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The fundamental solutions of the time-fractional diffusion equation

Article · December 2002

DOI: 10.1142/9789812776273_0020

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The fundamental solutions of the time-fractional diffusion equation

Francesco Mainardi * t Gianni Pagnini *

1. - Introduction

Time-fractional diffusion equations, obtained from the standard diffusion equa­tion by replacing the first order time-derivative by a fractional derivative (of order 0 < /? < 2, in Riemann-Liouville or Caputo sense), have been treated by a num­ber of authors, see, e.g. Engler [11], Fujita [15], Gorenflo, Luchko and Mainardi [18, 19], Hanyga [21], Mainardi [23, 24, 25, 26], Metzler and Klafter [34], Priiss [38], Saichev and Zaslavsky [39], Schneider and Wyss [41], Uchaikin and Zolotarev [43]. For other treatments of the time-fractional diffusion equations we refer the reader to the references cited therein. In this paper we intend to provide more insights for the fundamental solutions of the general time-fractional diffusion equation, based on the recent results by Mainardi, Luchko and Pagnini [28].

By time-fractional diffusion equation we mean the evolution equation

(1) d? d2

-^u{x,t) = —u{x,t), 0 < / 3 < 2 , xeR, teR+,

where the time-fractional derivative is intended in the Caputo sense, see Appendix A. When P is not integer (/? ± 1, 2) the L.H.S of (1) reads:

1 ft

(2) wu(x,t):=\ 1

r(2 - p) Jo

d_ ~d~T

dr2

U(X,T)

U(X,T)

dr

(t-T)"'

dr

(t-T)»-

if 0 < P < 1,

if 1 < p < 2 .

* G. Gentili (1961-2000) was the best student of the course of Mathematical Physics held by F. Mainardi during the academic year 1983/84. After getting a degree in Physics in 1985 and a PhD in Mathematics in 1989 (supervisor M. Fabrizio), he started a smart academic career becoming in 1999 Associate Professor of Mathematical Physics.

fDipartimento di Fisica, Universita di Bologna, Via Irnerio 46, 1-40126 Bologna, Italy E-mail: mainardi@bo.lnfn.it

'Istituto per le Scienze dell'Atmosfera e del Clima del CNR, Via Gobetti 101, 1-40129 Bologna, Italy E-mail: g.pagnini8isac.cnr.it

207

M. Fabrizio, B. Lazzari, A. Morro (Editors):MATHEMATICAL MODELS AND METHODS FOR SMART MATERIALS,Series on Advances in Mathematics for Applied Sciences — Vol. 62World Scientific, Singapore (2002).

208

When P = 1, 2 we recover well-known evolution equations, namely, for /3 = 1, the diffusion equation:

(3) d

t(x,t) d2

u(x,t). dt~K~'"' dx2

for f3 = 2, the D'Alembert wave equation:

i e J i , te 1R+;

(4) dt2 u(x,t)

dx2 i(x,t), x € Ft, t e IR£ .

For 1 < j5 < 2 the fractional equation (1) is expected to interpolate (3) and (4), thus in this case it can be referred to as the time-fractional diffusion-wave equation.

Suitable processes of integration allow us to eliminate the time-fractional deriva­tive in the L.H.S of (1); recalling the definition of the Caputo derivative we easily obtain the following integro-differential equations: if 0 < p < 1,

1 r* u(x,t) = u(x,0+) + —jjo dx

(5)

if 1 < P < 2 ,

(6) u{x,t)=u(x,0+)+tut(x,0+) + ~~ fQ

2U(X,T) {t - rf-1 dr ;

dx2 U(X,T) {t - rf-1 dr .

In order to formulate and solve the Cauchy problem for (1) we have to select explicit initial conditions concerning u(x, 0+) if 0 < P < 1 and u(x, 0 + ) , ut{x, 0+) if 1 < P < 2 . If (p(x) and tp(x) denote sufficiently well-behaved real functions x defined on JR, the Cauchy problem consists in finding the solution of (1) subjected to the additional conditions:

(7a) u(x, 0+) = cj>(x), x&R, if 0 < / ? < ! ;

(7ft) u(x,0+) = ut(x,0+)

<f>(x), xe R, if l< P < 2.

We note that if we set ip(x) = 0 in (7b) we ensure the continuous dependence of the corresponding solution with respect to the parameter P in the transition from P = 1~ to P = 1 + , as it turns out by comparing the representations (5) and (6).

The paper is divided as follows. In Section 2 we treat the Cauchy problem for the equation((1) by making use of the Fourier and Laplace transforms with respect to the space and time variables. We state the concept of fundamental solutions (the so-called Green functions) for which we derive the general scaling properties in

'In what follows we shall meet only functions that are defined and continuous in x 6 1R and/or t 6 (0, T), VT > 0 except, possibly, at isolated points where these functions can be infinite. Following Marichev [31] we restrict our attention to the classes of such functions for which the Riemann improper integrals in x and in t absolutely converge on 1R and (0, T), respectively. We denote these classes as LC(1R), Lc(0,T).

209

terms of a similarity variable. In Section 3 we provide a general representation for the (reduced) Green functions in terms of Mellin-Barnes integrals in the complex plane, which allow us to obtain them in computational form. We note that these functions are peculiar "higher transcendental" functions of the Wright-type, that, in their turn, are special cases of the more general Fox H functions. Finally, Section 4 is devoted to concluding discussions, and a summary of the results in which we present plots of the (reduced) Green functions for a number of cases. After the Appendix A devoted to the Caputo fractional derivative, we report in Appendix B some historical notes on the Italian mathematician S. Pincherle, who can be considered the pioneer of the Mellin-Barnes integrals.

2. - The Green functions: scaling and similarity properties

The Cauchy problems stated in the Introduction can be conveniently treated by making use of the most common integral transforms, i. e. the Fourier transform (in space) and the Laplace transform (in time) whose notations are briefly recalled in 2. Indeed, the combined Fourier-Laplace transforms of the solutions of the two Cauchy problems:

(a) {(l) + (7a)} if 0 < / 3 < 1, (b) {(1) + (7b)} if 1 < 0 < 2 ,

turn out to satisfy the following algebraic equations

(8a) -K2U{K,S) = S?U(K,S)-S^~1${K), 0 < / ? < 1 ,

(8b) ~K2U{K,S) = S^{K,S)-S0-1${K)-S^-2^{K), 1 < /3 < 2 ,

^Let

/(«) =F{J(X);K}= / e+lKXf{x)dx, t e f f i , J — oo

be the Fourier transform of a function f(x) € LC(1R), and let

f(x)=F-l{f{K);x) = — J e - m x / ( K ) c k , xeR,

be the inverse Fourier transform. Let

f(s)=C{f(t);s}= e~stf(t)dt, %(a)>af, Jo

be the Laplace transform of a function f(t) € Lc(0,T), VT > 0 and let

/ ( t ) = £ - 1 {/(«); *} = — y estf(s)ds, » ( 8 ) = 7 > o / ,

with t > 0, be the inverse Laplace transform. Above a/ denotes the abscissa of convergence: for its existence a sufficient condition is that the original function is of exponential type. We remind that if the original functions are piecewise differentiable, then the two inversion formulas hold true where the functions are continuous and the corresponding integrals must be understood in the sense of the Cauchy principal value.

210

from which we obtain

(9a) Z(K,S)= 24>{K), 0 < / 9 < l ,

(96) 2 ( « ) S ) = _ _ _ 0 ( / c ) + _ _ _ ^ ( « ) > l < / ? < 2 .

By fundamental solutions (or Green functions) of the above Cauchy problems we mean the (generalized) solutions corresponding to the initial conditions

(lOo) Gy(x, 0+) = 6(x), if 0 < P < 1;

Gf{x,Q+) = S{x), (Gf>(x,0+) = 0,

(106) { „ i a if 1 < /3 < 2. ^ G « ( x , 0 + ) = 0 , [ ^ G ^ ( x , 0 + ) = 5 ( x ) ,

Here <5(a;) is the delta-Dirac generalized function whose (generalized) Fourier trans­form is known to be identically 1. Thus, the combined space-Fourier and time-Laplace transforms of these Green functions turn out to be

(11a) G%\K,S) = ^ ~ , 0 < / ? < 2 ,

(lib) G P ) ( K ) S ) = _ _ _ i l < / 3 < 2 .

We note that the function Gp (x,t) along with its combined Fourier-Laplace trans­form is well defined also for 0 < ft < 1 even if it loses its meaning of being a fundamental solution of (1).

Then, by recalling the Fourier convolution property in the inversion of the Fourier-Laplace transforms of (9a)-(9b), we note that the Green functions allow us to represent the solutions of the above two Cauchy problems through the relevant integral formulas

(12a) u(x,t)= [+°°G$\z,t)<l>{x-S)dS, 0 < / ? < l ; J — 00

(126) u(x,t)= / + l G ^ ( 4 , t ) 0 ( i - O + G f ( e , t ) V - ( i - O ] ^ . K / 3 < 2 .

By using the known scaling rules for the Fourier and Laplace transforms,

(13) / ( m J ^ a - ' / W d ) , a > 0 , f(bt) A b'1 f{s/b), 6 > 0 ,

211

we infer directly from (13) (thus without inverting the two transforms) the following scaling properties of the Green functions,

G(p\ax, bt) = b-"Gf{ax/V , t),

Gf\ax , bt) = b-»+1Gf(axlbv , t), (14) I ~f21 ^'"-'-'' 7 f

v.r'" "•" u = 0/2.

Consequently, introducing the similarity variable xjt^l2 , we can write

1 j \Gf{x,t)=t-H™Kf){x/tV*),

where the one-variable functions Kp (x) , /Q (x) are referred to as the reduced Green functions. We note that both the Green functions are symmetric with respect to x and

(16) K{p{x)=Gf{x,l) = K{p{-z), j = 1,2.

In view of (15) and (12), the knowledge of the reduced Green functions is sufficient to provide the complete solutions of the Cauchy problems.

3. - Mellin-Barnes integral representation of the Green functions

To determine the two Green functions in the space-time domain we can follow two alternative strategies related to the different order in carrying out the inversion of the combined Fourier-Laplace transforms in (11)-(12). Indeed we can

(51) : invert the Fourier transforms getting Gg (x, s), Gg (x, s), and then invert these Laplace transforms,

(52) : invert the Laplace transforms getting Gp' (K, t), Gp (K, t), and then invert these Fourier transforms. Strategy (SI)

Recalling the Fourier transform pair,

(171 -^— 6 ^ - e - N * 1 / 2 6 > 0

and setting a,- = s^~J' , b — s13 we get

(18) ^ W ) = ^ e H ^ / 2 , J = 1,2.

Strategy (S2) Recalling the Laplace transform pair, see e.g. [12], [20], [36]

(19) ± . L £ t>-i El)j(-ct?), c > 0 ,

212

where Epj denotes the two-parameter Mittag-Leffler function3 and set t ing c = n2

we get

(20) G^\K,t) = lP-1Epj(-Kltp), j = 1,2.

The s t ra tegy (Si) has been followed by Mainardi [23, 24, 25, 26] to obtain the

first Green function as

(21) G$\x, t) = i r ^ 2 Mm (\x\/t»2) , - o o < x < + o o , t > 0 ,

inc

noteworthy case of the Wright function4 .

2

where Mp/2 denotes the so-called M function of order /3/2, see also [36], which is a

3The Mittag-Leffler function Efftli with 0, p. > 0 is an entire transcendental function of order p = 1/(8, defined in the complex plane by the power series

0 0 n

n—0 v '

Originally, at the beginning of 1900, Mittag-Leffler introduced and investigated (in five notes from 1902 to 1905) the function

OO n

n=0 y '

as an instructive example of an entire function that generalises the exponential. For detailed information on the Mittag-Leffler-type functions the reader may consult e.g. [10],[12], [20], [22], [27], [36], [40].

4The function Mv(z) is defined for any order v e (0,1) and Vz g W by

M „ ( z ) : = V — - — ^ '— - , 0 < I / < 1 , zeW. v ' *-> n\T\-vn + (l-v]\

It turns out that M„(z) is an entire function of order p = \/(l — v), which provides a generalization of the Gaussian and of the Airy function. In fact we obtain

M1/2(z) = -J=exp ( - z 2 / 4 ) , M1/3(z) = 32/3Ai (z/31 /3) .

The M function is a special case of the Wright function defined by the series representation, valid in the whole complex plane,

0 0 n

Indeed, we recognize M„(z) = * _ „ , i _ „ ( - z ) , 0 < ! / < l .

Originally, Wright introduced and investigated this function with the restriction A > 0 in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions. Only later, in 1940, he considered the case - 1 < A < 0. For detailed information on the Wright-type functions the interested reader may consult, e.g. [12] (where, presumably for a misprint, A is restricted to be non negative), [18, 19], [22].

213

As far as the second Green function is concerned, we note from (18) that

Gf{x,s) = G^] {x, s)/s, so

(22) Gf{x,t) = £G$\X,T) dr.

Closed form solutions are found in the special case /3 = 1 (diffusion equation) and in the limiting case f3 — 2 (D'Alembert wave equation). We easily recognize for (5 = 1 :

(23) Gi'W) = g!,-.-«, 0 f . t e „ . ^ . - « « . | r t ( ^ 5 ) ,

where erfc denotes the complementary error function, and, for p = 2 :

(24) <#>(*,*) = fo + t>+ '<*-*>, (ff(s,Q = g(j + t ) - 4 ^ ,

where 9 denotes the unit-step Heaviside function. The strategy (S2) has been followed by Gorenflo, Iskenderov & Luchko [17] and

by Mainardi, Luchko & Pagnini [28] to obtain the first Green function of space-time fractional diffusion equations. These authors have inverted the relevant Fourier transforms by using the Mellin transform5.

5 Let

Mi [ {/(r); s} = / • ( * ) = / / M r - 1 * , 7 i < K ( s ) < 7 2 Jo

be the Mellin transform of a sufficiently well-behaved function f(r), and let

M i p'y+ico

1{r(s);r} = f(r) = 1— r(s)r-

be the inverse Mellin transform, where r > 0, 7 = K (s), 71 < 7 < 72. We refer to specialised treatises and/or handbooks, see e.g. [13], [31], [37], for more details and tables on the Mellin transform. Here, for our convenience we recall the main rules that are useful to adapt the formulae from the handbooks and, meantime, are relevant in the following. Denoting by <-> the juxtaposition of a function f(r) with its Mellin transform f(s), the main rules are:

f(ar) tt a-°r(s),a>0; ra f(r) U f(s + a); f(r")Uy-r{s/p), p ^ O ;

CO

h(r) = j-f(p)g(rlp)dp U h*(s) = f'(s)g'(s).

0

The Mellin convolution formula is useful in treating integrals of Fourier type for x = \x\ > 0 as Ic{x) = i /0°° / ( K ) cos (nx)dK, when the Mellin transform f*(s) of / ( K ) is known.

Referring to [28] for details, we get

/ t ( i ) = - r / / * ( s ) r ( l - s ) s m ( — ) x'ds, x > 0 , 0 < 7 < 1 • 7T X Z7TZ J^-ioo \ Z J

214

Here we follow the same approach based on Mellin transform. For this purpose we note that the Mittag-Leffler function admits a Mellin transform type representation, see e.g. [31], which will be used to get the Fourier anti-transform of (20) (with t = 1), namely

(25) ^ ( - ^ 2 ) ^ ^ r ( r ^ ! ( ^ 2S ) / 2 ) • 0 < / 3 < 2 , 0 < ! f t ( S ) < 2 .

For the determination of the reduced Green functions Kp{x) = Gp{x, 1) we can restrict our attention to x > 0, and thus write in view of (20) and (25)

(26) K{P (x) = - f°° cos (KX) ED J (-K2) <1K . IT JO ' \ J

So, using the final result in footnote '6 ' and the reflection formula for the Gamma function, we obtain

The integral at the RHS of (27), in that it contains only gamma functions in the fraction multiplying x", is a particular Mellin-Barnes integral according to a usual terminology. In this respect the interested reader can find in [12] the discussion on the general conditions of convergence for the typical Mellin-Barnes integral, based on the asymptotic representation (Stirling formula) of the gamma function. The names refer to the two authors, who in the first 1900's developed the theory of these integrals using them for a complete integration of the hypergeometric differential equation. However, as pointed out in [12], these integrals were first used by the Italian mathematician S. Pincherle in 1888, see Appendix B.

Readers acquainted with the "higher transcendental" H functions (introduced by Fox [14] in 1961) can recognize in the R.H.S of (27) the representation of a certain function of this class see e.g. [22], [31], [32], [37], [40], [42]. Unfortunately, as far as we know, computing routines for this general class of special functions are not yet available. Here, following the approach adopted by Mainardi, Luchko & Pagnini [28], we intend to compute the (reduced) Green functions in any space domain by matching a convergent power series (suitable for small \x\) with an asymptotic representation (suitable for large |a;|).

In order to obtain the convergent power series we transform the original contour in (27) to the loop L+00 encircling all the poles sn = 1 + n, n € iV0 of the function T(l — s) and apply the residue theorem. We obtain

<28> ^"M°i£n:r[-^r4-OTl- '-1-2-The asymptotic representation can be obtained by using the arguments by Braaksma [3] (see also [28]), and turns out to be

(29) Kf (x) ~ Aj x ai exp ( -6 x c ) , x ->• +oo ,

215

where

(30) Aj = {2^(2 - ft 2f-40-D]/(2-« /?[2(2J-l)-2/3]/(2^)|-1/2

(31) a3 = 2P J}2j^ l)~, 6=(2^/3)2-2/(2-«/3^2^, c= 2

2(2-/?) ' v ^' ^ ' 2 - £ '

We now can complement Mainardi's result (21) providing also the second Green function in the space-time domain in terms of a Wright-type function. Indeed, using (15) and (28) we can write

(32) Gf)(x,t) = ^t1-V2P(i/2(\x\/tV2) , ~oo<x<+oo, t>0,

where Pp/2 denotes a suitable Wright-type function briefly discussed in 6.

4. - Concluding discussion and plots

We conclude with a discussion about some general features occurring in the Cauchy problem of our time fractional diffusion equation (l)-(2). A first general feature concerns the scaling property of the two Green functions which allows us to express them in terms of functions of a single variable, the reduced Green functions K$\x), j = 1,2, see (15). In this paper we have focused our attention on deriving a computational form for Kp (x) in all of M. In this respect the representation of Kp\x) through the Mellin-Barnes integral, see (27), was found useful. More precisely, to compute the functions Kp (x) we used the series expansions (28) and asymptotic representations (29)-(31), which were derived from (27).

Hereafter we shall exhibit some plots of the reduced Green functions KJ'(x) for some " characteristic" values of the parameter (3. All the plots were drawn by using the MATLAB system for the values of the independent variable x in the range \x\ < 5. To give the reader a better impression about the behaviour of the tails, the logarithmic scale was adopted. Both the reduced Green functions, being non-negative and normalized are of the greatest interest in view of their interpretation as probability densities. However, only the first Green function Gp (x,i) keeps the normalization when it evolves in time, see Mainardi and Pagnini [30].

6The function Pv(z) is defined for any order v 6 (0,1) and Vz 6ff" by

Pv{z):=Y ,„ ,„ u> 0 < £/ < 1, ze(T.

It turns out that Pv{z] is an entire function of order p = 1/(1 — v), and is a special case of the Wright function being

P„(2) = * - „ , 2 - „ ( - . z ) , 0 < i / < l .

216

5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Fig. 3 Fig. 4

Fig. 5 Fig. 6

217

(is 1.50

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 -S -4 -3 -2 -1 0 1 2 3 4 5

Fig. 7 Fig. 8

H psl .75

X

- 5 - 4 - 3 - 2 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Fig. 9 Fig. 10

)'

1

K'^'OO pr f .00

•

X

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 ] 2 3 4 5

Fig. 11 Fig. 12

218

Acknowledgment Research performed under the auspices of the National Group of Mathematical Physics (G.N.F.M. - I.N.D.A.M.) and partially supported by the Italian Ministry of University (M.I.U.R) through the Research Commission of the University of Bologna. The authors are grateful to Prof. R. Gorenflo for discussions and helpful com­ments.

Appendix A: The Caputo time-fractional derivative

Now we present an introduction to the Caputo fractional derivative starting from its representation in the Laplace transform domain and contrasting it to the standard Riemann-Liouville fractional derivative.

Let

(A la ) f(s)=C{f(t);s} = e-stf(t)dt, S (s) > af, Jo

be the Laplace transform of a function /(£) € Lc(0, T), VT > 0 and let

{Alb) f{t) = C-1 {/(a); t} = — est f(s) ds, 5ft (s) = 7 > af ,

with t > 0 , be the inverse Laplace transform. For a sufficiently well-behaved function f(t) we define the Caputo time-fractional

derivative of order /3 (m — 1 < /3 < m, m £ IN) through its Laplace transform

771 — 1

(A.2) £ { t ^ / ( t ) ; s } = S ^ / ( S ) - £ s ^ 1 - * / « ( 0 + ) , m - K / 3 < m .

This leads us to define, see e.g. [6], [20],

1 ft f(m){r)dT

(A3) tD?f(t) I T(m - P) Jo (t - r)/3+1-

dm

1 < P < m,

P--

The operator defined by (A.2)-(A.3) has been referred to as the Caputo fractional derivative since it was introduced by Caputo in the late 1960's for modelling the energy dissipation in some anelastic materials with memory, see [5, 6]. A former review of the theoretical aspects of this derivative with applications in viscoelasticity was given in 1971 by Caputo and Mainardi [9], with special emphasis to the long-memory effects.

The reader should observe that the Caputo fractional derivative differs from the usual Riemann-Liouville fractional derivative which, defined as the left inverse of the Riemann-Liouville fractional integral, is here denoted as tD& f(t). We have, see e.g. [40],

(A4) tDffit):--

dm

dtm

dm

dtm

\ 1 /" f(r)dr 1 T{m-P) Jo {t-Ty+1-m_

(t),

, m- 1< ft < m

P = m.

219

When the order is not integer, Gorenflo and Mainardi have shown the following relationships between the two fractional derivatives (when both of them exist), see e.g. [20],

(A.5) tD!f(t)=tDf> m-\ j.k

/ ( i ) - £ / < * ) ( 0 + ) L m — 1 < B < m,

or

k~/3 (A6) ^ / ( t ) = ^ / ( i ) - £ / ( * > ( 0 + ) - — - — , m-\<B<m.

fc=o T{k-() +1)

The Caputo fractional derivative, practically ignored in the mathematical trea­tises7, represents a sort of regularization in the time origin for the Riemann-Liouville fractional derivative. Recently, it has been extensively investigated by Gorenflo and Mainardi [20] and by Podlubny [36] in view of its major utility in treating physical and engineering problems which require standard initial conditions. Several appli­cations have been treated by Caputo himself up to nowadays, see e.g. [7, 8] and references therein.

We point out that the Caputo fractional derivative satisfies the relevant property of being zero when applied to a constant, and, in general, to any power function of non-negative integer degree less than m, if its order /? is such that m — \ < f3 < m. Furthermore, since

(A.7) tDf>v= J{l + l ) - t ^ , 8>0, 7 > - l , t>0, r ( 7 + 1 - p)

we note that

m

(A8) tDl> f(t) = t£>" g(t) < = • f(t) = g(t) + £ Cj t ^ , j=l

whereas, using also (A.5) or (A.6),

m

(A.9) tDi f{t) = tI%g{t) <=> f{t) = git) + j : CJ tm^ .

In these formulae the coefficients c3- are arbitrary constants. We also note the dif­ferent behaviour of tD% at the end points of the interval (m — 1, m ) ,

(A.10) lim tDU(t) = f{m^)(t)-f{m-1)(0+), lim t£>f/(*) = / ( m ) ( * ) . /3->(m-,l)+ f)-*m-

The last limit can be formally obtained by recalling the formal representation of the m-th derivative of the Dirac function, 5<m>(t) = t~m-1/T{-m), t > 0, see [16]. As a consequence of (A. 10), with respect to the order, the Caputo derivative is an operator left-continuous at any positive integer.

7According to Samko, Kilbas and Marichev[40] and Butzer and Westphal[4] the "regularized" fractional derivative was considered by Liouville himself (but then disregarded).

220

Appendix B: Pincherle and the Mellin-Barnes integrals

In Vol. 1, p. 49 of Higher Transcendental Functions of the Bateman Project [12] we read "Of all integrals which contain gamma functions in their integrands the most important ones are the so-called Mellin-Barnes integrals. Such integrals were first introduced by S. Pincherle, in 1888 [35]; their theory has been developed in 1910 by H. Mellin (where there are references to earlier work) [33] and they were used for a complete integration of the hypergeometric differential equation by E.W. Barnes [2]."

For a revisited analysis of the pioneering work of Pincherle we refer the interested reader to our recent paper [29]. Here we limit ourselves to give some biographical notes and to report the original quotations by Barnes and Mellin. This may help to recall the attention of mathematicians towards Pincherle.

Salvatore Pincherle (1853 - 1936) was Professor of Mathematics at the Univer­sity of Bologna from 1880 to 1928. He retired from the University just after the International Congress of Mathematicians that he had organized in Bologna, follow­ing the invitation received at the previous Congress held in Toronto in 1924. He wrote several treatises and lecture notes on Algebra, Geometry, Real and Complex Analysis. His main book related to his scientific activity is entitled "Le Operazioni Distributive e loro Applicazioni all'Analisi"; it was written in collaboration with his assistant, Dr. Ugo Amaldi, and was published in 1901 by Zanichelli, Bologna.

Pincherle can be considered one of the most prominent founders of Functional Analysis, as pointed out by J. Hadamard in his review lecture "Le developpement et le role scientifique du Calcul fonctionnel", given at the Congress of Bologna (1928).

A description of Pincherle's works, requested to the author by Mittag-Lefrler, the Editor of Acta Mathematica, appeared (in French) in Vol. 46, pp. 341-362 (1925) of this prestigious journal under the title "Notice sur les travaux de S. Pincherle". Furthermore, a collection of selected papers (38 from 247 notes plus 24 treatises) was edited by Unione Matematica Italiana (UMI) on the occasion of the centenary of his birth, and published by Cremonese, Roma 1954. Note that S. Pincherle was the first President of UMI, from 1922 to 1936.

Here we point out that the 1888 pioneering work of S. Pincherle on Generalized Hypergeometric Functions led him to introduce the later so named Mellin-Barnes integral to represent the solution of a hypergeometric differential equation investi­gated by Goursat in 1883. Pincherle's priority was explicitly recognized by Mellin and Barnes themselves, as reported below.

In 1907 Barnes, see p. 63 in [1], wrote: "The idea of employing contour integrals involving gamma functions of the variable in the subject of integration appears to be due to Pincherle, whose suggestive paper was the starting point of the investigations of Mellin (1895) though the type of contour and its use can be traced back to Riemann."

In 1910 Mellin, see p. 326 in [33], devoted a section (§10: Beweis eines Satzes von Pincherle = Proof of Theorems of Pincherle) to revisiting the original work of Pincherle; in particular, he wrote "Ehe wir zum Beweise dieses Satzes schreithen, welcher einen speziellen Fall eines noch allgemeinerem Satzes von Herrn Pincherle bildet, wollen wir die Linien L naher angeben, fiber welche die Integration vorzugs-

weise erstreckt wird.:

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E.W. Barnes, A new development of the theory of the hypergeometric functions, Proc. London Math. Soc. (ser. 2) 6 (1908) 141-177.

B.L.J. Braaksma, Asymptotic expansions and analytical continuations for a class of Barnes-integrals, Compositio Mathematica 15 No 3 (1962) 239-341.

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