Integrals of Frullani type and the method of brackets

Abstract The method of brackets is a collection of heuristic rules, some of which have being made rigorous, that provide a flexible, direct method for the evaluation of definite integrals. The present work uses this method to establish classical formulas due to Frullani which provide values of a specific family of integrals. Some generalizations are established.


Introduction
The integral and Frullani's theorem states that The identity (3) holds if, for example, f 0 is a continuous function and the integral in (3) exists. Other conditions for the validity of this formula are presented in [3,13,16]. The reader will find in [1] a systematic study of the Frullani integrals appearing in [12]. The goal of the present work is to use the method of brackets, a new procedure for the evaluation of definite integrals, to compute a variety of integrals similar to those in (1). The method itself is described in Section 2. This is based on a small number of heuristic rules, some of which have been rigorously established [2,8]. The point to be stressed here is that the application of the method of brackets is direct and it reduces the evaluation of a definite integral to the solution of a linear system of equations.

The method of brackets
A method to evaluate integrals over the half-line OE0; 1/, based on a small number of rules has been developed in [6,[9][10][11]. This method of brackets is described next. The heuristic rules are currently being placed on solid ground [2]. The reader will find in [5,7,8] a large collection of evaluations of definite integrals that illustrate the power and flexibility of this method.
For a 2 R, the symbol is the bracket associated to the (divergent) integral on the right. The symbol is called the indicator associated to the index n. The notation n 1 n 2 n r , or simply 12 r , denotes the product n 1 n 2 n r .

Rules for the production of bracket series
Rule P 1 . If the function f is given by the power series with˛;ˇ2 C, then the integral of f over OE0; 1/ is converted into a bracket series by the procedure n a n h˛n Cˇi: Rule P 2 . For˛2 C, the multinomial power .a 1 C a 2 C C a r /˛is assigned the r-dimension bracket series X n 1 X n 2 X n r n 1 n 2 n r a n 1 1 a n r r h ˛C n 1 C C n r i . ˛/ :

Rules for the evaluation of a bracket series
Rule E 1 . The one-dimensional bracket series is assigned the value X n n f .n/han C bi D 1 jaj f .n /. n /; where n is obtained from the vanishing of the bracket; that is, n solves an C b D 0. This is precisely the Ramanujan's Master Theorem. The next rule provides a value for multi-dimensional bracket series of index 0, that is, the number of sums is equal to the number of brackets.
Rule E 2 . Assume the matrix A D .a ij / is non-singular, then the assignment is X n 1 X n r n 1 n r f .n 1 ; ; n r /ha 11 n 1 C C a 1r n r C c 1 i ha r1 n 1 C C a rr n r C c r i D 1 jdet.A/j f .n 1 ; n r /. n 1 / . n r / where fn i g is the (unique) solution of the linear system obtained from the vanishing of the brackets.
Rule E 3 . The value of a multi-dimensional bracket series of positive index is obtained by computing all the contributions of maximal rank by Rule E 2 . These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded.

The formula in one dimension
The goal of this section is to establish Frullani's evaluation (3) by the method of brackets. The notation k D . 1/ k =.k C 1/ is used in the statement of the next theorem. Then, independently of˛.
Proof. Introduce an extra parameter and write Then, The method of brackets gives The result follows from the expansions ."=˛/ D˛=" In the examples given below, observe that C.0/ D f .0/ and that f .1/ D 0 is imposed as a condition on the integrand.
Note 3.3. The method of brackets gives a direct approach to Frullani style problems if the expansion (1) is replaced by the more general one withˇ¤ 0 and if the function f does not necessarily have a limit at infinity.

Example 3.4. Consider the evaluation of
for a; b > 0. The integral is evaluated directly as and since a; b > 0, both integrals are =2, giving I D 0. The classical version of Frullani theorem does not apply, since f .x/ does not have a limit as x ! 1. Ostrowski [15] shows that in the case f .x/ is periodic of period p, the value f .1/ might be replaced by In the present case, f .x/ D sin x has period 2 and mean 0. This yields the vanishing of the integral. The computation of (7) by the method of brackets begins with the expansion Here p F q Â a 1 ; : : : ; a p b 1 ; : : : with .a/ n D a.a C1/ .a Cn 1/, is the classical hypergeometric function. The integrand has the series expansion that yields The vanishing of the bracket gives n D 1=2 and the bracket series vanishes in view of the factor a 2nC1 b 2nC1 .
Example 3.5. The next example is the evaluation of for a; b > 0. The expansion Example 3.6. The integral is evaluated next. The expansion of the integrand is Therefore, and from here it follows that Thus, the integral is

A first generalization
This section describes examples of Frullani-type integrals that have an expansion of the form withˇ¤ 0.
Proof. The method of brackets gives S.a; bI "/ D Example 4.2. The integral appears as entry 4:536:2 in [12]. It is evaluated directly by the classical Frullani theorem. Its evaluation by the method of brackets comes from the expansion Therefore,˛D 2;ˇD 1 and The functions ff k .x/g are assumed to admit a series representation of the form

A second class of Frullani type integrals
where˛> 0 is independent of k and C k .0/ ¤ 0. The coefficients C k are assumed to admit a meromorphic extension from n 2 N to n 2 C.
Theorem 5.1. The integral I is given by where Proof. The proof begins with the expansion and the bracket series for the integral is The result follows by letting " ! 0.
Example 5.2. Entry 3:429 in [12] states that where > 0 and .x/ D 0 .x/= .x/ is the digamma function. This is one of many integral representation for this basic function. The reader will find a classical proof of this identity in [14]. The method of brackets gives a direct proof.
The functions appearing in this example are and where . / n D . The reader will find information about these integrals in [4,17]. Theorem 5.1 is now used to establish the value Here D 0 .1/ is Euler's constant. The first step is to compute series expansions of each of the terms in the integrand. The exponential term is easy: . ax 2 / n 1 n 1 Š D X n 1 n 1 a n 1 x 2n 1 ; (13) and this gives C 1 .n/ D a n : For the first elliptic integral, Therefore, where the term . 1/ n has been replaced by cos. n/. A similar calculation gives A direct calculation gives The result now comes from the values Example 5.4. Let a; b 2 R with a > 0. Then To apply Theorem 5.1 start with the series and In both expansions˛D 2 and the coefficients are given by C 1 .n/ D a n and C 2 .n/ D .n C 1/ .2n C 1/ b 2n : Then, C 0 The value (17) follows from here.
Example 5.5. The next example in this section involves the Bessel function of order 0 and Theorem 5.1 is used to evaluate This appears as entry 6:693:8 in [12]. The expansions show˛D 2 and C 1 .n/ D 1 .n C 1/ 2 2n and C 2 .n/ D .n C 1/ .2n C 1/ a 2n : Differentiation gives and Then, and the result now follows from Theorem 5.1. The reader is invited to use the representation to verify the identity Example 5.6. The final example in this section is The evaluation begins with the expansions Then, and e x 2 cos x D X k;n k;n p 4 k k C 1 2 x 2kC2n : Integration yields h2k C 2n C "i: The method of brackets now gives The term corresponding to k D 0 gives lim "!0 and the terms with k 1 as " ! 0 give Therefore, No further simplification seems to be possible.

A multi-dimensional extension
The method of brackets provides a direct proof of the following multi-dimensional extension of Frullani's theorem.
where the j are linear functions of the indices given by n D˛n 1`1 C C˛n n`n Cˇn: Then, 1 a n " n . ` 1 / . ` n /C.` 1 ; ;` n /; where A D ˛i j is the matrix of coefficients in (2) and` j ; 1 Ä j Ä n is the solution to the linear system ˛n 1`1 C C˛n n`n Cˇn n C " D 0: Proof. The proof is a direct extension of the one-dimensional case, so it is omitted.
with parameters 1 D 1 2 ; 2 D 1; b 1 D a 2 = ; b 2 D =a; a 1 D b 2 = ; a 2 D =b. The solution to the linear system is n 1 D 1 2 and n 2 D " 2 and j det Aj D 2. Then The double integral (4) has been evaluated. as claimed.

Conclusions
The method of brackets consists of a small number of heuristic rules that reduce the evaluation of a definite integral to the solution of a linear system of equations. The method has been used to establish a classical theorem of Frullani and to evaluate, in an algorithmic manner, a variety of integrals of Frullani type. The flexibility of the method yields a direct and simple solution to these evaluations.