aa r X i v : . [ m a t h - ph ] S e p A FAMILY OF EXPONENTIAL INTEGRALSSUGGESTED BY STELLAR DYNAMICS
Luca [email protected] of Physics and Astronomy, University of Bolognavia Gobetti 93/3, I-40129 Bologna, Italy(September 14, 2020)
Abstract
While investigating the generalization of the Chandrasekhar (1943) dynamical fric-tion to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error func-tion occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010).Here we show that both the integrals are very special cases of the family of (real) functions I ( λ, µ, ν ; z ) := Z z x λ E ν ( x µ ) dx = γ (cid:16) λµ , z µ (cid:17) + z λ E ν ( z µ )1 + λ + µ ( ν − , µ > , z ≥ , (1)where E ν is the Exponential Integral, γ is the incomplete Euler gamma function, and forexistence λ > max {− , − − µ ( ν − } . Only in one of the consulted tables a relatedintegral appears, that with some work can be reduced to eq. (1), while computer algebrasystems seem to be able to evaluate the integral in closed (and more complicated) formonly provided numerical values for some of the parameters are assigned. Here we showhow eq. (1) can in fact be established by elementary methods.
1. Introduction
Two interesting integrals, that can be expressed in closed form in terms of the ErrorFunction and of the Exponential Integral, were encountered while generalizing the Chan-drasekhar (1943) dynamical friction formula to the case of a test mass moving in a fieldof stars with a power-law mass spectrum, and equipartition Maxwell-Boltzmann velocitydistribution (eqs. [30]-[31] in Ciotti 2010). They both belong to the family of functions ineq. (1): quite surprisingly, this simple-looking identity is not found in the most importanttables of integrals (e.g., Erd´elyi et al. 1953, Gradshteyn and Ryzhik 2007, Prudnikov etal. 1990), and neither the latest releases of Mathematica and Maple seem to be able torecover the general result, but only particular cases for numerical values of some of theparameters. In the following I show how the identity in eq. (1) can be established withelementary methods. 1 . Some preliminary material
For succesive use, we report the relevant identities obeyed by the Exponential Inte-grals. They are defined for ℜ ( z ) > ν ( z ) := Z ∞ t − ν e − tz dt = z ν − Γ(1 − ν, z ) , (2)(e.g., Abramowitz & Stegun, Chapter 5; Arfken & Weber 2005, Exercise 8.5.8; Erd´elyi etal. 1953, Vol.2, Chapter 9; see also https://functions.wolfram.com, https://dlmf.nist.gov).The last expression above, where Γ(1 − ν, z ) is the incomplete right Euler Gammafunction, is obtained with an obvious change of integration variable. The Euler incompleteleft and right Gamma functions (over the reals) can be expressed in integral form as γ ( a, x ) := Z x t a − e − t dt, Γ( a, x ) := Z ∞ x t a − e − t dt, (3)where ℜ ( a ) > γ function , therefore they obey the relations ofeasy proof: γ ( a + 1 , x ) = aγ ( a, x ) − x a e − x , Γ( a + 1 , x ) = a Γ( a, x ) + x a e − x . (4)and γ ( a, x ) + Γ( a, x ) = Γ( a ) = Z ∞ t a − e − t dt = γ ( a, ∞ ) = Γ( a, , (5)where Γ( a ) is the complete Gamma function.About the Exponential Integrals E ν , from their integral expression in eq. (2) it is asimple exercise to show thatE ( z ) = e − z z , d E ν ( z ) dz = − E ν − ( z ) . (6)Moreover, from integration by parts of eq. (2), by using the first and the second functionin the integrand as differential factor, for z = 0 one obtain respectivelyE ν ( z ) = e − z − z E ν − ( z ) ν − − z − ν E ν +1 ( z ) z , (7)where of course ν = 1 in the first identity. Finally, from standard asymptotic expansion itfollows that, at the leading order for z → ν ( z ) ∼ ν − , ν > , − ln z, ν = 1 , Γ(1 − ν ) z − ν , ν < , (8) As we do not use the continuation of the functions to the Complex plane, from nowon all the quantities are intended reals. 2nd in particular it follows that the divergence of the Exponential Integrals near the origingets worse for decreasing ν ≤
1, an obvious consequence of eq. (2). The leading-orderexpansions in eq. (8) will be used in the next Section to determine the limitations on thevalues of the parameters ( λ, µ, ν ) required for existence of the function I ( λ, µ, ν ; z ); nospecial difficulties are encountered to evaluate higher order terms, and they can be alsoused to check the consistency of the recursion identities in eq. (7) for z →
3. The parameter space
Before proceeding to prove eq. (1), it is convenient to determine the restrictions on thevalues of the parameters ( λ, µ, ν ) to assure existence of the function I . Equation (8) showsthat we must consider three different cases as a function of the value of ν (a generic realnumber), and in fact elementary integration shows that at the leading order for z → + and µ > I ( λ, µ, ν ; z ) ∼ z λ +1 ( λ + 1)( ν − , ν > , − µλ + 1 z λ +1 ln z, ν = 1 , Γ(1 − ν )1 + λ + µ ( ν − z λ + µ ( ν − , ν < , (9)provided the conditions λ > ( − , ν ≥ , − − µ ( ν − , ν ≤ {− , − − µ ( ν − } , (10)are satisfied (see Figure 1).We finally notice that an obvious and useful transformation of the function I in eq. (1)can be obtained with the change integration variable y = x r and r > I ( λ, µ, ν ; z ) = 1 r I (cid:18) λ − r + 1 r , µr , ν ; z r (cid:19) : (11)in particular, by setting r = µ it is always possible to reduce to the case of integration ofeq. (1) with E ν depending linearly on the integration variable, and this case is evaluatedby Mathematica. 3 igure 1 The region in the ( λ, µ ) parameter space for existence of the function I ( λ, µ, ν ; z ), as determined by eq. (10). For values of ν > ν the existence region reduces to thepoints above the dashed line, here represented for ν = 0 and ν = − /
2. Notice that for allpoints above the ν = 0 line, also the expression in eq. (15) can be used, and that the value ν = − / H in eqs. (16)-(17)when λ = µ = 2 (solid dot).
4. A proof of identity (1)
We are now in position to prove the indentity in eq. (1) by elementary methods. First,as µ >
0, and considering that from eq. (10) certainly λ > −
1, we can integrate by partswith x λ as differential factor, obtaining a recursion identity I ( λ, µ, ν ; z ) = z λ +1 E ν ( z µ ) + µ I ( λ + µ, µ, ν − z ) λ + 1 . (12)The first term follows from eq. (9) and the limitations in eq. (10), while the second termfrom the second identity in eq. (6). Then, from the first identity in eq. (6), and restricting(for the moment) to ν = 1 we have x µ E ν − ( x µ ) = e − x µ − ( ν − ν ( x µ ) . (13)We now multiply the identity above for x λ and integrate over x , so that µ I ( λ + µ, µ, ν − z ) = γ (cid:18) λµ , z µ (cid:19) − µ ( ν − I ( λ, µ, ν, z ) : (14)notice that the identity also holds for ν = 1, so we can relax the restriction ν = 1. Thereforewe have a second identity that can be used with eq. (12) to obtain the function I ( λ, µ, ν ; z )and finally prove eq. (1), QED. 4otice that the procedure is the same used (for example) in standard exercises tointegrate products of trigonometric functions and exponentials. The correctness of eq. (1)can be verified with some work from the second of eq. 1.2.1.1 of Prudnikov et al. (1990,Volume 2). In particular, first express the Exponential Integral in terms of the incompleteGamma function from eq. (2), then change the parameters in Prudnikov’s equation as λ → λ + µ ( ν − α → − ν , a →
1, and ν → µ , and finally combine two Gamma functionsin the incomplete γ function from indentity (5). Reassuringly, notice how the limitationson the parameters given in Prudnikov, once expressed in terms of our parameters, coincidewith those given in eq. (10).Note that for (1 + λ ) /µ >
1, i.e. λ > − µ , it is possible to apply the first recursionformula in eq. (4) to the the incomplete γ function appearing in eq. (1), and sucessivelyreduce the resulting formula from the second identity in eq. (7), obtaining I ( λ, µ, ν ; z ) = λ − µµ γ (cid:16) λ − µµ , z µ (cid:17) − νz λ − µ E ν +1 ( z µ )1 + λ + µ ( ν − . (15)Of course, if λ > − µ , the argument can be applied again to eq. (15), and so on, butthe resulting formulae become increasingly complicated (even if of trivial construction),and not reported here.With the aid of eq. (15) we can easily prove eqs. (30)-(31) in Ciotti (2010), thatwere derived by using “ad hoc” integration based on the properties of the Error function.Starting from eqs. (16)-(21)-(29) in Ciotti (2010), the two integrals to be solved can bewritten as H ( y ) := ac a √ π Z ∞ c r − ν dr Z y t e − rt dt, a > , c = 1 − a , (16)and the two functions H and H of interest in Stellar Dynamics correspond to ν = a − / ν = a − /
2, respectively. In the original work the integration was performedconsidering first the inner integral, and then integrating by parts over r a term involvingthe Error function. Here instead we invert order of integration in eq. (16) so that H ( y ) = ac a − ν − / √ π Z √ cy x E ν ( x ) dx = ac a − ν − / √ π I (2 , , ν ; √ cy ) , (17)where in the integral we changed variable as x = √ cy . As λ = 2 > − µ = 1, it is thenpossible to use eq. (15), and finally from the identity1 √ π γ (cid:18) , z (cid:19) = Erf( z ) , (18)eqs. (30)-(31) in Ciotti (2010) are recovered. Figure 1 immediately shows (solid dot) that a > H , and a > H .5 . Conclusions Prompted by a problem of Stellar Dynamics, an elementary derivation is presentedfor the closed-form expression of a family of indefinite integrals involving powers andExponential Integrals. Well known computer algebra systems seem unable to obtain theprimitive in closed form in the general case, and also for numerical values of (some) ofthe parameters the resulting formulae can be quite complicated and not easily simplifiedto the compact expressions in eqs. (1)-(15), though the numerical values are in perfectagreement. However, from the two last identities and eqs. (4), (7) and (11), it is expectedthat general and uniform simplification procedures for the integrals I ( λ, µ, ν ; z ) could beeasily implemented in computer algebra systems.
6. References
Abramowitz, M., and Stegun, I.A., 1970
Handbook of Mathematical Functions , NinthEdition (Dover, New York)Arfken, G.B., and Weber, H.J., 2005
Mathematical Methods for Physicists , SixthEdition (Elsevier Academic Press, Burlington, MA, USA)Chandrasekhar, S., 1943,
The Astrophysical Journal , , 263Ciotti, L., 2010 Proceedings of the Symposium Plasmas in the Laboratory and in theUniverse: Interactions, Patterns, and Turbulence , G. Bertin et al. eds, AIP Conf.Ser.,vol.1242, p.117Erd´elyi, A., Magnus, W., Oberhettinger, and F., Tricomi, F.G., 1953,
Higher Tran-scendental Functions , (McGraw-Hill, New York)Gradshteyn, I.S., and Ryzhik F.G., 2007,
Table of Integrals, Series, and Products -7th Edition , Alan Jeffrey and Daniel Zwillinger, Eds., (Elsevier, Burlington)Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I. 1990,