definitions
We are now ready to show equivalence between our finite limit and infinitelimit definitions. Equation (41) is almost the same form as (2) and Eq. (9)of [Bli12], the only difference being that ζ ( x, b ) is a function of b , unlike thetermination functions described in [Bli12]. For a particular choice of ψ ( x ),however, the dependence of ζ ( x, b ) on b vanishes. In that case, the infinitelimit integral corresponding to our finite limit improper integral, (41), isidentical to the infinite limit integral presented in [Bli12].Choosing, with α > ψ ( x ) = e − αx we get(54) ψ ′ ( x ) = − αe − αx . Substituting these into (35) we can write(55) ζ ′ ( x, b ) = − αw ′ (cid:0) e − αx (cid:1) e − αx ≡ z ′ ( x ) . MICHAEL A. BLISCHKE
Since this is not a function of b , it satisfies the requirements for a terminationfunction. Using this transformation, we can see that for every finite limitimproper integral using the general definition (6), there is a correspondinginfinite-limit improper integral using the definition from [Bli12]. We alsohave that for any termination function z ( x ), the initialization function canbe explicitly found as(56) w ′ (cid:0) e − αx (cid:1) = − z ′ ( x ) e αx α (57) w ′ ( u ) = − z ′ (cid:16) − ln ( u ) α (cid:17) αu . We thus have, when the change of variable (20) is given by (53), that(58) Ξ Z ∞ − ln ( β ) α f ( x ) dx = Z Z ∞ − ln ( β ) α f ( x ) dx and the definition for the infinite limit case is seen to be equivalent to thedefinition for the finite limit case, so(59) Z Z β g ( u ) du = Z Z ∞ − ln ( β ) α f ( x ) dx We find that c and ǫ are related as(60) ǫ = e − αc . The equivalence between the finite limit and infinite limit generalizeddefinitions given by (53) and the correspondence between initialization andtermination functions given by (54) means that the properties found for theinfinite limit case in [Bli12] all have corresponding properties in the finitelimit case. We will list the properties here.If w ( ν ) is an initialization function for g ( u ), then for any other initial-ization function w ( ν ), w ( ν ) given by (15) is also an initialization functionfor g ( u ), and gives the same value for the integral.The integral defined using (6) produces a unique value for all initializationfunctions for which the limit exists.When the integral exists using the conventional definition, our generaldefinition gives the same answer,(61) Z Z β g ( u ) du = Z β g ( u ) du. Our general definition satisfies linearity,(62) a Z Z β g ( u ) du + b Z Z β h ( u ) du = Z Z β [ ag ( u ) + bh ( u )] du when the integrals on the LHS both exist. ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 11
Differentiation under the integral sign can be performed when both of theintegrals exist, and we have(63) ddy (cid:20) Z Z β g ( u, y ) du (cid:21) = (cid:20) Z Z β ∂∂y g ( u, y ) du (cid:21) . When one or the other integrals exists with the conventional definition, wealso get(64) ddy (cid:20)Z β g ( u, y ) du (cid:21) = (cid:20) Z Z β ∂∂y g ( u, y ) du (cid:21) and(65) ddy (cid:20) Z Z β g ( u, y ) du (cid:21) = (cid:20)Z β ∂∂y g ( u, y ) du (cid:21) . Interchange of the order of iterated integrations is also allowed. Here weassume that(66) Z Z β g ( u, y ) dx exists over the domain γ ≤ y ≤ δ for some initialization function w ( v, y ),and that(67) h ( u, t ) ≡ Z t g ( u, y ) s ( y ) dy exists, using the Riemann definition, over the domain γ ≤ t ≤ δ , for 0
When integrals in (66) and (68) exist, and the Riemann inte-gral in (67) exists, (69) Z δγ s ( y ) (cid:20) Z Z β g ( u, y ) du (cid:21) dy = Z Z β (cid:20)Z δγ s ( y ) g ( u, y ) dy (cid:21) du. Proof.
From [Bli12] and (58), we have that(70) Z δγ s ( y ) " Ξ Z ∞ − ln ( β ) α f ( x, y ) dx dy = Ξ Z ∞ − ln ( β ) α (cid:20)Z δγ s ( y ) f ( x, y ) dy (cid:21) dx. MICHAEL A. BLISCHKE
Using (67) and (58), we getΞ Z ∞ − ln ( β ) α (cid:20)Z δγ s ( y ) f ( x, y ) dy (cid:21) dx = Ξ Z ∞ − ln ( β ) α (cid:20)Z δγ − s ( y ) g ( ψ ( x ) , y ) ψ ′ ( x ) dy (cid:21) dx = Ξ Z ∞ − ln ( β ) α − h ( ψ ( x ) , y ) ψ ′ ( x ) (cid:12)(cid:12) δγ dx = Z Z β h ( u, y ) (cid:12)(cid:12)(cid:12) δγ du = Z Z β (cid:20)Z δγ s ( y ) g ( u, y ) dy (cid:21) du. (71)Using (58) and (59),(72) Z δγ s ( y ) " Ξ Z ∞ − ln ( β ) α f ( x, y ) dx dy = Z δγ s ( y ) (cid:20) Z Z β g ( u, y ) du (cid:21) dy. (cid:3) When one side exists using the conventional definition, we also get either(73) Z δγ s ( y ) (cid:20) Z Z β g ( u, y ) du (cid:21) dy = Z β (cid:20)Z δγ s ( y ) g ( u, y ) dy (cid:21) du or(74) Z δγ s ( y ) (cid:20)Z β g ( u, y ) du (cid:21) dy = Z Z β (cid:20)Z δγ s ( y ) g ( u, y ) dy (cid:21) du. A change of variable of integration of the form u ′ = cu for nonzero con-stant c is valid. An arbitrary change of the variable of integration is notnecessarily valid.
5. Examples
The following two examples show the evaluation of integrals which do notexist using the conventional definition.
Example 1. g ( u ) = sin(1 /u ) u G ( u ) = Z u sin(1 /s ) s ds = cos(1 /u ) Using (14) we have Z Z /a sin(1 /u ) u du = G (1 /a ) − lim δ → + (cid:26) δ Z δ G ( u ) w ′ ( u/δ ) du (cid:27) . We can use w ( v ) = (cid:26) (2 v −
1) 1 / ≤ v ≤ v < / ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 13 so w ′ ( v ) = (cid:26) / ≤ v ≤ elsegiving us Z Z /a sin(1 /u ) u du = cos( a ) − lim δ → + ( δ Z δδ/ /u ) du ) . Evaluating the integral on the RHS gives Z δδ/ cos(1 /u ) du = − δ ( u cos(1 /u ) + Si (1 /u )) (cid:12)(cid:12)(cid:12)(cid:12) δδ/ where Si is the sine integral. Expanding this to the necessary order of argu-ment in the large argument ( u → + ) limit,Si (1 /u ) ≈ π − u cos(1 /u ) − u sin(1 /u ) . The cosine terms cancel, and the π/ term doesn’t contribute when the limitsare taken. Thus, the integral is of order δ , and we get for the limit lim δ → + { O ( δ ) } = 0 so we obtain that Z Z /a sin(1 /u ) u du = cos( a ) . Example 2. g ( u ) = cos(1 /u ) u G ( u ) = Z u cos(1 /s ) s ds = − cos(1 /u ) − sin(1 /u ) u Using (14) we have Z Z /a cos(1 /u ) u du = G (1 /a ) − lim δ → + (cid:26) δ Z δ G ( u ) w ′ ( u/δ ) du (cid:27) . We will use w ( v ) = (cid:26) v − − v − / ≤ v ≤ v < / so w ′ ( v ) = (cid:26) v − − v ) 1 / ≤ v ≤ elsegiving Z Z /a cos(1 /u ) u du = − cos( a ) − a sin( a ) +lim δ → + ( δ Z δδ/ (cid:18) cos(1 /u ) + sin(1 /u ) u (cid:19) (2 u/δ − − u/δ ) du ) . MICHAEL A. BLISCHKE
Evaluating the integral on the RHS gives Z δδ/ (cid:18) cos(1 /u ) + sin(1 /u ) u (cid:19) (2 u/δ − − u/δ ) du = − δ h u (cid:0) δ − uδ + 4 u + 4 (cid:1) cos(1 /u ) + u u − δ ) sin(1 /u ) + 3 δ Ci (1 /u ) + 23 Si (1 /u ) (cid:21) δδ/ where Si and Ci are the sine integral and cosine integral. Expanding theseto the necessary order in the argument in the large argument ( u → + ) limit,Ci (1 /u ) ≈ u sin(1 /u ) − u cos(1 /u ) − u sin(1 /u ) Si (1 /u ) ≈ π − u cos(1 /u ) − u sin(1 /u ) + 2 u cos(1 /u ) . With a little algebra there is much cancellation of terms, and the integralcan be shown to be at least of order δ . We thus get for the limit lim δ → + { O ( δ ) } = 0 so we get Z Z /a cos(1 /u ) u du = − cos( a ) − a sin( a ) .
6. Conclusion
A generalized definition for an improper integral with finite bounds hasbeen presented. The definition presented here is a more powerful alternativeto the conventional definition. The range of functions which are integrableunder this definition is expanded as compared with the conventional defini-tion.The new definition presented here gives the same result as the conven-tional definition when that applies, and preserves uniqueness and linearity.The generalized definition allows interchange of the order of differentiationand integration whenever the two integrals exist under the new definition.Also allowed is interchange of the order of integration of iterated integra-tion, again when the integrations exist under this definition. The abilityto rigorously interchange order of integrations, or order of integration anddifferentiation, in cases where integrals under the conventional definition donot converge, provides an added tool for manipulation of complicated inte-grals. An arbitrary change of the variable of integration is not necessarilyvalid, although scaling the variable of integration by a constant is.The generalized finite limit definition presented here has been shown tobe equivalent to the generalized infinite limit definition presented in [Bli12].For a particular change of variable transforming between finite limit andinfinite limit integrals, the existence of either the finite or the infinite limitintegral implies existence of the other, with the same value.
ENERALIZED IMPROPER INTEGRAL DEFINITION FOR FINITE LIMIT 15
References [Bli12] Michael Blischke. Generalized improper integral definition for infinite limit. arXiv:0805.3559 [math.CA] , 2012., 2012.