Binet's second formula, Hermite's generalization, and two related identities

: Legendre was the ﬁ rst to evaluate two well - known integrals involving sines and exponentials. One of these integrals can be used to prove Binet ’ s second formula for the logarithm of the gamma function. Here, we show that the other integral leads to a speci ﬁ c case of Hermite ’ s generalization of Binet ’ s formula. From the analogs of Legendre ’ s integrals, with sines replaced by cosines, we obtain two integration iden - tities involving logarithms and trigonometric functions. Using these identities, we then subsequently derive generalizations of Binet ’ s and Hermite ’ s formulas involving the integral of a complex logarithm.


Summary of results
The following two results are due to Legendre [1]. The left-hand sides of these two equations vanish when a 0 = . In the limit that a 0 → , the right-hand sides also vanish (as can be shown by taking their Taylor expansions), and so these formulas are in fact valid for all a ∈ if the right-hand sides are extended by continuity. Equations (1) and (2) appear as Exercises 41 and 42 on page 71 of Watson's contour-integration book [2]. Watson attributes these results to Legendre, and moreover, he states that they hold for all real a. An alternative method to derive (1) and (2), based on infinite series and term-by-term integration, is illustrated in Examples 1 and 2 of Section 176 of Bromwich's infiniteseries book [3].
The analogous integrals with cosines appearing in the integrand instead of sines can be expressed (cf. Section 2.5.34 of [4]) in terms of the "digamma" function ψ (for which, see, for instance, Section 12.3 of [5]).
Here, however, we shall express them in the following forms: In Sections 177 and 180 of [3], equation (1) is elegantly used to derive Binet's second formula. Here, we shall show that, when the same technique is applied to equation (2), the result is Hermite's formula [7] for z logΓ 1 , and this is also true if z is real and positive. Thus, the principal value of h x z , ( ) is holomorphic in z (for real nonnegative x) and belongs to the same right-hand quadrant as z.
does not take values on the negative real axis, and so the principal value of the logarithm is also holomorphic on the right-hand half-plane.
We shall also show that equations (7) and (8) can be used to deduce the following results: The integrand in equation (13) extends continuously to x 0 = . In both (13) and (14), b π | | < ensures that the right-hand side has a positive denominator and the integral on the left-hand side converges.
The dominated convergence theorem [8] may thus be applied, and so the integral and the limit ε 0 → + may be interchanged. After taking the limit ε 0 → + on the right-hand side of equation ( The result above appears on page 189 of [1]. Rearranging this equation and simplifying it leads to equation (1). Applying the same procedure to equation ( Rearranging this equation and simplifying it leads to equation (2). In Section 44, page 186 of [1], Legendre also has two integral identities involving cosines. However, for each identity, the relative signs in the integrand are the same for the numerator and denominator. Thus, the identities obtained by Legendre involving cosine do not lead to the integrals expressed in (3) and (4). In Section 2, we present a contourintegral-based derivation of equations (1)-(4).
The closed contour is traversed in an anticlockwise direction and has vertices in the complex plane at ε R R ε ε , 0 , , 0 , , 1 , , 1 , 0, 1 where the contour is indented, for instance, by clockwise circular quadrants of radius ε at 0, 0 ( ) and 0, 1 ( ). By Cauchy's theorem, J 0 = . There are six contributions to the integral, and they are given as follows: The indentations about 0, 1 ( ) and 0, 0 ( ) have contributions, respectively, given by The second line is the result in equation (1). To write the result in the first line in the more convenient form

Second set of integrals
To prove equations (2) and (4), consider the contour integral The closed contour is traversed in an anticlockwise direction and has vertices at = . There are five contributions to the integral, and they are given as follows:  The clockwise indentation about 0, 1 2 ( ) is given by Next, we set both the real and imaginary parts of J equal to zero and then simplify. For the imaginary part, the result obtained is equation (2). For the real part, however, we note that the integrand in the Cauchy principal value integral is discontinuous at x from the integrand of the Cauchy principal value integral above, we ensure that the integrand has a finite limit as x 1 2 → . We can then continuously extend the domain of integration to [0,1] without computing a Cauchy principal value integral. After this modification, the result in equation (4) is obtained.

Hermite's generalization of Binet's second formula for ( ) z logΓ
An interesting application of equation (1) is to use it to prove Binet's second formula [6] for the logarithm of the gamma function. For convenience, we restate this formula here. Let The proof is outlined in Sections 177 and 180 of [3]. In this section, we shall apply Bromwich's method to the other Legendre sine integral (2), and we shall find that the result is Hermite's formula [7] for z logΓ In this article, all integrals with infinite domains of integration and continuous integrand are absolutely convergent, and thus there is no difficulty in reversing the order of integration in repeated integrals of this kind. This fact was used in obtaining (16). Further discussion can be found in Section 177 of [3]. To evaluate the integral on the left-hand side of the first line in equation (16), we use the integral representations of the logarithm (see Section 6.222, page 116, Example 6 of [5]) and of the digamma function ψ (see Section 12.3, page 247 of [5]). In both cases, the integrand extends continuously to a 0 = . By using equations (17) and (18), the left-hand side of equation (16) Here, we have again reversed the order of integration. The arctangent here is defined on the right-hand halfplane, with value 0 on the real axis (cf. line 11, page 251 of [5]): where the path of integration is a straight line and u i i , ≠ + − . Thus, the limit of x w arctan( ) / as ξ → ∞ is 0 (x being positive real), and its derivative with respect to w is x x w For the integral of the right-hand side of (19), we can use the fact that ψ w w logΓ Evaluating the anti-derivative at the lower limit of integration is straightforward, and for the upper limit, we rewrite the anti-derivative in a more convenient form, which results in the following expression: Note that, in the above expression, w ξ iη = + . A corollary of Binet's first formula (see the last paragraph of page 249 of [5]) is that, if z x iy = + , then, for large x, Thus, when x is large, the terms z z z π log log 2 Hence, using this result (or, equivalently, Stirling's series for the logarithm of the Gamma function; see page 252 of [5]), as ξ → ∞, the term on the second line of the right-hand side of (21) tends to zero. The term on the third line of the right-hand side of (21) tends to π log 2 1 4 ( ), and hence, ψ w w ξ π z z z z 1 2 Finally, since (20) is the result for the integral of the left-hand side of (19), and (23) is the result for the integral of the right-hand side of (19), we can equate these two expressions and rearrange to find, for The result in (24) corresponds to the specific case a 1 2 = in Hermite's general formula [7] for z a logΓ( ) + ; see also the expression below equation (1.10) in [9], where z a logΓ( ) + , which is written in terms of two integrals, reduces to (24) for the case a 3.2 Proof of Theorem 1.1, equation (7) We now apply the procedure from the previous section to the integrals in equation (3). First, we multiply the left-hand side of equation ( Performing the same procedure on the right-hand side of equation ( Note that, as mentioned in Section 1.1, the integrand on the right-hand side of equation ( The above final equality can be obtained by decomposing the integrand into partial fractions and performing the integration over ξ . Now perform the same procedure on (26) to obtain The above final equality is again obtained by decomposing the integrand into partial fractions and performing the ξ integration. The integrand on the right-hand side of (28) extends continuously to 0, 1 [ ]. Interchanging the order of integration in the previous repeated integrals is permissible in all cases since, as in the previous section, the integrals are absolutely convergent. Since the expressions in (25) and (26) are equal, the same is true for the expressions in (27) and (28), and therefore this proves equation (7).

Proof of Theorem 1.1, equation (8)
We now analyze equation (4). First, we multiply the left-hand side of equation (4) Performing the same procedure on the right-hand side of equation ( For . Thus, the left-hand side of equation (7) can be expressed as f z f z 1 ( ) ( ) − + ; see the comments after equation (7). By extension, for any k 0 ∈ ≥ , Theorem 1.1 provides the following expression for f z k f z k 1 Here, N 1 ≥ is a natural number. The summation of the right-hand side of equation (33) is To calculate the sum of the S x z , k ( ), the summation of the general expression z k y log( ) + + , where y 0 ≥ is a nonnegative real number, needs to be performed. The appropriate value for y can then be substituted at the end. To perform the summation, we use the identity z z z , which is valid when z is not a negative integer (see page 237 of [5] In the last step, we used the identities z z z ) ( ) ( ) + + = + + . After equating our previous results, the summation over k of both sides of equation (33) is The integrands extend continuously to the endpoints of the interval. The above result is valid for all N ∈ . The term on the first line of the right-hand side of this equation is a function only of z and the second line is a function of z N + . To show that these terms exactly correspond to f z ( ) and f z N ( ) + , we take the limit as N → ∞. As noted previously, f z 0  In view of (22), the limit and the integral may be interchanged, by the dominated convergence theorem [8] for instance, and for each  The integrands extend continuously to the whole interval 0, 1 [ ], and in particular, they have a finite limit as In view of (22), the limit and the integral may be interchanged, by the dominated convergence theorem [8] for instance, and for each In equations (38) and (44), in general, z is a complex number with z Re 0 ( ) > . If we restrict z to be a positive real number, z