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The Eulerian Functions

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The Eulerian Functions Appendix A The Eulerian Functions Here we consider the so-called Eulerian functions, namely the well-known Gamma function and Beta function together with some special functions that turn out to be related to them, such as the Psi function and the incomplete gamma functions. We recall not only the main properties and representations of these functions, but we also briefly consider their applications in the evaluation of certain expressions relevant for the fractional calculus. A.1 The Gamma Function The Gamma function .z/ is the most widely used of all the special functions: it is usually discussed first because it appears in almost every integral or series representation of other advanced mathematical functions. We take as its definition the integral formula Z 1 .z/ WD uz1 eu du ; Re z >0: (A.1.1) 0 This integral representation is the most common for , even if it is valid only in the right half-plane of C. The analytic continuation to the left half-plane can be done in different ways. As will be shown later, the domain of analyticity D of is D D C nf0; 1; 2;:::;g : (A.1.2) Using integration by parts, (A.1.1) shows that, at least for Re z >0, satisfies the simple difference equation .z C 1/ D z .z/; (A.1.3) © Springer-Verlag Berlin Heidelberg 2014 269 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 270 A The Eulerian Functions which can be iterated to yield .z C n/ D z .z C 1/::: .z C n 1/ .z/; n2 N : (A.1.4) The recurrence formulas (A.1.3 and A.1.4) can be extended to any z 2 D .In particular, since .1/ D 1,wegetfornon-negative integer values .n C 1/ D nŠ ; n D 0; 1; 2; : : : : (A.1.5) As a consequence can be used to define the Complex Factorial Function zŠ WD .z C 1/ : (A.1.6) By the substitution u D v2 in (A.1.1)wegettheGaussian Integral Representation Z 1 2 .z/ D 2 ev v2z1dv ; Re .z/>0; (A.1.7) 0 which can be used to obtain when z assumes positive semi-integer values. Starting from  à Z C1 1 2 p D ev dv D 1:77245 ; (A.1.8) 2 1 we obtain for n 2 N,  à Z  à C1 1 2 1 .2n 1/ŠŠ p .2n/Š n C D v vn vD D :1 e d n 2n (A.1.9) 2 1 2 2 2 nŠ A.1.1 Analytic Continuation A common way to derive the domain of analyticity (A.1.2) is to carry out the analytic continuation by the mixed representation due to Mittag-Leffler: Z X1 .1/n 1 .z/ D C eu uz1 du ; z 2 D : (A.1.10) nŠ.z C n/ nD0 1 This representation can be obtained from the so-called Prym’s decomposition, namely by splitting the integral in (A.1.1) into two integrals, one over the interval 1The double factorial mŠŠ means mŠŠ D 1 3 :::mif m is odd, and mŠŠ D 2 4 :::mif m is even. A.1 The Gamma Function 271 0 Ä u Ä 1 which is then developed as a series, the other over the interval 1 Ä u Ä 1, which, being uniformly convergent inside C, provides an entire function. The terms of the series (uniformly convergent inside D ) provide the principal parts of at the corresponding poles zn Dn. So we recognize that is analytic in the entire complex plane except at the points zn Dn (n D 0; 1; : : : ), which turn n out to be simple poles with residues Rn D .1/ =nŠ. The point at infinity, being an accumulation point of poles, is an essential non-isolated singularity. Thus is a transcendental meromorphic function. A formal way to obtain the domain of analyticity D is to carry out the required analytical continuation via the Recurrence Formula .z C n/ .z/ D ; (A.1.11) .z C n 1/ .z C n 2/:::.z C 1/ z that is obtained by iterating (A.1.3) written as .z/ D .z C 1/=z.Inthiswaywe can enter the left half-plane step by step. The numerator on the R.H.S. of (A.1.11) is analytic for Re z > n; hence, the L.H.S. is analytic for Re z > n except for simple poles at z D 0; 1;:::;.n C 2/; .n C 1/.Sincen can be arbitrarily large, we deduce the properties discussed above. Another way to interpret the analytic continuation of the Gamma function is provided by the Cauchy–Saalschütz representation, which is obtained by iterated integration by parts in the basic representation (A.1.1). If n 0 denotes any non- negative integer, we have Z Ä 1 1 1 .z/D uz1 eu 1 C u u2 CC.1/nC1 un du (A.1.12) 0 2 n in the strip .n C 1/ < Re z < n. To prove this representation the starting point is provided by the integral Z 1 uz1 Œeu 1 du ; 1<Rez <0: 0 Integration by parts gives (the integrated terms vanish at both limits) Z Z 1 1 1 1 uz1 Œeu 1 du D uzeu du D .z C 1/ D .z/: 0 z 0 z So, by iteration, we get (A.1.12). A.1.2 The Graph of the Gamma Function on the Real Axis Plots of .x/ (continuous line) and 1=.x/ (dashed line) for 4<xÄ 4 are shown in Fig. A.1 and for 0<xÄ 3 in Fig. A.2. 272 A The Eulerian Functions 6 Γ(x) 4 2 1/Γ(x) 0 −2 −4 −6 −4 −3 −2 −1 0 1 2 3 4 x Fig. A.1 Plots of .x/ (continuous line)and1=.x/ (dashed line) 3 2.5 Γ(x) 2 1/Γ(x) 1.5 1 X: 1.462 Y: 0.8856 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 x Fig. A.2 Plots of .x/ (continuous line)and1=.x/ (dashed line) Hereafter we provide some analytical arguments that support the plots on the real axis. In fact, one can get an idea of the graph of the Gamma function on the real axis using the formulas .x/ .x C 1/ D x.x/; .x 1/ D ; x 1 to be iterated starting from the interval 0<xÄ 1,where.x/ !C1as x ! 0C and .1/ D 1. A.1 The Gamma Function 273 For x>0the integral representation (A.1.1) yields .x/ > 0 and 00.x/ > 0 since Z Z 1 1 .x/ D eu ux1 du ;00.x/ D eu ux1 .log u/2 du : 0 0 As a consequence, on the positive real axis .x/ turns out to be positive and convex so that it first decreases and then increases, exhibiting a minimum value. Since .1/ D .2/ D 1, we must have a minimum at some x0, 1<x0 <2. It turns out that x0 D 1:4616 : : : and .x0/ Dp0:8856 : : :; hence x0 is quite close to the point x D 1:5 where attains the value =2 D 0:8862 : : :. On the negative real axis .x/ exhibits vertical asymptotes at x Dn.nD 0; 1; 2; : : :/; it turns out to be positive for 2<x<1, 4<x<3, :::,and negative for 1<x<0, 3<x<2, :::. A.1.3 The Reflection or Complementary Formula .z/.1 z/ D : (A.1.13) sin z This formula, which shows the relationship between the function and the trigonometric sin function, is of great importance together with the recurrence formula (A.1.3). It can be proved in several ways; the simplest proof consists in proving (A.1.13)for0<Rez <1and extending the result by analytic continuation to C except at the points 0; ˙1; ˙2;::: The reflection formula shows that has no zeros. In fact, the zeros cannot be in z D 0; ˙1; ˙2;:::and, if vanished at a non-integer z, then because of (A.1.13), this zero would be a pole of .1 z/, which cannot be true. This fact implies that 1 1= is an entire function. Loosely speaking, .1z/ collects the positive zeros of 1 sin .z/, while .z/ collects the non-positive zeros. A.1.4 The Multiplication Formulas Gauss proved the following Multiplication Formula nY1 k .nz/ D .2/.1n/=2 nnz1=2 .z C /; nD 2;3;::: ; (A.1.14) n kD0 which reduces, for n D 2,toLegendre’s Duplication Formula 1 1 .2z/ D p 22z1=2 .z/.z C /; (A.1.15) 2 2 274 A The Eulerian Functions and, for n D 3,totheTriplication Formula 1 1 2 .3z/ D 33z1=2 .z/.z C /.z C /: (A.1.16) 2 3 3 A.1.5 Pochhammer’s Symbols Pochhammer’s symbols .z/n are defined for any non-negative integer n as .z C n/ .z/ WD z .z C 1/ .z C 2/:::.z C n 1/ D ;n2 N ; (A.1.17) n .z/ with .z/0 D 1. In particular, for z D 1=2 ; we obtain from (A.1.9)  à 1 .n C 1=2/ .2n 1/ŠŠ WD D : n 2 n .1=2/ 2 We extend the above notation to negative integers, defining .z C 1/ .z/ WD z .z 1/ .z 2/:::.z n C 1/ D ;n2 N : (A.1.18) n .z n C 1/ A.1.6 Hankel Integral Representations In 1864 Hankel provided a complex integral representation of the function 1=.z/ valid for unrestricted z;itreads: Z 1 1 et .z/ D t; z 2 C ; z d (A.1.19a) 2i Hat where Ha denotes the Hankel path defined as a contour that begins at t D1ia (a>0), encircles the branch cut that lies along the negative real axis, and ends up at t D1Cib (b>0).
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