The lensing properties of the Einasto profile
aa r X i v : . [ a s t r o - ph . C O ] A ug The lensing properties of the Einasto profile
E. Retana-Montenegro and F. Frutos-Alfaro
Escuela de F´ısica, Universidad de Costa Rica, San Pedro 11501, Costa Ricae-mail: [email protected]
Abstract.
In recent high resolution N-body CDM simulations, it has been had found that nonsingular three-parameter models, e.g. the Einasto profile has a better performance better than the singular two-parameter models,e.g. the Navarro, Frenk and White in the fitting of a wide range of dark matter halos. A problem with this profile isthat the surface mass density is non-analytical for general values of the Einasto index. Therefore, its other lensingproperties have the same problem. We obtain an exact analytical expression for the surface mass density of theEinasto profile in terms of the Fox H-function for all values of the Einasto index. With the idea of facilitate theuse of the Einasto profile in lensing studies, we calculate the surface mass density, deflection angle, lens equation,deflection potential, magnification, shear and critical curves of the Einasto profile in terms of the Meijer G-functionfor all rational values of the Einasto index. The Meijer G-function have been implemented in several commercialand open-source computer algebra systems, thus the use of the lensing properties of the Einasto profile in strongand weak lensing studies is straighforward. We also compare the S´ersic and Einasto surface mass densities profilesand found di ff erences between them. This implies that the lensing properties are not equal for both profiles.
1. Introduction
The Cold Dark Matter (CDM) theory has become the standard theory of cosmological structuralformation. Its variant the Λ CMD with ( Ω m , Ω Λ ) = (0 . , .
7) seems to be in agreement withthe observations on cluster-sized scales (Primack 2003). On galaxy / sub-galaxy scales has sev-eral problems, such as the discrepancy between observations and the results of numerical sim-ulations. The high resolution observations of rotation curves of low surface brightness (LSM)and dark matter dominated dwarf galaxies (de Blok et al. 2001; van den Bosch & Swaters 2001;Swaters et al. 2003; Weldrake et al. 2003; Donato et al. 2004; Gentile et al. 2005; Simon et al.2005; Gentile et al. 2007; Banerjee et al. 2010) favor density profiles with a flat central core (e.g.Burkert 1995; Salucci & Burkert 2000; Gentile et al. 2004; Li & Chen 2009). In contrast N-bodyCDM simulations predict a two parameter functional form for the density profiles with too highdensities (cusps) in the galatic center (Navarro et al. 1996, 1997; Moore et al. 1999). This discrep-ancy is called the cusp-core problem.Gravitational lensing is one of the most powerful tools in observational cosmology for probing thedistribution of matter of collapsed objects like galaxies and clusters in strong (Kochanek et al.1989; Wambsganss & Paczynski 1994; Bartelmann 1996; Chae et al. 1998; Kochanek et al.2000; Keeton & Madau 2001; Sand et al. 2002; Keeton 2002, 2003; Keeton & Zabludo ff observational data in strong and weak lensing studies the most accurate density profile must beused. The first step before using a profile in lensing studies is to investigate the lensing propertiesof the profile.Recently N-body CDM simulations (Navarro et al. 2004; Merritt et al. 2006; Gao et al. 2008;Hayashi & White 2008; Stadel et al. 2009; Navarro et al. 2010) have found that the three-parameterprofiles fit better to a wide range of dark matter halos. One of this profiles is the Einasto (1965)profile, a 3D version of the 2D Sersic (1968) model used to described the surface brightness ofgalaxies. The S´ersic profile can be written as: Σ S ( R ) = Υ I e exp − b n RR e ! / n − , (1)where R is the distance in the sky plane, n the S´ersic index, Υ the mass-to-light ratio, I e the luminos-ity density at the e ff ective radius R e , b n is a function of n that can be determined from the conditionthat the luminosity inside R E equals half of the total luminosity, for example Prugniel & Simien(1997) found b n = n − . + . / n .The Einasto profile is given by: ρ ( r ) = ρ E exp − d α " rr E ! α − (2)where r is the spatial radius, α is the Einasto index that determines the shape of the profile, d α isa function of α which allows the calculation of the density ρ E inside an e ff ective radius r E . In thecontext of dark matter halos this can expressed as: ρ ( r ) = ρ − exp − α " rr − ! α − (3)where ρ − and r − are the density and radius at which ρ ( r ) ∝ r − . Both radius and densities arerelated by ρ − = ρ E exp (2 /α − d α ) and r − = r E ( α d α / α . First Navarro et al. (2004) found thatfor haloes with masses from dwarfs to clusters 0 . . α . .
22 with an average value of α = . α E to increase withmass and redshift, with α ∼ .
17 for galaxy and α ∼ .
23 for cluster-sized haloes in the MillenniumSimulation (MS) (Springel et al. 2005). Navarro et al. (2010) found similar results for galaxy-sizedhaloes in the Aquarius simulation (Springel et al. 2008). Also, Gao et al. (2008) found that α ∼ . α in terms of the Meijer G-function. Thisfunction can be automatically handed by numerical routines implemented in computer algebrasystems (CAS) such as the commercial Mathematica R (cid:13) and Maple R (cid:13) and the free open-source S age and in the Python library mpmath . This paper is organized as follows. In Section 2 we derive the surface mass density of the Einastoprofile in terms of the Fox H-function and the Meijer G-function for values α = n and α = n with n integer, and rational values of the Einasto index. In Section 3 we evaluate the total massenclosed by this class of models using the density profile and the surface mass density obtained inthe previous section. In Section 4 we use the result for the projected surface mass density to calcu-late the deflection angle, the lens equation and deflection potential for a spherically symmetric lensdescribed by the Einasto profile in terms of the Meijer G-function. In Section 5 we derive expres-sions for the magnification, shear and the critical curves of the Einasto profile. We summary ourconclusions in Section 6. We give an brief description of the Mellin transform-method in AppendixA. In Appendix B we formulate all the properties of the Fox H-functions and Meijer G-functionsthat are used in this work.
2. Analytical expression for the surface mass density of the Einasto profile
The projected surface mass density of a spherically symmetric lens is given by integrating alongthe line of sight the 3D density profile: Σ ( ξ ) = Z + ∞−∞ ρ ( ξ, r ) dz , (4)where ξ is the radius measure from the centre of the lens and r = p ξ + z . This expression canalso be written as an Abel integral (Binney & Tremaine 1987): Σ ( ξ ) = Z ∞ ξ ρ ( r ) rdr p r − ξ (5)By inserting equation (3) into the above expression Σ ( x ) = ρ − r − e α Z ∞ x exp (cid:16) − s α α (cid:17) sds √ s − x (6)having introduced the quantities x = ξ/ r − and s = r / r − .The integral (6) can not be expressed in terms of ordinary functions for all the values of α E .However, using the Mellin-transform method (Marichev 1982; Adamchick 1996; Fikioris 2007)is possible the exact calculation of one dimensional definite integrals. The most powerful featureof this method is that the result is a Mellin-Barnes integral. This integral for a certain combinationof coe ffi cients is the integral representation of a Fox H-function or a Meijer G-function (for detailssee Appendix A) .Using the Mellin-transform method with the integral (6), with z = f ( s ) = ρ − r − e α exp − s α α ! (7) g ( s ) = s p − ( sx ) ≤ s ≤ x − elsewhere (8)and its Mellin transforms: {M f } ( u ) = ρ − r − e α α − α ! − u /α Γ (cid:18) u α (cid:19) (9) {M g } ( u ) = √ π x u − Γ (cid:16) u − (cid:17) u Γ (cid:16) + u (cid:17) (10)Combining equations (9), (10) and (A.4) with u = y and m = /α yields: Σ ( x ) = √ πρ − r − e m x π i Z C Γ (cid:16) − + y (cid:17) Γ (1 + my ) Γ (1 + y ) h (2 m ) m x i − y dy (11)Comparing the last equation with (B.11) is possible to obtain an analytical expression in terms ofthe Fox H-function for the surface mass density of the Einasto profile: Σ ( x ) = √ πρ − r − e α x H , , (1 , (cid:16) , α (cid:17) , (cid:16) − , (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ! α x (12)Writting the surface mass density as a Fox H-function has an inconvenient. The Fox H-functiondespite having a great potential for analytical work in Mathematics, sciences and engineering nonumerical routines has been implemented yet. We prefer to describe the lensing properties of theEinasto profile in terms of analytical functions that have numerical routines already implementedto facilitate its use in strong and weak lensing studies. The Meijer G-function meets the requirement pointed out before. A list of the relevant properties ofthe Meijer G-function can be found in Appendix B. We can use this function to write expressions inanalytical form for most of the lensing properties of the Einasto profile. The Meijer G-function hadbeen implemented in several commercial and free available CAS. This means that using the MeijerG-function in lensing studies is just as simple as use other special functions like Hypergeometric,Gamma and Bessel functions for example.Using a similar procedure to the one used by Baes & Gentile (2011) to obtain an analytical expres-sion for the luminosity density in terms of the Meijer G-function for all rational values of the S´ersicindex we proceed to do the same to derive an expression for the surface mass density of the Einastoprofile for all values of the Einasto index. α = n and α = n with n integer The equation (11) can be written in terms of the Meijer G-function for the Einasto index withvalues α = n and α = n with n integer. But first, one substitution is required using the GaussMultiplication formula (Abramowitz & Stegun 1970): N − Y j = Γ (cid:18) z ′ + jN (cid:19) = (2 π ) N − N − Nz ′ Γ (cid:0) Nz ′ (cid:1) , (13)with z ′ = z / N , N = m and z = my , we get: Γ (1 + my ) = (2 m ) + my (2 π ) − m Γ (1 + y ) m − Y j = Γ (cid:18) j m + y (cid:19) (14)Substituting the last equation into (11), we obtain: Σ ( x ) = ρ − r − √ m (2 π ) m − e m x π i Z C Γ − + y ! m − Y j = Γ (cid:18) j m + y (cid:19) h x i − y dy (15) Comparing with the integral representation of the Meijer G-function (B.1) we found an analyticalexpression for the surface mass density of the Einasto profile: Σ ( x ) = ρ − r − e α (2 π ) α − √ α x G α , , α − b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (16)where b is a vector of size α given by: b = ( α , α , ..., α − ! α , − ) (17)This result indicates that the form of the surface mass density of the Einasto profile di ff ers from thesurface mass density of the S´ersic model (equation 1) in functional form. Also is possible to write this expressions for all rational values of the Einasto index. Using m = p / q with p and q both integer numbers equation (11) becomes: Σ ( x ) = ρ − r − √ π e m x π i Z C q Γ (cid:16) − + qy (cid:17) Γ (1 + py ) Γ (1 + qy ) pq ! p x q − y dy (18)Substituting the three Gamma functions in equation (18) using the equation (14), we obtain anintegral and compare it with the definition of the Meijer G-function, we find: Σ ( x ) = ρ − r − e pq (2 π ) p − r pq x G p + q − , q − , p + q − ab (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (19)where a and b are vectors of size q − p + q − a = ( q , q , ..., q − q ) (20) b = ( p , p , ..., p − , − q , q , q , ..., q − q ) (21)It is immediate to verify that the equation (19) is equivalent to the equation (16) for Einasto indexwith values α E = n . Using the properties of the Meijer G-function (B.3, B.4) and (B.6) is possibleto demonstrate that equations (19) and (16) are equal for Einasto index with values α E = n . α E = and α E = α = α = α are outside the range favoredby the N-body CDM simulations, but are practical to check the consistency of our calculations.For the case α = ρ ( r ) = ρ − exp − " rr − − (22)Calculating the projected surface mass density using the equation (5), we find: Σ ( x ) = ρ − r − e x K (2 x ) (23)where K ( x ) is the modified Bessel of the second kind. Setting α = Σ ( x ) = ρ − r − e x G , , − , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (24)Substituting the equation (B.10) into (24) we obtain the equation (23).In a similar way with the case α = ρ ( r ) = ρ − exp − rr − ! − (25)The projected surface mass density can be found using the equation (5): Σ ( x ) = √ πρ − r − e − x + (26)Setting α = Σ ( x ) = √ πρ − r − e x G , , −− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (27)Using (B.9) in the last equation this one reduces to equation (26).It is interesting to compare these two cases with the surface mass density of the S´ersic profile Σ S ( R ) with the same values for the S´ersic index 1 / m that for the Einasto index α . We also includethe cases α = . m =
2) and α = . Σ S ( R ) forfour values of m and Figure 2 displays Σ ( x ) for four values of α . In both it can be seen clearlythat the respective index is very important in determining the overall behavior of the curves. TheS´ersic profile is characterized by a more steeper central core and extended external wing for largervalues of the S´ersic index m . For low values of m the central core is more flat and the externalwing is sharply truncated. The Einasto profile has a similar behavior, with the di ff erence that theexternal wings are most spread out. Also in the inner region for both profiles with low values ofthe respectively index we obtain larger values of Σ S and Σ . However, the Einasto profile seems tobe less sensitive to the value of the surface mass density for a given α and radius and in the innerregion than the S´ersic profile. It is in this region where the lensing e ff ect is more important and thedi ff erence in the surface mass density determines the lensing properties of the respectively profiles.Given this di ff erence, we see that the lensing properties of the S´ersic and Einasto profile are notequal. Studies of the lensing properties of the S´ersic profile had been done by Cardone (2004) andEl´ıasd´ottir & M¨oller (2007).
3. The total mass enclosed
The total mass enclosed in a halo described by the Einasto profile can be found by: M tot = π Z ∞ ρ ( r ) r dr (28)Combining equations (3) and (28), we get: M tot = πρ − r − e α α (cid:18) α (cid:19) α Γ α ! (29)This result was also obtained by Cardone et al. (2005). m =0.5 1 2 50 20 40 60 80 1000.01110010 Rad iu s l og S s The Sérsic Profile
Fig. 1.
S´ersic profile where Υ I e and R e are held fixed for four values of the S´ersic index m .We can get the same result calculating the total mass projected on the sky plane: M tot = π Z ∞ Σ ( ξ ) ξ d ξ (30)Inserting equation (19) into (30), we find: M tot = ρ − r − e pq q (2 π ) p − r pq Z ∞ G p + q − , q − , p + q − ab (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ′ q p (cid:0) x ′ (cid:1) q − dx ′ (31)Integrating the last equation using the formula (B.7) for indefinite integration of the Meijer G-function: M tot = ρ − r − e pq q (2 π ) p − r pq (cid:16) q p (cid:17) q Q p + q − j = Γ (cid:16) q + b j (cid:17)Q q − j = Γ (cid:16) q + a j (cid:17) (32)We can write both products appearing in the numerator and denominator in equation (32) using theGauss multiplication formula (13) respectively as: Q p + q − j = Γ q + b j ! = Q p − j = Γ (cid:18) q + j p (cid:19) Q q − j = Γ (cid:16) + jq (cid:17) Γ (cid:16) q (cid:17) = √ π (2 π ) p + q − (2 p ) − pq Γ (cid:16) pq (cid:17) q Γ (cid:16) q (cid:17) (33) Α =2 1 0.5 0.20 20 40 60 80 1000.01110010 Rad iu s l og S The Projected Ein asto Profile
Fig. 2.
Projected Einasto profile where ρ − r − and r − are held fixed for four values of the Einastoindex α . q − Y j = Γ q + a j ! = Q q − j = Γ (cid:18) + jq (cid:19) Γ (cid:16) q (cid:17) = √ π q − (2 π ) q − Γ (cid:16) q (cid:17) (34)Substituting equations (33) and (34) into (32) with α = qp we obtain the same result (29) for thetotal mass enclosed of the Einasto profile. This confirms that our calculations for the surface massdensity of the Einasto profile are correct.
4. The deflection angle, lens equation and the lensing potential
In the thin lens approximation, the lens equation for a axially symmetric lens is: η = D S D L ξ − D LS ˆ α (35)where the quantities η and ξ are the physical positions of the of a source in the source plane andan image in the image plane, respectively, ˆ α is the deflection angle, and D L , D S and D LS are theangular distances from observer to lens, observer to source, and lens to source, respectively.With the dimensionless positions y = D L η/ D S r − and x = ξ/ r − , and dimensionless α = D L D LS ˆ α/ D S ξ the lens equation reduces to: y = x − α ( x ) (36) The deflection angle for a spherical symmetric lens is (Schneider et al. 1992): α ( x ) = x Z x x ′ Σ ( x ′ ) Σ crit dx ′ = x Z x x ′ κ (cid:0) x ′ (cid:1) dx ′ (37)where κ ( x ) = Σ ( x ) / Σ crit is the convergence and Σ crit is the critical surface mass density defined by: Σ crit = c D S π GD L D LS (38)where c is the speed of light, G is the gravitational constant.Inserting equation (19) into (37), we find the deflection angle of the Einasto profile: α ( x ) = ρ − r − e pq (2 π ) p − q Σ crit r pq x G p + q − , q , p + q − q , ab , − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (39)Introducing the central convergence, κ c , a parameter that determinate the lensing properties of theEinasto profile, defined by: κ c ≡ Σ ( x = Σ crit = ρ − r − e pq Γ (cid:16) pq (cid:17) Σ crit q p ! pq − = ρ − r − e α Γ (cid:16) α (cid:17) Σ crit (cid:18) α (cid:19) α − (40)and use it to write α ( x ) in terms of κ c : α ( x ) = κ c (2 π ) p − Γ (cid:16) pq (cid:17) q r pq pq ! pq − x G p + q − , q , p + q − q , ab , − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (41)For Einasto index with values α = n and α = n with n integer, the last equation can be written as: α ( x ) = κ c (2 π ) α − Γ (cid:16) α (cid:17) √ α α ! α − x G α , , α + − , ab , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (42)The lens equation for the Einasto profile is then: y = x − κ c (2 π ) p − Γ (cid:16) pq (cid:17) q r pq pq ! pq − x G p + q − , q , p + q − q , ab , − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (43)which can be simplified to: y = x − κ c (2 π ) α − Γ (cid:16) α (cid:17) √ α α ! α − x G α , , α + − , ab , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (44)for Einasto index with values α = n and α = n with n integer.For a spherically symmetric lens being capable of forming multiple images of the source a su ffi cientcondition is κ c > κ c ≤ κ c > | y |≤ y crit (Li & Ostriker 2002), where y crit is the the maximum value of y when x < x >
0. For singular profiles such as the NFW profile, the central convergence always is divergent,hence the condition κ c > κ c > The deflection potential ψ ( x ) for spherically symmetric lens is given by: α ( x ) = d ψ dx (45)We see from equation (45) that can find the lensing potential simply integrating the deflectionangle: ψ ( x ) = Z x α (cid:0) x ′ (cid:1) dx ′ (46)Inserting the equation (41) into (46) and using the identity (B.7), we found: ψ ( x ) = κ c π ) p − Γ (cid:16) pq (cid:17) q r pq pq ! pq − x G p + q − , q + , p + q + − q , − q , ab , − q , − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (47)which can be reduced to : ψ ( x ) = κ c π ) α − Γ (cid:16) α (cid:17) √ α α ! α − x G α , , α + − , − , ab , − , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (48)for Einasto index with values α = n and α = n with n integer.
5. Magnification, shear and the critical curves
The gravitational lensing e ff ect preservers the surface brightness but causes variations in the shapeand the solid angle of the source. Thereby, the source luminosity is amplified by (Schneider et al.1992): µ = − κ ) − γ (49)where κ ( x ) is the convergence and γ ( x ) is the shear. The amplification has two contributions onefrom the convergence which describes an isotropic focusing of light rays in the lens plane and theother is an anisotropic focusing caused by the tidal gravitational forces acting on the light rays,described by the shear. For a spherical symmetric lens, the shear is given by (Miralda-Escude1991): γ ( x ) = ¯ Σ ( x ) − Σ ( x ) Σ crit = ¯ κ − κ (50)where¯ Σ ( x ) = x Z x x ′ Σ (cid:0) x ′ (cid:1) dx ′ (51)is the average surface mass density within x .The magnification of the Einasto profile can be found combining equations (19), (49), (50) and(51). We get: µ = [(1 − ¯ κ ) (1 + ¯ κ − κ )] − (52)where κ ( x ) = κ c (2 π ) p − Γ (cid:16) pq (cid:17) r pq pq ! pq − x G p + q − , q − , p + q − ab (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (53) ¯ κ ( x ) = κ c (2 π ) p − Γ (cid:16) pq (cid:17) q r pq pq ! pq − x G p + q − , q , p + q − q , ab , − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x q q p (54)The last equations reduce to: κ ( x ) = κ c (2 π ) α − Γ (cid:16) α (cid:17) √ α α ! α − x G α , , α ab (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (55)¯ κ ( x ) = κ c (2 π ) α − Γ (cid:16) α (cid:17) √ α α ! α − x G α , , α + − , ab , − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (56)for Einasto index with values α = n and α = n with n integer.The magnification may be divergent for some image positions. The loci of the divergent magnifica-tion in the image plane are called the critical curves. For the Einasto profile we see from equation(52) that has one pair of critical curves. The first curve 1 − ¯ κ = + ¯ κ − κ =
6. Summary and Conclusions
In this paper, we have derived an analytical expression for the surface mass density of the Einastoprofile using the Mellin transformed. This expression can be written in terms of the Fox H-functionfor general values of the Einasto index α . The same expression can be written in terms of the MeijerG-function for all rational values of the Einasto index, with a simplification for values α = n and α = n with n integer of the Einasto index. One we obtained an analytical expression for the surfacemass density we also derived in terms of the Meijer G-function other lensing properties: deflectionangle, lens equation, deflection potential, magnification, shear and critical curves of the Einastoprofile for all rational values of the Einasto index, with a simplification for values α = n and α = n with n integer of the Einasto index. Our analytical results can be used to investigate further thelensing properties of the Einasto profile taking advantage of the fact that the Meijer G-function isa very well studied function in the literature.We compared the S´ersic and Einasto surface mass density profiles using the equivalent values forthe S´ersic m and Einasto α indexes and where the quantities Υ I e , R e and ρ − r − , r − are held fixed.We found that both profiles have similar behavior determined by the index value. However, wenoted that for the Einasto profile the external wings are most spread out and seems to be lesssensitive to the value of the surface mass density for a given Einasto index and radius in the innerregion than the S´ersic profile. This feature is key because it is in this region where the lensinge ff ect is more important and the di ff erence of the surface mass densities implies a di ff erence in thelensing properties of the two profiles.Our results can be used in strong and weak lensing studies of galaxies and clusters where darkmatter is to believed the main mass component and the mass distribution can be assumed to be givenby the Einasto profile. The implementation of this results is easy because the Meijer G-function isavailable in several commercial and open-source CAS. The performance of this nonsingular three-parameter model in fitting the 3D spatial densities in high resolution N-body CDM simulations is better than the singular two-parameter NFW profile makes very promising its use in strong andweak lensing studies. The constant increasing computational power available opens the possibilityof using most realistic and sophisticated profiles like the Einasto profile for lensing studies andmarks a route to obtain a satisfactory solution to the cusp-core problem. Acknowledgements : The authors wish to thank H. Morales and R. Carboni for critical reading.This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
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Appendix A: The Meijer transform-method
The Mellin transform-method (Marichev 1982; Adamchick 1996; Fikioris 2007) uses the Mellinintegral transform for the integral evaluation.The Mellin transform of a function f ( z ) is an integral transform defined by: {M f } ( u ) = Z ∞ z u − f ( z ) dz (A.1)if the integral exits.It is clear from the definition that the Mellin transform does not exist for all functions such as thepolynomials, the integral does not converge. The Mellin transform when it does exits it convergesin a vertical strip in the complex z -plane. This strip is called the strip of analyticity (SOA). The inverse Mellin transform is defined by: f ( z ) = π i Z C z − u {M f } ( u ) du (A.2)where the contour of integration C is a vertical line in the complex z -plane and must be placed inthe SOA of f ( z ).Given two functions f ( z ) and g ( z ) the Mellin convolution is defined by:( f ⋆ g ) ( z ) = Z ∞ f ( y ) g zy ! dyy (A.3)It is well know that the Laplace or Fourier transform of the product of two di ff erent functions is theconvolution of the respectively transform. In the case of the Mellin transform we have: Z ∞ f ( y ) g zy ! dyy = π i Z C z − u {M f } ( u ) {M g } ( u ) du (A.4)if z = I ( z ) = Z ∞ f ( y ) g zy ! dyy (A.5)can be written as an inverse Mellin transform. With the requirement that f and g should be of thehypergeometric type and consequently their Mellin transforms can be written as products with theform Γ ( a + Au ) or [ Γ ( a + Au )] − with the A ’s being real numbers, the resulting integrals are of theMellin-Barnes type and then can be written in terms of the Fox H-function for A , A = Appendix B: The Meijer G function and its properties
The Meijer G-function is a very general, analytical function introduced by Meijer (1936) whichincludes most of the special functions as specific cases. It is defined in terms of the inverse Mellintransform (Erd´elyi 1954) by: G m , np , q ab (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ≡ G m , np , q a , ..., a p b , ..., b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = π i Z C Q mj = Γ (cid:16) b j + s (cid:17) Q nj = Γ (cid:16) − a j − s (cid:17)Q qj = m + Γ (cid:16) − b j − s (cid:17) Q pj = n + Γ (cid:16) a j + s (cid:17) z − s ds (B.1)where C is a contour in the complex plane, Γ ( s ) is the Gamma function and a and b are vectors ofdimension p and q , respectively.The basic properties of the Meijer G-function are too numerous to be mention here. We only pro-vide a short list of the most relevant properties for this work.A Meijer G-function with p > q can be transformed to another G-function with p < q : G m , np , q ab (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = G m , nq , p − b , ..., − b q − a , ..., − a p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (B.2) Other property is that if one the parameters of a and b appears in both the numerator and denomi-nator of the integrand, the order of the Meijer G-function may decrease and the fraction simplified.The positions of the parameters dictates which order m or n will decrease. For example if a k = b j for some k = , , ..., n and j = m + , m + , ..., q , the orders p , q and n of the Meijer G-functionwill decrease: G m , np , q a , a , ..., a p b , ... b q − , a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = G m , n − p − , q − a , ..., a p b , ..., b q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (B.3)In the other case if a k = b j for some k = n + , n + , ..., p and j = , , ..., m , the orders p , q and m of the Meijer G-function will decrease: G m , np , q a , ..., a p − , b b , b , ... b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = G m − , np − , q − a , ..., a p − b , ..., b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (B.4)The order reduction formula for the Meijer G-function is: G m , np , q a , ..., a p b , ..., b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = k + v + ( p − q ) / (2 π ) ( k − δ (B.5) × G km , knkp , kq a / k , ..., ( a + k − / k , ..., a p / k , ..., (cid:16) a p + k − (cid:17) / kb / k , ..., ( b + k − / k , ..., b q / k , ..., (cid:16) b q + k − (cid:17) / k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z k k k ( p − q ) The multiplication by powers of z is another property: z α G m , np , q a , ..., a p b , ..., b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = G m , np , q a + α, ..., a p + α b + α, ..., b q + α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (B.6)Among the indefinite and definite integrals of the Meijer G-function one has the following: Z G m , np , q a , ..., a p b , ..., b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α z z α − dz = z α G m , np , q − α, a , ..., a p b , ..., b q , − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (B.7) R ∞ G m , np , q a , ..., a p b , ..., b q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β z z α − dz = Q mj = Γ (cid:16) b j + α (cid:17) Q nj = Γ (cid:16) − a j − α (cid:17)Q qj = m + Γ (cid:16) − b j − α (cid:17) Q pj = n + Γ (cid:16) a j + α (cid:17) β − α (B.8)A short list of relations between the Meijer G-function and some elementary and special functionsis: G , , − b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = exp ( − z ) z b (B.9) G , , − b , b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = z ( b + b ) K b − b (cid:16) √ z (cid:17) (B.10)A more complete list can found in Bateman & Erd´elyi (1953) and the Wolfram Functions Site . http: // functions.wolfram.com / HypergeometricFunctions / MeijerG / The Fox H-function is a generalization of the Meijer G-function introduced by Fox (1961). It isdefined in terms of an Mellin inverse transform: H m , np , q ( a , A )( b , B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ≡ H m , np , q ( a , A ) , ..., ( a p , A p )( b , B ) , ..., ( b q , B q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = π i Z C Q mj = Γ (cid:16) b j + B j s (cid:17) Q nj = Γ (cid:16) − a j − A j s (cid:17)Q qj = m + Γ (cid:16) − b j − B j s (cid:17) Q pj = n + Γ (cid:16) a j + A j s (cid:17) z − s dsds