Some dual definite integrals for Bessel functions
aa r X i v : . [ m a t h . C A ] S e p Journal JOURNAL * (201*), ***, ?? pages Some dual definite integrals for Bessel functions
Howard S. COHL, ∗ Sean J. NAIR, † and Rebekah M. PALMER ‡∗ Applied and Computational Mathematics Division, National Institute of Standards and Tech-nology, Gaithersburg, MD, 20899-8910, USA
E-mail: [email protected]
URL: † Mathematics, Science, and Computer Science Magnet Program, Montgomery Blair High School,Silver Spring, MD, 20901, USA
E-mail: [email protected] ‡ Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
E-mail: [email protected]
Received XX July 2012 in final form ????; Published online ???? doi:10.3842/JOURNAL.201*.***
Abstract.
Based on known definite integrals of Bessel functions of the first kind, we obtainexact solutions to unknown definite integrals using the method of integral transforms fromHankel’s transform.
Key words:
Definite integrals; Bessel functions; Associated Legendre functions; Hypergeo-metric functions; Struve functions; Chebyshev polynomials; Jacobi polynomials
In Cohl (2012) [1], orthogonality and Hankel’s transform are used to generate solutions tonew definite integrals based on known integrals. In this paper, we use the method of integraltransforms to create new integrals from a variety of known integrals containing Bessel functionsof the first kind J ν . In this method, we use the closure relation for Bessel functions of thefirst kind to generate a guess for a function to use in Hankel’s transform. This guess may beincorrect if it does not satisfy the first condition for Hankel’s transform, in which case a newdefinite integral is not generated. If the guess satisfies the condition, restrictions on ν must thenbe adjusted to satisfy the other condition of Hankel’s transform. For the functions used in thispaper, superscripts and subscripts refer to lists of parameters. Any exceptions to this will beclear from context.As far as we are aware, the 40 definite integrals over Bessel functions of the first kind thatwe present in this manuscript, do not currently appear in the literature. An extension of thesurvey presented in this manuscript can be used to mechanically compute new definite integralsfrom pre-existing definite integrals over Bessel functions of the first kind. We use the following result where for x ∈ (0 , ∞ ) we define F ( r ±
0) := lim x → r ± F ( x );see Watson (1944) [7, p. 456]: H. S. Cohl, S. J. Nair, & R. Palmer Theorem 1.
Let F : (0 , ∞ ) → C be such that Z ∞ √ x | F ( x ) | dx < ∞ , (1) and let ν ≥ − . Then
12 ( F ( r + 0) + F ( r − Z ∞ uJ ν ( ur ) Z ∞ xF ( x ) J ν ( ux ) dx du (2) provided that the positive number r lies inside an interval in which F ( x ) has finite variation. The effort described in this paper was motivated by the large collection of Bessel functiondefinite integrals which exist in the book “Table of Integrals, Series, and Products” [4]. Notcounting Theorem 2 (which stands alone), the method of integral transforms was applied to theBessel function definite integrals appearing in Sections 6.51 and 6.52 of [4]. This method can beapplied to many definite integrals appearing in the remainder of sections appearing in Sections6.5-6.7 of [4].For the definite integrals presented in this manuscript, we have directly verified that (1) issatisfied. This is easily accomplished by analyzing the behavior of the integrands in a smallneighborhood of the endpoints { , ∞} . For this paper, this technique produced 30 theoremsincluding 40 definite integrals which are given below. The method of integral transforms doesnot always succeed in producing new definite integrals because the conditions on the Hankeltransform (1) is not satisfied. Some cases of this are shown in Section 7. Theorem 2.
Let b , c > , ν > − , t ∈ C \ ( −∞ , . Then Z ∞ (∆( a, b, c )) ν − a − ν J ν ( at ) da = 2 − ν √ π Γ (cid:0) ν + (cid:1) (cid:0) bct (cid:1) ν J ν ( bt ) J ν ( ct ) . (3)∆ : [0 , ∞ ) → [0 , ∞ ) (Heron’s formula [5]), defined by ∆( a, b, c ) := p s ( s − a )( s − b )( s − c ) , where s = ( a + b + c ) / , is the area of a triangle with sides of length a, b, and c. Proof.
We apply Theorem 1 to the function F b,cν : (0 , ∞ ) → C defined by F b,cν ( t ) := 2 − ν √ π Γ (cid:0) ν + (cid:1) (cid:0) bct (cid:1) ν J ν ( bt ) J ν ( ct ) , where Γ : C \− N → C is Euler’s gamma function is defined in [3, (5.2.1)], and J ν : C \ ( −∞ , → C , (order) ν ∈ C , is the Bessel function of the first kind defined in [3, (10.2.2)]. The desiredresult is obtained from Sonine’s formula [6] Z ∞ J ν ( at ) J ν ( bt ) J ν ( ct ) t − ν dt = 2 ν − (∆( a, b, c )) ν − √ π Γ (cid:0) ν + (cid:1) ( abc ) ν , where Re a > b , c >
0, Re ν > − . (cid:4) ome dual definite integrals for Bessel functions 3 Theorem 3.
Let ν > , µ > , α , β > , z ∈ C \ ( −∞ , . Then Z α b ν J ν − ( bz ) db = α ν z − J ν ( αz ) , (4) Z ∞ β a − µ J µ ( az ) da = β − µ z − J µ − ( βz ) . (5) Proof.
By applying Theorem 1 to the functions F αν : (0 , ∞ ) → C and G βµ : (0 , ∞ ) → C definedby F αν ( x ) := α ν x − J ν ( αx ), G βµ ( x ) := β − µ x − J µ − ( βx ), we obtain the desired results from theknown integral [4, (6.512.3)] Z ∞ J ν ( αx ) J ν − ( βx ) dx = α − ν β ν − if β < α, (2 β ) − if β = α, β > α, where Re ν > (cid:4) Theorem 4.
Let ν ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ c ν +1 c J ν ( cz ) dc = K ν ( z ) . (6) Proof.
We are given the integral [4, (6.521.2)] Z ∞ xK ν ( ax ) J ν ( bx ) dx = b ν a ν ( b + a ) , where Re a > b >
0, Re ν > −
1. By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := a ν K ν ( ax ), we obtain the following integral Z ∞ b ν +1 b + a J ν ( bx ) db = a ν K ν ( ax ) , where Re a > ν ≥ − , x ∈ C \ ( −∞ , z = ax and c = b/a , weobtain the desired result. (cid:4) Note that when the method of integral transforms is applied to F a ( x ) := aK ( ax ) given theintegral [4, (6.521.7)] Z ∞ xK ( ax ) J ( bx ) = ba ( a + b ) , where a > b >
0, we obtain the integral generated from [4, (6.521.2)] when ν = 1. Theorem 5.
Let z ∈ C \ ( −∞ , . Then Z ∞ c (1 + c ) J ( cz ) dc = z K ( z ) . (7) H. S. Cohl, S. J. Nair, & R. Palmer Proof.
We are given the integral [4, (6.521.12)] Z ∞ x K ( ax ) J ( bx ) = 2 a ( a + b ) , where a > b >
0. By applying Theorem 1 to the function F a : (0 , ∞ ) → C defined by F a ( x ) := x a K ( ax ), we obtain the following integral Z ∞ bJ ( bx )( a + b ) db = x a K ( ax ) , where a > x ∈ C \ ( −∞ , z = ax and c = b/a , we obtain the desiredresult. (cid:4) Theorem 6.
Let z ∈ C \ ( −∞ , . Then Z ∞ c (1 + c ) J ( cz ) dc = z K ( z ) . (8) Proof.
We are given the integral [4, (6.521.12)] Z ∞ x K ( ax ) J ( bx ) dx = 2 b ( a + b ) , where a , b >
0. By applying Theorem 1 to the function F a : (0 , ∞ ) → C defined by F a ( x ) := x K ( ax ), we obtain the following integral Z ∞ b J ( bx )( a + b ) db = x K ( ax ) , where a > x ∈ C \ ( −∞ , z = ax and c = b/a , we obtain the desiredresult. (cid:4) Theorem 7.
Let γ > , ν ≥ − , Re α > | Im β | , z ∈ C \ ( −∞ , . Then Z ∞ bJ ν ( bz ) l ν l ν ( l − l ) db = K ( αz ) J ν ( γz ) , (9) Z ∞ cJ ν ( cz ) l ν l ν ( l − l ) dc = K ( αz ) J ν ( βz ) , (10) where l and l are defined as l = 12 hp ( b + c ) + a − p ( b − c ) + a i , (11) l = 12 hp ( b + c ) + a + p ( b − c ) + a i . (12) Proof.
By applying Theorem 1 to the function F a,cν : (0 , ∞ ) → C and G a,bν : (0 , ∞ ) → C defined by F a,cν ( x ) := K ( ax ) J ν ( cx ), G a,bν ( x ) := K ( ax ) J ν ( bx ), we obtain the desired result fromthe known integral [4, (6.522.12)] Z ∞ xK ( ax ) J ν ( bx ) J ν ( cx ) dx = l ν l ν ( l − l ) , where c >
0, Re ν > −
1, Re a > | Im b | . (cid:4) ome dual definite integrals for Bessel functions 5 Theorem 8.
Let Re b > Re a , z ∈ C \ ( −∞ , . Then Z ∞ cJ ( cz )( a + b + c − a b + 2 a c + 2 b c ) − / dc = I ( az ) K ( bz ) . (13) Let Re b > Re c , z ∈ C \ ( −∞ , . Then Z ∞ aJ ( az ) l − l da = I ( cz ) K ( bz ) . (14) where l and l are defined in (11) and (12) respectively. Proof.
By applying Theorem 1 to the function F ba : (0 , ∞ ) → C and G bc : (0 , ∞ ) → C definedby F ba ( x ) := I ( ax ) K ( bx ), G bc ( x ) := I ( cx ) K ( bx ), we obtain the desired results from the knownintegrals (see [4, (6.522.4)]) Z ∞ xI ( ax ) K ( bx ) J ( cx ) dx = ( a + b + c − a b + 2 a c + 2 b c ) − / , where Re b > Re a , c >
0, and Z ∞ xI ( cx ) K ( bx ) J ( ax ) dx = 1 l − l , where Re b > Re c , a >
0, respectively. (cid:4)
Theorem 9.
Let γ > , ν ≥ − , Re α > | Im β | , z ∈ C \ ( −∞ , . Then Z ∞ b ν +1 ( l − l ) ν +1 J ν ( bz ) db = z ν ( αγ ) − ν √ π ν Γ( ν + ) K ν ( αz ) J ν ( γz ) , (15) Z ∞ c ν +1 ( l − l ) ν +1 J ν ( cz ) dc = z ν ( αβ ) − ν √ π ν Γ( ν + ) K ν ( αz ) J ν ( βz ) , (16) where l and l are defined in (11) and (12) , respectively. Proof.
By applying Theorem 1 to the functions F a,cν : (0 , ∞ ) → C , G a,bν : (0 , ∞ ) → C , definedby F a,cν ( x ) := x ν ( ac ) − ν √ π ν Γ( ν + ) K ν ( ax ) J ν ( cx ) ,G a,bν ( x ) := x ν ( ab ) − ν √ π ν Γ( ν + ) K ν ( ax ) J ν ( bx ) , we obtain the desired results from the known integral [4, (6.522.15)] Z ∞ x ν +1 J ν ( bx ) K ν ( ax ) J ν ( cx ) dx = 2 ν ( abc ) ν Γ( ν + ) √ π ( l − l ) ν +1 , where Re a > | Im b | , c > (cid:4) H. S. Cohl, S. J. Nair, & R. Palmer
Theorem 10.
Let γ > , Re β ≥ | Im α | , Re α > , Re p > | Re q | , Re q > , z ∈ C \ ( −∞ , .Then Z ∞ a J ( az )( a + β − γ ) (cid:2) ( a + β + γ ) − a γ (cid:3) − / da = zK ( βz ) J ( γz ) , (17) Z ∞ cJ ( cz )( α + β − c ) (cid:2) ( α + β + c ) − α c (cid:3) − / dc = z α J ( αz ) K ( βz ) . (18) Z ∞ J ( bz ) 2 b ( p + b − γ )( l − l ) db = zK ( pz ) J ( γz ) , (19) Z ∞ J ( cz ) c ( p + q − c )( l − l ) dc = z q J ( qz ) K ( pz ) , (20) where l and l are defined in (11) and (12) , respectively. Proof.
By applying Theorem 1 to the functions F cb : (0 , ∞ ) → C , G ba : (0 , ∞ ) → C , H ca :(0 , ∞ ) → C , I ab : (0 , ∞ ) → C defined by F cb ( x ) := xK ( bx ) J ( cx ), G ba ( x ) := x a J ( ax ) K ( bx ), H ca ( x ) := xK ( ax ) J ( cx ), I ab ( x ) := x b J ( bx ) K ( ax ), we obtain the desired results from theknown integrals (see [4, (6.525.1)]) Z ∞ x J ( ax ) K ( bx ) J ( cx ) dx = 2 a ( a + b − c ) (cid:2) ( a + b + c ) − a c (cid:3) − / , where c >
0, Re b ≥ | Re a | , Re a > Z ∞ x J ( bx ) K ( ax ) J ( cx ) dx = 2 b ( a + b − c )( l − l ) , where c >
0, Re a > | Im b | , Re b > (cid:4) Theorem 11.
Let Re a > , ν ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( bz ) √ b + 4 a db = I ν/ ( az ) K ν/ ( az ) . (21) Proof.
By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := I ν/ ( ax ) K ν/ ( ax ) , we obtain the desired result from the known integral [4, (6.522.9)] Z ∞ xI ν/ ( ax ) K ν/ ( ax ) J ν ( bx ) dx = b − ( b + 4 a ) − / , where b >
0, Re a >
0, Re ν > − (cid:4) Theorem 12.
Let a > , ν ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ a J ν ( bz ) √ b − a db = − π J ν/ ( az ) Y ν/ ( az ) . (22)ome dual definite integrals for Bessel functions 7 Proof.
By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := − π J ν/ ( ax ) Y ν/ ( ax ) , we obtain the desired result from the known integral [4, (6.522.10)] Z ∞ xJ ν/ ( ax ) Y ν/ ( ax ) J ν ( bx ) dx = ( < b < a, − π − b − ( b − a ) − / if 2 a < b, where Re ν > − (cid:4) Theorem 13.
Let Re a > , ν ≥ − , Re µ ≥ , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( bz ) √ b + 4 a h b + ( b + 4 a ) / i µ db = 2 µ a µ I ( ν − µ ) / ( az ) K ( ν + µ ) / ( az ) . (23) Proof.
By applying Theorem 1 to the function F µ,aν : (0 , ∞ ) → C defined by F µ,aν ( x ) := 2 µ a µ I ( ν − µ ) / ( ax ) K ( ν + µ ) / ( ax ) , we obtain the desired result from the known integral [4, (6.522.12)] Z ∞ xI ( ν − µ ) / ( ax ) K ( ν + µ ) / ( ax ) J ν ( bx ) dx = 2 − µ a − µ b − ( b + 4 a ) − / h b + ( b + 4 a ) / i µ , where Re a > b >
0, Re ν > −
1, Re ( ν − µ ) > − (cid:4) Theorem 14.
Let | Re a | < Re b , x ∈ C \ ( −∞ , . Then Z ∞ cJ ( cx )( b + c − a ) (cid:2) ( a + b + c ) − a b (cid:3) − / dc = x b I ( ax ) K ( bx ) . (24) Proof.
By applying Theorem 1 to the function F ba : (0 , ∞ ) → C defined by F ba ( x ) := x b I ( ax ) K ( bx ) , we obtain the desired result from the known integral [4, (6.525.2)] Z ∞ x I ( ax ) K ( bx ) J ( cx ) dx = 2 b ( b + c − a ) (cid:2) ( a + b + c ) − a b (cid:3) − / , where Re b > | Re a | , c > (cid:4) Theorem 15.
Let ν > , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( cz ) J ν (cid:0) √ c (cid:1) dc = 1 z J ν (cid:18) z (cid:19) . (25) Proof.
We are given the integral [4, (6.514.1)] Z ∞ J ν (cid:16) ax (cid:17) J ν ( bx ) dx = b − J ν (cid:16) √ ab (cid:17) , where Re ν > a , b >
0. By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := x − J ν (cid:0) ax − (cid:1) , we obtain the following integral Z ∞ J ν ( bx ) J ν (cid:16) √ ab (cid:17) db = x − J ν (cid:16) ax (cid:17) , where ν > a > x ∈ C \ ( −∞ , x = az , c = ba , we obtainthe desired result. (cid:4) H. S. Cohl, S. J. Nair, & R. Palmer
Theorem 16.
Let − ≤ ν < , z ∈ C \ ( −∞ , . Then Z ∞ cJ ν ( cz ) h e i ( ν +1) π/ K ν (cid:16) e iπ/ √ c (cid:17) + e − i ( ν +1) π/ K ν (cid:16) e − iπ/ √ c (cid:17)i dc = 1 z K ν (cid:18) z (cid:19) . (26) Proof.
We are given the integral [4, (6.514.3)] Z ∞ J ν (cid:16) ax (cid:17) K ν ( bx ) dx = b − e i ( ν +1) π/ K ν h e iπ/ √ ab i + b − e − i ( ν +1) π/ K ν h e − iπ/ √ ab i , where a >
0, Re b > | Re ν | < . By applying Theorem 1 to the function F bν : (0 , ∞ ) → C defined by F bν ( x ) := bx − K ν ( bx − ) , we obtain the following integral Z ∞ aJ ν ( ax ) h e i ( ν +1) π/ K ν (cid:16) e iπ/ √ ab (cid:17) + e − i ( ν +1) π/ K ν (cid:16) e − iπ/ √ ab (cid:17)i da = bx K ν (cid:18) bx (cid:19) , where Re b > − ≤ ν < , x ∈ C \ ( −∞ , x = bz , c = ba , we obtainthe desired result. (cid:4) Theorem 17.
Let | ν | < , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( cz ) h K ν (cid:0) √ c (cid:1) − π Y ν (cid:0) √ c (cid:1)i dc = − π z Y ν (cid:18) z (cid:19) . (27) Proof.
We are given the integral [4, (6.514.4)] Z ∞ Y ν (cid:16) ax (cid:17) J ν ( bx ) dx = − b − π h K ν (cid:16) √ ab (cid:17) − π Y ν (cid:16) √ ab (cid:17)i , where a , b > | Re ν | < . By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := − π x Y ν (cid:16) ax (cid:17) , we obtain the following integral Z ∞ J ν ( bx ) h K ν (cid:16) √ ab (cid:17) − π Y ν (cid:16) √ ab (cid:17)i db = − π x Y ν (cid:16) ax (cid:17) , where a > | ν | < , x ∈ C \ ( −∞ , x = az , c = ab , we obtain thedesired result. (cid:4) Theorem 18.
Let ν ≥ − , µ > − , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( cz ) J ν (cid:18) c (cid:19) c dc = 2 J ν (cid:0) z (cid:1) , (28) Z ∞ J µ ( cz ) J µ (cid:18) c (cid:19) dc = z − J µ (cid:0) √ z (cid:1) . (29)ome dual definite integrals for Bessel functions 9 Proof.
We are given the integral [4, (6.516.1)] Z ∞ J ν (cid:0) a √ x (cid:1) J ν ( bx ) dx = b − J ν (cid:18) a b (cid:19) , where Re ν > − , a , b >
0. By applying Theorem 1 to the functions F bν : (0 , ∞ ) → C and G aµ : (0 , ∞ ) → C defined by F bν ( x ) := 2 bJ ν (cid:0) bx (cid:1) , G aµ ( x ) := x − J µ ( a √ x ), we obtain thefollowing integrals Z ∞ aJ ν ( ax ) J ν (cid:18) a β (cid:19) da = 2 βJ ν (cid:0) βx (cid:1) , Z ∞ J µ ( bx ) J µ (cid:18) α b (cid:19) db = x − J µ ( α √ x ) , where α , β > ν ≥ − , µ > − , z ∈ C \ ( −∞ , z = bx , c = a/ √ b ,and √ z = a √ x , c = b/a respectively, we obtain the desired results. (cid:4) Theorem 19.
Let ν > − , µ ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ J ν/ ( cz ) J ν/ (cid:18) c (cid:19) dc = z − J ν (cid:0) √ z (cid:1) , (30) Z ∞ cJ µ ( cz ) J µ/ (cid:18) c (cid:19) dc = 2 J µ/ ( z ) . (31) Proof.
We are given the integral [4, (6.526.1)] Z ∞ xJ ν/ ( ax ) J ν ( bx ) dx = 12 a J ν/ (cid:18) b a (cid:19) , where a, b >
0, Re ν > −
1. By applying Theorem 1 to the function F bν : (0 , ∞ ) → C defined by F bν ( x ) := x − J ν ( b √ x ), we obtain the following integrals Z ∞ J ν/ ( ax ) J ν/ (cid:18) β a (cid:19) da = x − J ν (cid:0) β √ x (cid:1) , Z ∞ bJ µ ( bx ) J µ/ (cid:18) b α (cid:19) db = 2 αJ µ/ ( αx ) , where α , β > ν > − µ ≥ − , x ∈ C \ ( −∞ , √ z = β √ x , c = a/β ,and z = αx , x = z √ α , we obtain the desired results. (cid:4) Theorem 20.
Let ν ≥ − , x ∈ C \ ( −∞ , . Then Z ∞ a J ν ( ax ) J ν +1 / ( a ) da = x J ν − / (cid:18) x (cid:19) . (32) Proof.
By applying Theorem 1 to the function F ν : (0 , ∞ ) → C defined by F ν ( x ) := x J ν − / (cid:16) x (cid:17) ,we obtain the desired result from the known integral [4, (6.527.1)] Z ∞ J ν (2 ax ) J ν − / ( x ) dx = 12 a J ν +1 / ( a ) , where a >
0, Re ν > − . (cid:4) Theorem 21.
Let ν ≥ − , x ∈ C \ ( −∞ , . Then Z ∞ a J ν ( ax ) J ν − / ( a ) da = x J ν +1 / (cid:18) x (cid:19) . (33) Proof.
By applying Theorem 1 to the function F ν : (0 , ∞ ) → C defined by F ν ( x ) := x J ν +1 / (cid:16) x (cid:17) ,we obtain the desired result from the known integral [4, (6.527.1)] Z ∞ J ν (2 ax ) J ν +1 / ( x ) dx = 12 a J ν − / ( a ) , where a >
0, Re ν > − (cid:4) Theorem 22.
Let ν ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ cJ ν ( cz ) H ν/ (cid:18) c (cid:19) dc = − Y ν/ ( z ) . (34) Proof.
We are given the integral [4, (6.526.4)] Z ∞ xY ν/ ( ax ) J ν ( bx ) dx = − a H ν/ (cid:18) b a (cid:19) , where a >
0, Re b >
0, Re ν > − H ν : C → C , for ν ∈ N , is the Struve functiondefined in [3, (11.2.1)]. By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := − aY ν/ ( ax ), we obtain the following integral Z ∞ bJ ν ( bx ) H ν/ (cid:18) b a (cid:19) db = − aY ν/ ( ax ) , where a > ν ≥ − , x ∈ C \ ( −∞ , z = ax , c = b/ √ a , we obtainthe desired result. (cid:4) Theorem 23.
Let ν ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( cz ) e − /c c − dc = 2 J ν (2 √ z ) K ν (2 √ z ) . (35) Proof.
We are given the integral [4, (6.526.4)] Z ∞ xJ ν (2 √ ax ) K ν (2 √ ax ) J ν ( bx ) dx = 12 b − e − a/b , where Re a > b >
0, Re ν > −
1. By applying Theorem 1 to the function F aν : (0 , ∞ ) → C defined by F aν ( x ) := 2 J ν (2 √ ax ) K ν (2 √ ax ), we obtain the following integral Z ∞ b − J ν ( bx ) e − a/b db = 2 J ν (2 √ ax ) K ν (2 √ ax ) , where Re a > ν ≥ − , x ∈ C \ ( −∞ , z = ax , c = b/a , we obtainthe desired result. (cid:4) ome dual definite integrals for Bessel functions 11 Theorem 24.
Let a > , z ∈ C \ ( −∞ , . Then Z a J ( bz ) ln (cid:18) − b a (cid:19) db = − πz − Y ( az ) . (36) Proof.
We apply Theorem 1 to the function F a : (0 , ∞ ) → C defined by F a ( x ) := − πx − Y ( ax ),where Y ν : C \ ( −∞ , → C , (order) ν ∈ C , is the Bessel function of the second kind defined in[3, (10.2.3)]. We obtain the desired result from the known integral [4, (6.512.6)] Z ∞ J ( bx ) Y ( ax ) dx = − b − π ln (cid:18) − b a (cid:19) , where 0 < b < a . (cid:4) Theorem 25.
Let z ∈ C \ ( −∞ , . Then Z ∞ J ( cz ) ln(1 + c ) dc = 2 z − K ( z ) . (37) Proof.
We are given the integral [4, (6.512.9)] Z ∞ K ( ax ) J ( bx ) dx = 12 b ln (cid:18) b a (cid:19) , where a , b > K ν : C \ ( −∞ , → C , (order) ν ∈ C , is the modified Bessel function ofthe second kind defined in [3, (10.25.3)]. We apply Theorem 1 to the function F a : (0 , ∞ ) → C defined by F a ( x ) := 2 x − K ( ax ), and obtain the following integral Z ∞ J ( bx ) ln (cid:18) b a (cid:19) db = 2 x − K ( ax ) , where a > x ∈ C \ ( −∞ , z = ax and c = b/a , we obtain the desiredresult. (cid:4) Theorem 26.
Let a > , z ∈ C \ ( −∞ , . Then Z a sin − (cid:18) b a (cid:19) J ( bz ) db = π z [ J ( az ) − J (2 az )] . (38) Proof.
By applying Theorem 1 to the function F a : (0 , ∞ ) → C defined by F a ( x ) := π x J ( ax ) , we obtain the desired result from the known integral [4, (6.513.9)] Z ∞ J ( ax ) J ( bx ) dx = b − if 0 < a < b, πb sin − (cid:18) b a (cid:19) if 0 < b < a. (cid:4) Theorem 27.
Let n ∈ N , µ > , ν > , t ∈ C \ ( −∞ , . Then Z α J ν − n − ( bt ) F (cid:18) ν, − nν − n ; b α (cid:19) b ν − n db = n ! α ν − n Γ( ν − n ) J ν + n ( αt ) t Γ( ν ) , (39) Z ∞ β J µ + n ( at ) F (cid:18) µ, − nµ − n ; β a (cid:19) a − µ + n +1 da = n ! β − µ + n +1 Γ( µ − n ) J µ − n − ( βt ) t Γ( µ ) . (40) Proof.
By applying Theorem 1 to the functions G ν,αn : (0 , ∞ ) → C and H µ,βn : (0 , ∞ ) → C defined by G ν,αn ( t ) := n ! α ν − n Γ( ν − n ) J ν + n ( αt ) t Γ( ν ) ,H µ,βn ( t ) := n ! β − µ + n +1 Γ( µ − n ) J µ − n − ( βt ) t Γ( µ ) , we obtain the desired results from the known integral [4, (6.512.2)] Z ∞ J ν + n ( αt ) J ν − n − ( βt ) dt = β ν − n − Γ( ν ) α ν − n n ! Γ( ν − n ) 2 F (cid:18) ν, − nν − n ; β α (cid:19) if 0 < β < α, ( − n (2 α ) − if β = α, β > α, where Re ν > F : C × ( C \ − N ) × ( C \ [1 , ∞ )) → C is the hypergeometric functiondefined in [3, (15.2.1)]. (cid:4) Theorem 28.
Let ν ≥ − , ν > − µ − , z ∈ C \ ( −∞ , . Then Z ∞ P − µ − / ν/ r c ! Q − µ − / ν/ r c ! J ν ( cz ) dc = e − µπi Γ (cid:16) ν − µ +12 (cid:17) z Γ (cid:16) ν +2 µ +12 (cid:17) I µ ( z ) K µ ( z ) . (41) Proof.
We are given the integral [4, (6.513.3)] Z ∞ I µ ( ax ) K µ ( ax ) J ν ( bx ) dx = e µπi Γ (cid:16) ν +2 µ +12 (cid:17) b Γ (cid:16) ν − µ +12 (cid:17) P − µ − / ν/ r a b ! Q − µ − / ν/ r a b ! , where Re a > b >
0, Re ν > −
1, Re ν + 2 µ > − P µν : C \ ( −∞ , → C , for ν + µ / ∈ − N with degree ν and order µ , and Q µν : C \ ( −∞ , → C , for ν + µ / ∈ − N with degree ν and order µ , are the associated Legendre functions of the first [3, (14.3.6)] and second [3, (14.3.7)] kindrespectively. By applying Theorem 1 to the function F µ,aν : (0 , ∞ ) → C defined by F µ,aν ( x ) := e − µπi Γ (cid:16) ν − µ +12 (cid:17) x Γ (cid:16) ν +2 µ +12 (cid:17) I µ ( ax ) K µ ( ax ) , ome dual definite integrals for Bessel functions 13we obtain the following integral Z ∞ P − µ − / ν/ r a b ! Q − µ − / ν/ r a b ! J ν ( bx ) db = e − µπi Γ (cid:16) ν − µ +12 (cid:17) x Γ (cid:16) ν +2 µ +12 (cid:17) I µ ( ax ) K µ ( ax ) , where Re a > ν ≥ − , ν > − Re 2 µ − x ∈ C \ ( −∞ , z = ax , c = b/a , we obtain the desired result. (cid:4) Theorem 29.
Let ν > ∓ Re µ − , ν ≥ − , z ∈ C \ ( −∞ , . Then Z ∞ J ν ( cz ) " Q − µ − / ν/ r c ! dc = e − µπi Γ (cid:16) ν − µ (cid:17) z Γ (cid:16) ν +2 µ (cid:17) [ K µ ( z )] . (42) Proof.
We are given the integral [4, (6.513.5)] Z ∞ [ K µ ( ax )] J ν ( bx ) dx = e µπi Γ (cid:16) ν +2 µ (cid:17) b Γ (cid:16) ν − µ (cid:17) " Q − µ − / ν/ r a b ! , where Re a > b >
0, Re ( ν ± µ ) > − . By applying Theorem 1 to the function F µ,aν : (0 , ∞ ) → C defined by F µ,aν ( x ) := e − µπi Γ (cid:16) ν − µ (cid:17) x Γ (cid:16) ν +2 µ (cid:17) [ K µ ( ax )] , we obtain the following integral Z ∞ J ν ( bx ) " Q − µ − / ν/ r a b ! db = e − µπi Γ (cid:16) ν − µ (cid:17) x Γ (cid:16) ν +2 µ (cid:17) [ K µ ( ax )] , where Re a > ν > ∓ Re µ − ν ≥ − , x ∈ C \ ( −∞ , z = ax , c = b/a , we obtain the desired result. (cid:4) Theorem 30.
Let n ∈ N , ν > − n − , α , β > , z ∈ C \ ( −∞ , . Then Z ∞ β P ( ν, n (cid:18) − β a (cid:19) J ν +2 n +1 ( az ) a − ν da = z − β − ν J ν ( βz ) , (43) Z α P ( ν, n (cid:18) − b α (cid:19) J ν ( bz ) b ν +1 db = z − α ν +1 J ν +2 n +1 ( αz ) . (44) Proof.
By applying Theorem 1 to the functions F bν : (0 , ∞ ) → C and G a,nν : (0 , ∞ ) → C defined by F bν ( x ) := x − b − ν J ν ( bx ), G a,nν ( x ) := x − a ν +1 J ν +2 n +1 ( ax ), we obtain the desiredresults from the known integral [4, (6.512.4)] Z ∞ J ν +2 n +1 ( ax ) J ν ( bx ) dx = ( b ν a − ν − P ( ν, n (cid:0) − a − b (cid:1) if 0 < b < a, < a < b, where Re ν > − n − P ( α,β ) n : C → C is the Jacobi polynomial defined in [3, (18.3.1)]. (cid:4) Theorem 31.
Let a > , ν ≥ − , z ∈ C \ ( −∞ , . Then Z a J ν ( bz ) √ a − b T n (cid:18) b a (cid:19) db = π J ( ν + n ) / ( az ) J ( ν − n ) / ( az ) . (45) Proof.
By applying Theorem 1 to the function F n,aν : (0 , ∞ ) → C defined by F n,aν ( x ) = π J ( ν + n ) / ( ax ) J ( ν − n ) / ( ax ) , we obtain the desired result from the known integral [4, (6.522.11)] Z ∞ xJ ( ν + n ) / ( ax ) J ( ν − n ) / ( ax ) J ν ( bx ) dx = ( π − b − (4 a − b ) − / T n ( b a ) if 0 < b < a, a < b, where Re ν > − T n : C → C , for n ∈ N , is the Chebyshev polynomial of the first kindfound in [3, (18.3.1)]. (cid:4) In the following examples, the potential use of the method given by Theorem 1 fails becausethe condition (1) can not be satisfied. This has been verified by analyzing the well-understoodbehavior of the integrands in a small neighborhood of the endpoints { , ∞} . • The definite integral [4, (6.512.1)] with G µ,bν ( x ) := α ( ν, µ )Γ( ν +1) b − ν x − J ν ( bx ) , H µ,aν ( x ) := α ( ν, µ )Γ( ν + 1) a ν +1 x − J µ ( ax ) , where α ( ν, µ ) := Γ (cid:16) µ − ν +12 (cid:17) / Γ (cid:16) µ + ν +12 (cid:17) . • The definite integral [4, (6.514.1)] with G bν ( x ) := bx − J ν ( bx − ) . • The definite integral [4, (6.514.2)] with F bν ( x ) := bx − Y ν ( bx − ) . • The definite integrals [4, (6.516.2)], [4, (6.516.3)], [4, (6.516.4)], and [4, (6.516.7)] withrespectively F bν ( x ) := − bY ν ( bx ), F bν ( x ) := 4 bπ − K ν ( bx ), F aν ( x ) := x − Y ν ( a √ x ), and F aν ( x ) := π − x − sec( νπ ) K ν ( a √ x ) . • The definite integral [4, (6.522.2)] with F µ,aν ( x ) := e − µπi β ( ν, µ )[ K µ ( ax )] , where β ( ν, µ ) :=Γ( ν − µ ) / Γ(1 + ν + µ ) . • The definite integrals [4, (6.522.6)] and [4, (6.522.8)] with F a ( x ) := − π J ( ax ) Y ( ax ) , and G µ,aν ( x ) := e − µπi β ( ν, µ ) K µ − / ( ax ) K µ +1 / ( ax ) , respectively. • The definite integral [4, (6.522.16)] with F b,cν ( x ) := √ πx ν γ ( ν ) I ν ( cx ) K ν ( bx ), where γ ( ν ) :=(8 bc ) − ν / Γ (cid:0) ν + (cid:1) . • The definite integrals [4, (6.526.2)] and [4, (6.526.3)] with F bν ( x ) := 2 x − Y ν ( b √ x ) , G bν ( x ) :=cos( νπ ) K ν ( b √ x ) / (2 πx ) , respectively. • The definite integral [4, (6.526.6)] with F aν ( x ) := 4 aπ − K ν/ ( ax ) . • The definite integral [4, (6.527.3)] with F ν ( x ) := − xY ν +1 / (cid:0) x / (cid:1) / . ome dual definite integrals for Bessel functions 15 References [1] H. S. Cohl and H. Volkmer. Definite integrals using orthogonality and integral transforms.
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