Fractional calculus, completely monotonic functions, a generalized Mittag-Leffler function and phase-space consistency of separable augmented densities

0.39 MB·English

by  J. An

Preprintver.2012January30(19pp)
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
FRACTIONALCALCULUS,COMPLETELYMONOTONICFUNCTIONS,AGENERALIZEDMITTAG-LEFFLER
FUNCTIONANDPHASE-SPACECONSISTENCYOFSEPARABLEAUGMENTEDDENSITIES
JinH.An
[email protected]
ABSTRACT
Undertheseparabilityassumptionontheaugmenteddensity,adistributionfunctioncanbealwaysconstructed
forasphericalpopulationwiththespecifieddensityandanisotropyprofile.Then,aquestionarises,underwhat
2 conditionsthedistributionconstructedassuchisnon-negativeeverywhereintheentireaccessiblesubvolume
1 of the phase-space. We rediscovernecessary conditionson the augmented density expressed with fractional
0 calculus.TheconditionontheradiuspartR(r2)–whoselogarithmicderivativeistheanisotropyparameter–is
2 equivalenttow 1R(w 1)beingacompletelymonotonicfunctionwhereastheconditiononthepotentialpartis
− −
an vstaaltueedfaosritthsedearniivsaottirvoepuypptaorathmeeoterdr)e.rWnoetaglrseoadteerrtivheanth23e−seβt0obfesinugffincoienn-tnecgoantdiviteio(nwshoenrethβe0issepthaeracbelnetraaulglmimeintitnegd
J densityforthe non-negativityof thedistribution,whichgeneralizesthe conditionderivedforthegeneralized
0 Cuddeford system by Ciotti&Morganti (2010) to arbitrary separable systems. This is applied for the case
3 whentheanisotropyisparameterizedbyamonotonicfunctionoftheradiusofBaes&VanHese(2007). The
resultingcriteriaarefoundbasedonthecompletemonotonicityofgeneralizedMittag-Lefflerfunctions.

]
h
p 1. MODELSFORSPHERICALDYNAMICALSYSTEM where
h- 1.1. Distributionfunction E=Ψ(r)− 12v2; L=kLk=rvt,
at Supposethat (r;ut)isaphase-spacedistributionsothat are the two isotropic isolating integralsadmitted by all such
F |
m potentials, namely, the specific bindingenergyandthe mag-
d3rd3u nitudeofthespecificangularmomentum,respectively.Here,
[ Z F
S
1 is the number of tracers in any measurable phase-space vol- Φ(rout) Φ(r) ifrout isfinite
−
3v uramtieonSsaptatciemaentd. uHe=rer˙risisththeevpeoloscitiitoyn. vWeectoonrliyncthoensciodnefirgthue- Ψ(r)≡ΦΦ(∞(r))−Φ(r) iiffrroouutt ==∞aanndd|ΦΦ((∞))|<∞
11 isnydsteepmenidneenqt.uTilhiberdiuismtriabnudtitohnuosfthaespdhisetrriicbaultliyonsymmumstetbreictipmope-- istherelative−potentialwithrespectto∞thebound∞ary→ro∞ut. The
6 ulationin a steadystate is also invariantundertransformsin systemnotboundedbyafiniteboundaryradiusisrepresented
01. SrˆO=(3r/)rsaorethtahteFra(dri;aul|td)ist=anFce(ar;nvdr,uknuittk)vewcthoerrewhril=evkr=kuan·drˆ bfiynitreo,utth=en∞(wit<h 0Φ,(L∞2))==0libmerc→au∞sΦeb(ry).deIfifnriotuitonor Φ(∞0)foisr
12 aadnodputtt=heuc−anvrorˆnaicraelthspehreardiciaallapnodlatrancgoeonrtdiainlavteelo(rc,iθti,eφrs.),Ifthweye alltracersbFounEdtothesystem(andboundedbyr ≤Ero≥ut).
: aregivenby 1.2. Augmenteddensitiesofasphericalsystem
v
Xi v=kuk=(v2r +v2t)12 ; vt =kutk=(v2θ +v2φ)21, veIloncteitgyrastpinacgethreesusplthseirnicaalbitvwaori-aintetefgurnacltdiofnFo(fEΨ,La2n)dorv2e,rthe
ar where (vr,vθ,vφ) = (r˙,rθ˙,rφ˙sinθ) are the velocity compo-
nentsprojectedontotheassociatedorthonormalbasis. N(Ψ,r2) d3u =Ψ 1v2,L2 =r2v2 , (1)
Inorderforthedistributiontobeindeedtime-independent, ≡$ F E − 2 t
(cid:0) (cid:1)
itmustbeinvariantunderdynamicevolutionsoftracers,that
whichisreferredtoastheaugmenteddensity(AD).Theinte-
is,thedistributionisatime-independentsolutiontotheBoltz-
gral here is formally over the whole velocity subspace, but
mann1transportequation.Fortypicalstellardynamicalappli- if r or Φ( ) is finite, it is essentially within the sphere
out
cations,thetrajectoryofeachtracerisitsorbitundertheexter- v2 2Ψsinc∞e ( < 0,L2) = 0 for these cases. With Ψ(r)
nalpotential,whichmayormaynotbeself-consistentlygen- ≤ F E
specified,theADyieldsthelocaldensityν(r)via
eratedbythetracerpopulation.Thetransportequationforthis
case results in the collisionless Boltzmann equation (CBE), ν(r)=N[Ψ(r),r2].
whose solution is completely characterized by the theorem
duetoJ.Jeans2. TheJeanstheoremindicatesthatifthegiven Similarly,theaugmentedmomentfunctionsaregivenby
time-independentspherically-symmetricdistributionfunction
(df)isasolutiontoCBEwithagenericstaticsphericalpoten- m (Ψ,r2) d3uv2kv2n =Ψ 1v2,L2 =r2v2
tialΦ(r),itmustbeintheformof k,n ≡$ r t F E − 2 t
(cid:0) (cid:1)
F =F(E,L2) =4π " dvrdvtv2rkv2tn+1F Ψ− v2r +2v2t ,r2v2t . (2a)
1LudwigEduardBoltzmann(1844-1906) (v2 2Ψ) (cid:16) (cid:17)
2SirJamesHopwoodJeans(1877-1946) vr≥0≤,vt≥0
2 J.An
Changingtheintegrationvariablesto( ,L2),thesearerepre- ThebasiccompositerulefortheRiemann-Liouvilleopera-
E
sentedtobeasetofintegraltransformationsofthedf, torsisthat,foranypairofnon-negativerealsλandξ,
2π + ξ + λf =+ ξ+λf, (6)
mk,n = r2n+2" dEdL2Kk−12L2nF(E,L2) a>x (cid:16)a>x (cid:17) a>x
whichmaybeshownbydirectcalculationsusingtheFubini8
T
= r22nπ+2" dEdL2Θ(K)|K|k−12L2nF(E,L2). (2b) fthuenocrtieomn,atnhdattihse, Euler9 integralof the first kind for the beta
E≥E0,L2≥0
x y
HereΘ(x)istheHeaviside3unit-stepfunctionand dy(x y)ξ 1 dw(y w)λ 1f(w)
− −
Z − Z −
a a
0 ifr orΦ( )isfinite
out x x
E0 ≡(−∞ iflimr→∞Ψ∞(r)=−Φ(∞)→−∞ =Za dw f(w)Zw dy(x−y)ξ−1(y−w)λ−1
isthelowerboundofthebindingenergy. Thetransformker- x 1
nelandthedomainin( ,L2)spaceoverwhichtheintegralis = dw f(w)(x w)ξ+λ−1 dt(1 t)ξ−1tλ−1.
performedaregivenbyE Za − Z0 −
Nextforanyrealλandanon-negativeintegern
( ,L2;Ψ,r2) 2(Ψ ) r 2L2,
−
K E ( ,L2) ≡ ,−L2E −0, 0 . d + λf =+ λ−1f ; dn + λf =+ λ−nf. (7)
T ≡{ E |E≥E0 ≥ K ≥ } dxa>x a>x dxna>x a>x
Note isv2expressedasafunctionof4-tuple( ,L2;Ψ,r2). Here the latter follows the former (n = 1) by means of in-
K r E duction. Then = 1caseisprovenbydirectdifferentiationof
2. MATHEMATICALPRELIMINARY equation (3) for λ > 1 and the fundamentaltheorem of cal-
2.1. Fractionalcalculus culusforλ = 1whilethesamecasewithλ < 1isessentially
trivial from the definitions of fractional derivatives in equa-
Definition2.1 The Riemann4-Liouville5 integral operatorof
tions(4)and(5). Togethertheyalsoindicatethat
arbitrarynon-negativerealorderλ 0isgivenby
+a>xλf ≡Γf((1xλ))Zax(x−y)λ−1≥f(y)dy ((λλ=>00)), (3) for non-negat+aiv∂xeξ(cid:16)r+ae>axlsλfλ(cid:17),=ξ +a+a>∂0xxλξa−−nλξdff ar((bξξit≤≥rarλλy))f,unction f((x8)),
whereΓ(x)isthegammafunction. provided that all the integra≥ls in their respective definitions
absolutelyconverge.Nextweobserveforλ 0that
≥
Tpehaisteidsiantterigvriaatliognesn.erFaolriz0at<ionλo<ft1h,ethCiasuicshayls6oforermcouglnaifzoerdraes- +a xλ+1f′ =+a xλf − (xΓ−(λa)+λ1f()a), (9a)
thegeneralizedAbel7transformwiththeclassicalcasecorre- > >
spondingtotheλ= 1 case. Wealsodefine thanks to the fundamental theorem of calculus (λ = 0) and
2 integrationbypart.Bymeansofinduction,thisgeneralizesto
Definition2.2 thefractionalderivativeforλ 0suchthat
≥ + λ+nf(n) =+ λf n−1 (x−a)λ+kf(k)(a), (9b)
+a∂xλf ≡ ddx⌈λλ⌉ +a x⌈λ⌉−λf a>x a>x −Xk=0 Γ(λ+k+1)
⌈ ⌉ > wherenisanynon-negativeinteger,andwealsofindthat
1 dλ x f(y)dy
=Γdλ(1f(−y){λ})d=x⌈⌈λf⌉⌉(λZ)(ax)(x−y){λ} ((0λ<={λ0})<1) (4) ddxnn+a>xλf =+a>xλf(n)+Xk=n1 (x−Γ(a1)ξ+−kλf−(n−kk))(a) (10)
where λ, λdx,λan(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dy=xλ = λ λ arethein{te}gerceiling,the ffroarcλtio≥n0aladnedriavnaytivneosni-nneegqautaivtieonin(t4eg)earren.alTtehrenlaatsivteimlypgliiveesnthbayt
⌈ ⌉ ⌊ ⌋ { } −⌊ ⌋
integerfloorandthefractionalpartofλ,respectively.
Noteequation(4)isageneralizationofthedifferentiationfor +a∂xλf = ddx⌈λλ⌉−nn+a x⌈λ⌉−λf(n)+ n−1 (xΓ−(1a)+k−kλf(kλ)()a) (11)
positiverealorderasisequation(3)oftheintegration. These ⌈ ⌉− > Xk=0 −
definitionsextendtoincludeanegativeindexusing whereλ>0andn=0,1,..., λ .
⌈ ⌉
Usingtheseandequation(10),wecanalsoderivethat
Definition2.3 forarbitraryrealλ
λ
+a>x−λf =+a∂xλf andviceversa. (5) +a>xξ+a∂xλf =+a>xξ−λf −Xk⌊=⌋1Cξ+,k+a∂xλ−kf(a)(x−a)ξ−k
(12)
34OGleiovregrFHreieadvrisicidheB(e1r8n5h0a-r1d9R25ie)mann(1826-1866) +a∂xξ+a∂xλf =+a∂xξ+λf − ⌊λ⌋ Cξ−,k+a(x∂xλ−akf)k(+aξ)
5JosephLioville(1809-1882) Xk=1 −
6Augustin-LouisCauchy(1789-1857) 8GuidoFubini(1879-1943)
7NielsHenrikAbel(1802-1829) 9LeonhardEuler(1707-1783)
J.An 3
for non-negative reals λ,ξ 0 and arbitrary function f(x), Corollary2.7 if f(x)isright-continuousatx=aand f(a)is
providedagainthatallthein≥tegralsintheirrespectivedefini- finite,then+ λf(a)=0forλ>0.
tionsabsolutelyconverge.HereC aregivenby a x
ξ±,k >
Nextweexaminethebehaviorsoffractionalcalculusoper-
(ξ)
−k (+case) atorsundertheLaplace11 transform. Forthis, wefirstnotea
1 Γ(1+ξ) generalpropertyoftheLaplacetransformofthederivative,
whereC0ξ±,k =δΓ=(1ξ±ξξ−k<)1=ist(h−e1f)Γr⌊(ξa1⌋c+t−ki(oδδn))a+⌊ξl⌋+pkarto(−fξc,aasned) whicshn+i1sxL→vsa[lfid(xg)]iv=enxL→ths[aft(nl+im1)(xx)]+e−Psxnjf=(0n)s(jxf)(n−=j)(00)f,or su(1ffi6-)
≤ −⌊ ⌋ ciently large s (which is requir→e∞d for the Laplace transform
(a)−n = nj=1(a+1− j); (a)+n = nj=1(a−1+ j) toconverge).Equation(16)isprovenviaintegrationbypart,
Q Q
arethePochhammer10 symbol. Thefallingproduct(a)−n fol- ∞dxe sxdf(x) = f(0)+s ∞dxe sxf(x)
lows the combinatorist’s convention whereas the rising one Z − dx − Z −
(a)+doestheanalyst’s. Notethesearerelatedtoeachother, 0 0
n forn = 0andtheinductioncompletesitsproofforanynon-
(−a)−n =(−1)n(a)+n; (a)−n =(a−n+1)+n negative integer. In order to generalize equation (16) to in-
cludethefractionalderivative,wenextconsiderforλ 0
andalsotothegammafunctions,
≥
x
(a)+ = Γ(a+n); (a) = Γ(1+a) . ∞dxe−sx dy(x y)λ−1f(y)
n Γ(a) −n Γ(1+a n) Z0 Z0 −
−
Thelast maybe used to generalizethe Pochhammersymbol = ∞dy f(y) ∞dx(x y)λ 1e sx
− −
fornon-integern.Togetherequations(8)and(12)providethe Z0 Zy −
generalizationofequation(6)foranypairofrealsξandλ. e sy
Thesimplestspecificresultoffractionalcalculuswouldbe =Z ∞dy f(y) s−λ Z ∞duuλ−1e−su.
0 0
Lemma2.4 forrealλandα>0,
WiththeEulerintegralofthesecondkindforthegammafunc-
+ λxα 1 = Γ(α)xα+λ−1. (13) tion,wefindthat
0>x − Γ(α+λ) sλ + λf(x) = [f(x)]. (17)
Thisisformallyageneralizationoftheresult,namely xL→sh0>x i xL→s
TheLaplacetransformofanarbitraryreal-orderderivativeis
dnxα
=(α) xα n (n=0,1,...) (14) thenfoundbycombiningequations(16)and(17).
dxn −n −
althoughthelastisinfactvalidforanyα. 2.2. Post–Widderformula
Weformalizeanobviousbutimportantfact,namely Theorem2.8(Post–Widder) If φ(t) is continuous for t 0
≥
andthereexistreal A>0and b>0suchthat
Lemma2.5 for λ > 0 and x > a, if f 0 in [a,x], then ∃ ∃
+a xλf(x) ≥ 0. Moreover+a xλf , 0provid≥edthatthesupport e−bt|φ(t)|≤ A for∀t>0,
o>f f in(a,x)hasnon-zero>measure.
thentheLaplacetransform
Next,ifaisfinite,thenforξ >0
f(x)= [φ(t)] ∞dte xtφ(t). (18)
+a>xξf = (xΓ−(ξa))ξZ01dttξ−1f x−(x−a)t , convergesandisinfint→Litexlydiffe≡reZn0tiable−for x> b. Moreover,
(cid:2) (cid:3)
whilefor0<λ= ξ <1andn=1,equation(11)resultsin φ(t)fort > 0maybeinvertedfrom f(x)usingthedifferential
− inversionformula(Post1930;Widder1941),
+a>xξf =+a∂xλf = Γ(1f(−a)λ)(x−a)−λ++a>x1−λf′. φ(t)=Lx−1t[f(x)]=nlim (−n1!)n nt n+1f(n) nt . (19)
Itthenfollowsthat → →∞ (cid:16) (cid:17) (cid:16) (cid:17)
Lemma2.6 fora, , Inliterature,thelastformulaistypicallynamedafterE.Post12
±∞ ortogetherwithD.Widder13. Arigorousproof,whichisbe-
lim +a>xξf(x) = f(a) , (15) yoonntdhethLeaspclaocpeetorafnthsfisorpmap.eHr,omweavyebreitfsohuenudriisntiacsjutasntidfiacradtitoenxst
x→a+ (x−a)ξ Γ(ξ+1) aboundandareeasytoobserve. Forinstance,directcalcula-
which is valid for ξ 0 if f(x) is right-continuousat x = a tionsusingequation(18)indicatethat
or for ξ 1 if f(x)≥is right-differentiableat x = a. Equa-
ctioonnti(n1u5i)t≥yfow−rhξil=ef0orisξe=quiv1a,lietnbtetcoomtheesdleimfinxitiao+n(xofath)ef′r(ixg)h=t- f(n)(x)=(−1)nZ0∞dttne−xtφ(t)= (x−n1+)1nZ0∞dssne−sφ(cid:16)xs(cid:17).
0whichholdsif f′(a)is−finite. Equation(15→)impl−iesthat 11Pierre-SimonLaplace(1749-1827)
12EmilLeonPost(1897-1954)
10LeoAugustPochhammer(1841-1920) 13DavidVernonWidder(1898-1990)
4 J.An
andthuswefindthat Notethen
(−1)n n n+1f(n) n = ∞dsP(s;n)φ st . m=0mjm =n−k
n! (cid:16)t(cid:17) (cid:16)t(cid:17) Z0 (cid:16)n (cid:17) andthus jm ≥ 0indicPatesthat jm = 0form > n−k (n.b.,if
otherwise, mj > n k, which is contradictory). The
where m=0 m −
sn property6fPollowsthisbecause
P(s;n) e s
≡ n! − n−k− m=0 j2m+1 =2 m=0m(j2m+ j2m+1)
is the probability density of the Poisson14 distribution with iseven.ThatistPosay,if f iscm,PtheparityoftheBellpolyno-
a mean of s¯ = n. It follows that as n , the relative mialinequation(22)is( 1)n k,andthus,giventhatgisalso
dispersiondecreasesandsoφ(st/n) φ(s¯→t/n∞) = φ(t),which cm,theparityofeveryte−rmin−thesumontheright-handside
results in the Post–Widder formula.→Note however that the ofequation(22)is( 1)n. Equation(22)alsoindicatesthat
−
convergence of equation (19) by itself does not necessarily
dnexp[f(t)]
implythat f(x)istheLaplacetransformationofφ(t),whichis =exp[f(t)] B f (t), f (t),..., f(n k+1)(t)
ratherapartoftheconditionfortheformulatobevalid. dtn · n ′ ′′ −
(cid:2) ((cid:3)23)
2.3. Completelymonotonicfunctions whereBnisthen-thcompleteBellpolynomial,thatis,
Definition2.9 A smoothfunction f(x) of x > 0issaidto be B (x ,...,x )= n B (x ,...,x ).
n 1 n k=1 n,k 0 n k
completelymonotonic(cmhenceforth)ifandonlyif −
Note P
(−1)nf(n)(x)≥0 (x>0, n=0,1,2,...). (20) n− m=0 j2m =2 m=0m(j2m−1+ j2m).
is even. HencPe if f is cm,Pthe parity of the complete Bell
Thedefinitionextendsto x 0if f(x)isright-continuousat
x=0. Somebasicproperties≥ofcmfunctionsare: polynomialinequation(23)is( 1)nandsoexp(f)iscm.
The archetypal example of a−cm function is f(x) = e x.
−
Lemma2.10 Let f andgbecm. Then, Otherelementaryexamplesofcmfunctionsinclude:
1. ( 1)nf(n) foranynon-negativeintegerniscm. 1. f(t)=t δ(t>0)iscmifandonlyifδ 0.
− − ≥
2. IfF ≥0in(0,∞)and f =−F′,thenF iscm. 2. f(t)=ln(1+t−1)iscm.
3. ∞f(y)dyisacmfunctionofxifitconverges. Theseareproventhrough
x
4. Raf+bgwhereaandbarenon-negativeconstantsiscm. dnx−δ =( δ) x δ n =( 1)n(δ)+n, (24a)
dxn − −n − − − xn+δ
5. f giscm.
· dn+1ln(1+x−1) =( 1)n+1n! 1 1 . (24b)
6. IfF >0in(0,∞)and f = F′,theng◦F iscm. dxn+1 − (cid:20)xn+1 − (1+x)n+1(cid:21)
7. exp(f)iscm. FollowingthisandLemma2.10are
Items 1 and 2 are essentially trivial from Defintion 2.9 and Corollary2.11 Let g(t) be cm, then both t δg(t) with δ 0
−
item3issimplyaparticularcaseofitem2. Item4followsthe andg(tp)with0< p 1arecm. ≥
linearity of differentiations while item 5 is shown using the proof. Thefirstisobv≤iousthankstoLemma2.10-5. Thelast
Leibniz15rule,thatis,(here n isthebinomialcoefficient) follows Lemma 2.10-6 with F(t) = tp since F = ptp 1 for
k ′ −
(cid:16) (cid:17) 0< p 1iscm. q.e.d.
dn(f g) n n dkf dn kg ≤
(−1)n dxn· =Xk=0 k!(−1)kdxk (−1)n−kdx−n−k. (21) Corollary2.12 For0< p≤1anda,b≥0,thesearecm:
f(t)=t a(1+tp) b; f(t)=t a(1+t p)b.
ThelasttwomaybeshownusingtheFaa` diBruno16formula − − − −
(i.e.,thegeneralizedchainrule), proof.LetF(t)=c+tp. ThenF′ = ptp−1iscmfor0< p 1.
Hencefirst(g F)(t) = (1+tp) b withc = 1andg(w)=≤w b
(g◦F)(n)(t)=Pnk=0g(k)(cid:2)F(t)(cid:3)·Bn,k(cid:2)f(t), f′(t),..., f(n−k)((t2)(cid:3)2) bfolrn(01<+pw−≤1)1,◦waendfinbd≥th0atis(gcm.F−N)(etx)t=,wbitlhn(c1=+t0−pa)nidsgc(mw)fo−=r
whereF′(t)= f(t)andBn,kistheBell17polynomial,thatis, 0< p 1andb 0,andsois◦(1+t−p)b =exp[bln(1+t−p)].
Thefin≤alconclus≥ionfollowsCorollary2.11. q.e.d.
B (x ,...,x )= ′ n! x0 j0 x1 j1 .
n,k 0 n k
− (j0X,j1,...) j0!j1!···(cid:18)1!(cid:19) (cid:18)2!(cid:19) ··· (BTerhnestefiunn1d9a2m8e;nWtaildderers1u9lt41)chisardaucetetroizSin.gBercnmsteinfu18n,ctions
Herethesummationisoverallsequences(j , j ,...)ofnon-
0 1
negativeintegersconstrainedsuchthat Theorem2.13(Hausdorff–Bernstein–Widder)
A smooth function f(x) of x > 0 is completely monotonic if
j =k; (m+1)j =n.
14Sime´onDePnims=P0oismson(1781-1P84m0)=0 m asunrdeoonnly[0i,f f()x,)th=atRi0s∞,teh−exrtedeµx(its)tswahenroenµ-n(te)giastitvheedBisotrreilbumteioan-
∞
15GottfriedWilhelmLeibniz(1646-1716) φ(t) 0oft>0suchthatequation(18)holds.
16FrancescoFaa`diBruno(1825-1888) ≥
17EricTempleBell(1883-1960) 18Serg´ei˘ Nata´noviq Bernxt´ei˘n(1880-1968)
J.An 5
The‘if’-partiselementarysince Notethatfortheδ = 0,thelastresultsinequation(26). The
middle for the same case is consistent with the fundamental
f(n)(x)=( 1)n ∞dttne xtφ(t)=( 1)n [tnφ(t)]. theoremofcalculusgivenequation(26)indicating
−
Although the co−mpleZte0 proof of the ‘on−ly if’t→L-pxartis beyond xn+1f(n)(x)=✘xn+✘1f✘(n)✘(x)✘x✘=0+Z xynf(n+1)(y)dy (28)
ourscope,thepartialprooffollowsthePost–Widderformula. (cid:12) 0
(cid:12)
oTfhaatcism, iffutnhcetiionnvefrs(ex)Liasplwaceell-tdraenfisnfeodr,mthφe(nt)e=quLat−xio→1nt[(f1(x9))], providedthat f(n)(0)isfinite.(cid:12)Equations(26)and(27)imply
provided that it converges, indicates that φ(t) must be non- Corollary2.14 for a non-negativeintegern, if f (x) 0
(n+1)
negativeinthepositiverealdomain. forx>0,then+∂µ(xµf) 0forx>0andn µ n+1.≥
0 x ≥ ≤ ≤
2.4. Miscellaneous Infact, the successiveapplicationsofthiswitha descending
subscript furthermore suggest that, if f (x) 0 for x > 0
Wenoteanadditionalauxiliaryrelation,whichwillbeused andanon-negativeintegern,itfollowst(hn)at+∂≥µ(xµf) 0for
throughoutthispaper: thatis, foranynon-negativeintegern 0 x ≥
andarbitrarydifferentiablefunction f(x), x>0andany∀µ n.
≤
Corollary2.14withanintegerµmaybegeneralizedalter-
x2d n(xf)= xn+1dn(xnf), (25) natively,namely,
(cid:18) dx(cid:19) dxn
Theorem2.15 for a non-negative integer n, if xaf (x) is
whichmay beprovenvia theinductiononn (see An 2011b, (n+1)
cm,thenxaf (x)isalsocm.
theoremA3).Infactthisisalsoequivalenttoalemma (n)
proof. If xaf is cm, then bythe Bernstein theorem, there
(n+1)
xnf(n+1)(x)= d xn+1f(n)(x) (26) existsanon-negativefunctionh(u)≥0ofu>0suchthat
dx
where (cid:2) (cid:3) xaf(n+1)(x)= ∞due−xuh(u).
dn[xnf(x)] Z0
f (x) .
(n) ≡ dxn The complete monotonicity of xaf can then be shown di-
(n)
Thislemmamaybeprovendirectlyvia rectlyusingequation(28),whichindicatesthat
dn d(x xnf) dn d(xnf) x 1
f(n+1)(x)= dxn(cid:20) d·x (cid:21)= dxn(cid:20)xnf +x dx (cid:21) xaf(n) = xa−n−1Z0 dyynf(n+1)(y)=Z0 dttn−aZ0∞due−xtuh(u),
= dn(xnf) + n n dkx dn−k d(xnf) dk[xaf(n)] =( 1)k 1dttn+k a ∞due xtuukh(u) (cid:3).
dxn Xk=0 k!dxkdxn−k(cid:20) dx (cid:21) dxk − Z0 − Z0 −
d dn[xnf(x)] 1 d
= 1+n+x = xn+1f (x) Finally,wealsonote
(cid:18) dx(cid:19) dxn xndx (n)
wherewe also used thatdkx/dxk = 0 if k 2(cid:2)andthe Lei(cid:3)b- Lemma2.16 foranon-negativeintegern,if f(n+1)(a)isfinite
nizrule(eq.21). Thetheoreminequation≥(25)implyingthe and f(0)(a) = ··· = f(k)(a) = 0, then +a∂xn+δf(a) = 0 for
lemmainequation(26)hasbeenshowninAn(2011b,corol- 0 δ<1.
≤
lary A4) whereas the opposite implication may be deduced proof. Hereweassumea=0,butthesimilarargumentholds
becausetheinductionstepfortheproofofequation(25)fol- foranyfinite“a”accompaniedbyasimpletranslation.First,
lowsequation(26)as
dn+d1(xxnn++11f) = x1nddx(cid:20)(cid:16)x2ddx(cid:17)n(xf)(cid:21)= xn1+2(cid:18)x2ddx(cid:19)n+1(xf). +0>x11−δf =1Γd(nx1+11−−[δyδ1)Zδ0f1(y(f)1(]x−t)td)δtt;n+δdt (29a)
(26F)ragcetinoenraalliczaelsc.ulIunspaalsrtoicguelnare,rafolirzeasnthoen-lnemegmataivienienqteugaetironn +0∂xn+δf = Γ(1−δ)Z0 dyn−+1 (cid:12)(cid:12)(cid:12)y=xt(1−t)δ. (29b)
and0 δ<1, Herethelatterfollowstheformerbecaus(cid:12)(cid:12)e
≤
+0>x1−δ(xn+δf)= Γ(11−δ)Z0xyn(+xδ−f(yy))δdy (27a) dn+1[dxx1−n+δ1f(xt)] =tn+δdn+1[dyy1n−+δ1f(y)](cid:12)(cid:12)y=xt.
xn+1 1tn+δf(xt)dt (cid:12)(cid:12)
= Finally,giventheLeibnizrule, (cid:12)
Γ(1 δ)Z (1 t)δ
− 0 −
+0∂xn+δ(xn+δf)= Γ(11 δ)Z 1(d1ttn+t)δδdn+1[dxxn+n+11f(xt)] dn+1[dyy1n−+δ1f(y)] =y1−δf(n+1)(y)
− 0 −
= 1 xyn+δf(n+1)(y)dy (27b) +(1 δ) n ( 1)n k n+1 (δ)+ f(k)(y),
xn+1Γ(1−δ)Z0 (x−y)δ − Xk=0 − − k ! n−kyn+δ−k
= xn1+1 +0 x1−δ xn+δf(n+1)(x) which identically vanishes for y = 0 if the conditionpart of
> Lemma2.16witha=0holds.Heretheconclusionfollowsas
(cid:2) (cid:3)
xn+δf(n+1)(x)=+0∂x1−δ xn+1+0∂xn+δ(xn+δf) . (27c) theintegrandofequation(29b)with x=0isalsozero. q.e.d.
(cid:2) (cid:3)
6 J.An
3. FRACTIONALCALCULUSONTHEAUGMENTEDDENSITY whichare valid for any s > 1 and λ 0, providedthat all
− ≥
integralsontheright-handsidesconverge.
An (2011a) has shownthatthe Abeltransformationof the
We next find differentiationsof the integraltransform I ,
augmentedmomentfunctionofananisotropicsphericalsys- s
namely(hereX Ψorr2)
tem results in a similar integral transformation of the df as
≡
equation(2b)butwithdifferentpowerson andL2. Thisre- ∂
tesopauqfablWuittlrhaigesoitshiefosnsnmnteeaocror(natt3ril-eo)ibnzngyteeogeisscnaebotetqiyonrvuaseemalistdrrieteeeoaaarsbnnilulnssi(lgst20oshbf.t≤)toWthhoµeaeinmtp≤hfpΨr(alξsyc.oeteritoherne2qa.sil.nIKc3tnea6glfcaaruacnlltdu,os3w.p9eeT)rhcfaoaetorngraoenosay-fl Th∂e∂IX1s-f=actos12rZ"f0oL¯Tr2tddhLEe2ds∂∂LK=X2K0(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Ksc=−a01sG∂eK(cid:16)XiΨsGd−ue2Lrt2o2,L2(cid:17) ((ss>=00)). (31)
2
Is(Ψ,r2)≡" dEdL2KsG(E,L2) δ(K)= 21δ Ψ− 2Lr22 −E
T h i
where the Ψ and r2 dependencies of the integrable function whereδ(x)=Θ(x)istheDiracdelta. Inaddition,
′
G = G( ,L2)areonlythroughthetwointegralsofmotion
and L2 (Ehenceforththese trivial argumentsofG will be supE- L¯2 = 2r2Ψ ifE0 =0 .
pressedforthesakeofbrevity),theFubinitheoremimplies ( if 0 =
∞ E −∞
+ λI = d dL2G+ λ sΘ( ) , Giventhat
E+0>Ψλ sIs "=E≥E0,L2≥0 E d dLE02>GΨ+(cid:2)Kλ KKsΘ(cid:3)(K) , ∂∂KΨ =2; ∂∂Kr2 = Lr42,
0>r2 (cid:18)r2λ+2(cid:19) "E≥E0,L2≥0 E 0>r2 (cid:20) r2λ+2 (cid:21) equation(31)suggeststhatforanintegern≥0ands>−1,
otThfheAroE+0n>uurgl(2e2hλr(cid:16)0dri1ni21rstIeeacg,str(cid:17)aacp=laplfce"ounrldtaEihtx≥ieEoA0bn,Le)s2t≥etah0xfadcutEenapcdrtteLifo2obGnra,sd+0wi>icffear2elfiλlry(cid:2)enrndi2dtsteKahnrastgtiΘucam(Kleton)(cid:3)t.tshoaft ∂∂nΨIns =22nsΓ(s()1−n+"sT)Zd0EL¯2ddLL22GK(cid:16)sΨ−n−G2Lr22,L2(cid:17) ((nn=< ss++(113))2,a)
+ λ sΘ( ) r4 ∂ nI
E0>Ψ (cid:2)KΘ(K)(cid:3) Ψ L2 s (cid:18) ∂r2(cid:19) s
== ΘΓ((KKλ))ZKEs++2Lλr22Bd(Qλ,(sΨ+−1Q),)λ−1(cid:20)2(Q−E)− r2(cid:21) =(Γs()1−n"+sT)dEL¯d2dLL22KL2ss−+n2LG2nGΨ L2 ,L2 ((nn<= ss++11)).
+ λ KsΘΓ((Kλ)) 2λ  2 Z0 (cid:16) − 2r2 (cid:17) (32b)
0>r2 (cid:20) r2λ+2 (cid:21) Equations(30),(32)andN = m0,0 expressedasanintegral
Θ( ) r2 (r2 R2)λ 1 L2 s transformationofthedfasinequation(2b)resultin
= K dR2 − − 2(Ψ )
Γ(λ) Z2(QL2 ) R2λ+2 (cid:20) −E − R2(cid:21) ∂n + ξ−12 N (33)
= ΘΓ((Kλ))r2λ−L−2E2Kλs+λB(λ,s+1), ∂Ψn(cid:20)0>2rn2+1π(cid:16)23rr22ξξ−−13(cid:17)(cid:21) d dL2Kξ−n−1 ( ,L2) (n<ξ)
+0>r2λ(cid:2)r2sΘK(sΘ)(K)r(cid:3)2 L2 s =2ξΓπ(23ξr2−ξ−n3) "L¯2 dTL2E ΨL2ξ−L12 ,FL2E (n=ξ),
= Γ(Kλ) Z2(QL2−E)dR2R2s(r2−R2)λ−1(cid:20)2(Ψ−E)− R2(cid:21) r4 ∂n r2+ ξ−Z210N L2ξ−1F(cid:16) − 2r2 (cid:17) (34)
= Θ(K) Ks+λ B(λ,s+1). (cid:18) ∂r2(cid:19) (cid:16) E0>Ψ (cid:17)
Γ(λ) 2λ(Ψ−E)λ 232−ξπ23 d dL2 ξ n 1L2n ( ,L2) (n<ξ)
Hence,wehaveestablishedthat Γ(ξ n)" E K − − F E
+ λI = Γ(s+1) d dL2 s+λG, (30a) =212−ξ−π23 L¯2dTL2L2ξ Ψ L2 ,L2 (n=ξ).
E+0>Ψλ sIs 2λ=Γ(rs2+λ−λ2Γ+(s1+)"1)T Ed dKL2Ks+λG, (30b) where n is agaiZn0a non-neFga(cid:16)tive−i2nrte2ger(cid:17)and ξ ≥ 21. Both
0>r2 (cid:18)r2λ+2(cid:19) Γ(s+λ+1)" E L2λ equationsfurthergeneralizefromanintegerntoarealµ ≤ ξ
T using fractional order derivatives, and it can also be shown
+ λ r2sI = r2(s+λ)Γ(s+1) Ks+λGdEdL2, (30c) thattheyareinfactvalidforξ 0iftheextendeddefinition
0>r2 (cid:16) s(cid:17) 2λΓ(s+λ+1)"T (Ψ−E)λ inequation(5)isadopted. ≥
J.An 7
Inparticular,togeneralizeequation(33),wefirstfindthat Thisis also consistentwith the case n = ξ of equation(34),
againthankstoequation(25). Thatistosay,equation(38a)is
+E0>Ψλ(cid:20)+0>r2ξ−21(cid:16)r2Nξ−1(cid:17)(cid:21)= 22λπΓ(23ξr2+ξ−λ3)" dEdL2KLλ2+ξ−ξ−11F(E,L2) actFuianlalyllyv,acliodnfsoidrearnaypµpl≥yin0ginthceluidnitneggrianlteogpeerravtaolureins.equation
(3)onΨtoequation(38a),asin
T (35)
faolirzξat≥ion12oafnedqλua≥tio0n,(w3h3i)cihsfaorrlliovwedsbeyquapatpiolynin(g30e)q.uTahtieonge(n3e2r)-, +E0>Ψξ−µ(cid:20)+0∂r2µr2(cid:16)r2µ+E0>Ψµ−21N(cid:17)(cid:21)=+0∂r2µr2(cid:16)r2µ+E0>Ψξ−12N(cid:17)
thatis,foranyreals0 µ ξandξ 1 (thelatterrestriction
≤ ≤ ≥ 2 whereξ µ. Theactualcalculationsisaidedbyanalternative
thatξ 1 willbedroppedlaterinthissection), expressio≥nfortheright-handsideofequation(38a)
≥ 2
+E0∂Ψµ(cid:20)+0>2µr2+rξ12−π2123(cid:16)rr22ξNξ−−31(cid:17)(cid:21) d dL2Kξ−µ−1 ( ,L2) (µ<ξ()36) +0It∂trh2µe(cid:16)nr2fµo+Ell0o>wΨµs−t21hNat(cid:17)f=or(02π)23µZ<E0ΨξdE(Ψ−E)µF(cid:2)E,2r2(Ψ−(3E8)b(cid:3)).
prov=ided2tξhΓπa(23ξtrt2−hξ−eµ3i)Zn0te"L¯g2rLadT2lLξs−2Ec1oFnv(cid:16)ΨerLg−e2ξ.2−LEr122q,FuLat2Ei(cid:17)on(36)(fµor=ξξ=),1 Γ((2ξπ−)2Ψ3µ)+0∂r2µr2(cid:16)r2µ+E0>Ψ≤ξ−21NQ(cid:17)
2 = dQ(Ψ Q)ξ µ 1 d (Q )µ , 2r2(Q )
nowreducesto Z − − − Z E −E F E −E
21+µπ32 d dL2F(E,L2) (µ< 1) = E0Ψd ΨdQ(Ψ Q)Eξ0 µ 1(Q )µ (cid:2) , 2r2(Q )(cid:3)
r2+E0∂ΨµN=Γ√(221π−23 µ)L¯"2dLT2 EΨ KL2µ+,L122 (µ= 21). (37) = ZE01 EZEΨd 2r−2(Ψ−E)−dL−2 Ψ−E LF2(cid:2)Eξ−µ−1L2µ −(E,L(cid:3)2)
Htihneafrtae+Ecs0t>evΨttaξiln−igd21Nµfo==rξ+E120−∂ΨξZ012r0e−(sξnNu.bl.t.s,IFit0ni(cid:16)seqinun−faetr2iroreξn2d(at3nh4da(cid:17))tsweoqitiuhfan0tio=2n0ξ(3g4iv)e1ins, (2r2)µ+=1Z2Eξ0r12Eµ+Z20"T dEdL2K(cid:16) ξ−−µE−−1L22rµ2F(cid:17)(E,L2).F(3E8c)
thenn=0). ≥ ≤ ≤ ≤ ≤ 2 Equations(38a)and(38c)together,thatis,
n tAo asimreiallarµg(ecnf.e,reaqli.za2t5io)nanodftehqeuaextitoenns(i3o4n)offroemquaantioinnte(3g6e)r +0∂r2µr2(cid:16)r2µ+E0>Ψξ−21N(cid:17) (39)
tdoireξct≥ca0lcaurelatpioosnssibislecaolmthpoaurgahtivdeelmyonnosntrtraitviniagl.thIenmstethardo,uwghe r2µ+2232Γ−(ξξπ23 µ)" dEdL2Kξ−µ−1L2µF(E,L2) (ξ >µ)
dreecrtivroeutthee.Lgeenteursafilirzsattcioonnsoifdeerqcuoamtiobnin(i3n4g)efqoulalotiwonin(g30acn)iwnditih- = π23 − L¯2dLT2L2µ Ψ L2 , L2 (ξ =µ)
Gequ=atFio,nµ(3=4)sw+it1h>n=0a0nadnλdξ==1µ−>δ0w,hwerheicδh=reµsu−lts⌊µin⌋,and constitu2tµe−t12hre2µg+e2nZe0ralizationFof(cid:16)equ−at2ior2n(34(cid:17))fromaninteger
+0>r21−δ(cid:16)r2µ+E0>Ψµ−21N(cid:17)= 2Γ21−(1⌊µ⌋+π⌊32µr2⌋⌊)µ⌋"dEdL2K(⌊Ψµ⌋F−(EE),1−Lδ2) µ(n1t≤oaξξ.,re1Faolrµµ,0)ws≤ehniµcdhs≤iesqξvuaa≤ltiido12nf,o(tr3h9ea)niytnodpi(ac3ie6rs)otgfriaµvnesanfnoedrqmξuaw(tµiio,thnξ)0(5→≤).
T 2 − 2 −
forµ>0and0<δ<1. Nextequation(32b)indicatesthat Equations(36)and(39)thusarebothvalidforanyrealpairµ
andξwith0 µ ξ.
(cid:18)r4∂∂r2(cid:19)n+1(cid:20)r21⌊µ⌋+0>r21−δ(cid:16)r2µ+E0>Ψµ−12N(cid:17)(cid:21) ifeIsntaftaioctn,sbooft≤hthreess≤ualmtsearnedsuallts,otheaqtuiastitoonsa3y5,aredifferentman-
= π23r2−2δ L¯2dL2L2µ Ψ L2 ,L2 . r2 + λ+ ξ N (40)
2µ−21 Z0 F(cid:16) − 2r2 (cid:17) √2π32E0>Ψ 0>r2 (cid:16)r2ξ(cid:17)
foranon-negativeintegern=⌊µ⌋. However, 221−λr2ξ d dL2Kλ+ξ−12F(E,L2) (λ+ξ > 1)
Γ(λ+ξ+ 1)" E L2ξ −2
(cid:18)r4∂∂r2(cid:19)n+1(cid:20)r21⌊µ⌋+0=>rr221⌊−µδ⌋(cid:16)+r42(cid:18)µ∂+E∂r0>2Ψ(cid:19)⌊µµ−⌋+121N+0>(cid:17)(cid:21)r21−δ(cid:16)r2µ+E0>Ψµ−12N(cid:17) whi=cha(2rer2v)aξlZid0L¯f2o2dLrL2aξ2nFyTr(cid:16)eΨal−pa2Lirr22(,λL,ξ2(cid:17))suchthatλ+(λξ++ξ21=≥−0.21).
thankstoequation(25),andconsequently,wefindthat
+0∂r2µ(cid:16)r2µ+E0>Ψµ−12N(cid:17)= 2µ−π12r322µ+2Z0L¯2dL2L2µF(cid:16)Ψ− 2Lr22,L2(cid:17).
(38a)
8 J.An
4. MOMENTSEQUENCESANDAUGMENTEDDENSITIES Here note that (1)+ = Γ(k+ 1)/√π. This is basically equa-
2 k 2
Consider the moment sequence of the df in ( ,L2) space tion(13)ofDejonghe&Merritt(1992)–seealsoequation(8)
restrictedalong =0,givenasin E ofBaes&VanHese(2007),equation(A2)ofVanHeseetal.
K (2009),equation(5c)ofAn(2011b)andsoon. Equation(43)
Fµ(Ψ,r2)≡ (2(2r2π))µ23+1Z L¯2dL2L2µF Ψ− 2Lr22,L2 (41a) ipnldetieclaytefisxtehsaet,vegriyve(ninpportiennctiipalleΨo(brs)e,rsvpaebcleif)ynionng-tvhaeniAshDincgovme--
0 (cid:16) (cid:17) locitymomentsuchthat
1
Ψµ+1 dyyµF(yΨ;Ψ,r2) ( =0, L¯2 =2r2Ψ) m [Ψ(r),r2]
wher=e Z0∞FdZY0(YY;µΨF,(rY2);Ψ,(r22π))23 (Ψ(EEY00,=2r−2Y∞),. L¯2 =∞(4)1,b) Cmoµ+n1v,0er=se2lyµ,+1e(q21u)+µa+ti1vo+E2rn0k>vΨ(2t4nµ3+=1)NfoNakr,t[nΨa(k(fi,rxn)e,)dr2=r]r.e(dµu+ces1,to0), that is,
≡ F − µ!v2(µ+1)
Thenequations(36)and(39)indicatethat M (r) r (44a)
µ ≡ 2µ+1 1 +
2 µ+1
InparticuFlaµr,=ifµ+E+E+E000i>∂∂sΨΨΨaµξ21+−p−o2112µ+0+0+0s>i∂∂trrri222vξµµe(cid:16)(cid:0)(cid:0)rrriNn222µµξtNeN(cid:17)g(cid:1)(cid:1)er,(((tµ0ξhi=≤s≥r−µe12sµ)≤u≥lt12s)0i)n. (42) where =Z[Ψ0(∞(cid:0)rd)](cid:1)Qµ+Q1ZµP01d(qQq;µrP) (cid:2)qΨ(r);r(cid:3) ((EE00 ==−0)∞),
N[Ψ(r) Q,r2]
1 ∂ ΨN(Q,r2)dQ P(Q;r) − . (44b)
F = ≡ ν(r)
0 √π∂ΨZ √Ψ Q
E0 − Inotherwords,giventheknowledgesofthelocaldensityν(r)
1 Ψ ∂ n and the potential Ψ(r), the infinite set of the radial velocity
Fn = 1 + √πZ dQ(Ψ−Q)n−32(cid:18)∂r2(cid:19) r2nN(Q,r2) , momentsin everyorder consists in the momentsequence of
2 n−1 E0 (cid:2) (cid:3) theAD consideredasa distributionofΨ–overthecompact
wheren =(cid:0) 1(cid:1),2,.... Thatistosay,asetoffractionalcalculus supportif = 0orthehalf-openinterval[0, )if =
0 0
chains of the AD directly determine the entire moment se- –atfixedrE. Theproblemiscloselyrelatedtot∞heHaEusdor−ff∞19
quencesalongafixedsectionallinein( ,L2)space. Inother (for = 0) or the Stieltjes20 (for = ) momentprob-
0 0
words, the AD is similar to the momenEtgeneratingfunction lemsE. With the infinite sequence oEf the−ra∞dial velocity mo-
(orthecharacteristicfunction)forthedfasaprobabilityden- ments as functionsof r, the AD can then be uniquelydeter-
sity. With varying(Ψ,r2), the = 0lineseventuallysweep mined at least formally by such means as e.g., the Hilbert21
thewholeaccessible( ,L2)spKace,andthusN(Ψ,r2)inprin- basisorthe Laplaceand/orFourier22 transform(cf., the mo-
cipleuniquelydetermiEnethetwo-integraldf, f( ,L2). Afew mentgeneratingfunctionandthecharacteristicfunction)etc.
explicit inversion algorithms from N(Ψ,r2) toEf( ,L2) are The final informationrequired for the full specification of
already available in the literature utilizing either tEhe known thesystemisthenthedeterminationofthepotential. Clearly
inverse of named integral transforms (see e.g., Lynden-Bell thepotentialmaybedeterminedthroughthePoissonequation
1962; Dejonghe 1986; Baes&VanHese 2007) or complex 2Φ=4πGρ,whichunderthesphericalsymmetryreducesto
∇
contourintegrals(see e.g.,Hunter&Qian 1993; An 2011a).
1 d dΨ
Since the definition of the AD in equation (1) provides the r2 = 4πGΥν. (45)
explicitformulafrom f( ,L2)toN(Ψ,r2),theknowledgeof r2dr(cid:18) dr(cid:19) −
N(Ψ,r2) is therefore maEthematically equivalent to knowing Hence if Υ ρ(r)/ν(r) is assumed to be constant, Ψ(r) can
f( ,L2). OncethepotentialΨ = Ψ(r)isspecified,thespeci- befixedbys≡olvingtheordinarydifferentialequationonΨ(r)
E
ficationoftheADthuscompletelydetermineauniquespher- thatresultsfromsettingν = N(Ψ,r2)inequation(45). Alter-
icaldynamicsystemin equilibrium. Althoughthisapproach natively,fromequation(43),wededucefork 1that
to the df f( ,L2) through the AD N(Ψ,r2) is advantageous ≥
E ∂m
as the observables constrain the AD more directly than the k,n =(2k 1)m ;
df,thisproceduresuffersasignificantdrawbackinthatthedf ∂Ψ − k−1,n
(46a)
recoveredassuchisindeedphysical,thatis,non-negativeev- ∂(r2n+2m )
e–rythweh“eprehainset-hspeaaclel accocnessissitbenlecys”u,bwvohliucmheisotfhtehesupbhjeacset-osfpathcee ∂r2 k,n =(cid:0)k− 12(cid:1)r2nmk−1,n+1.
reminderofthispaperfollowingthecurrentchapter.
Next, we considerwhatinformationon the physicalprop-
ertiesofthesystemissufficienttospecifyauniqueAD.First,
wefindfromequation(39)thatthe(augmented)velocitymo-
mentsoftheevenordersarerelatedtotheADasin
m (Ψ,r2)= 2k+nΓ(k+ 12) r4 ∂ n r2+ n+kN 2109TFehloixmHasauJosdanonrffes(1S8ti6e8lt-j1es94(128)56-1894)
k,n √πr2n+2 (cid:18) ∂r2(cid:19) E0>Ψ (43) 21DavidHilbert(1862-1943)
(cid:0) (cid:1) 22JeanBaptisteJosephFourier(1768-1830)
=2k+n 1 ++ k+n +∂ n(r2nN) .
2 kE0>Ψ 0 r2
(cid:0) (cid:1) (cid:2) (cid:3)
J.An 9
Thetotalradialderivativeofm fork 1thenresultsin which is necessary for the corresponding df to be non-
k,n
≥
negative as noted by An (2011b). It is clear that equation
dm 2m ∂log(r2n+2m ) dΨ∂m (52) implies equation(53) as the latter is a restriction of the
k,n = k,n k,n (n+1) + k,n
dr r (cid:20) ∂logr2 − (cid:21) dr ∂Ψ formerforanintegerµ=n. Theoppositeimplicationfollows
Corollary2.14):equation(53)forapositiveintegernimplies
2(n+1)m (2k 1)m
= k,n− − k−1,n+1 equation (52) for µ [n 1,n] and thus equation (52) for
− r µ 0followsequatio∈n(53−)forallpositiveintegersn.
dΨ ≥Wefindmoreequivalentstatementsofequation(53). First
+(2k 1)m . (46b)
− k−1,n dr equation(25)indicatesthat
stWioolinvtheodΨffro=radreΨΨ/k(drn)rowaifnntd.hemFrkoe,nrq[uΨthir(eerd)s,ivrm2e]plolec=sittyνcvma2rskoevm2t(nek,n,tnths)ias=sma(af1yu,n0bc)e-, R(n)(x)= xn1+1(cid:16)x2ddx(cid:17)n(cid:2)xR(x)(cid:3)=(−1)nwn+1dndRw(nw)(cid:12)(cid:12)(cid:12)(cid:12)w=x−(15,4)
thisreducestothespherical(second-ordersteady-state)Jeans where (cid:12)
R(w 1)
equation, (w) − . (55)
1d(νv2) 2v2 v2 dΨ R ≡ w
r + r − t = , (47)
Henceequation(53)isequivalentto
ν dr r dr
that is, the spherically-symmetric hydrostatic equilibrium d n
x2 xR(x) 0 (x>0, n=0,1,2,...), (56)
equationwithananisotropicvelocitydispersiontensor. dx ≥
(cid:16) d(cid:17)n(cid:2) (w) (cid:3)
5. NECESSARYCONDITIONFORSEPARABLEAUGMENTED ( 1)n R 0 (w>0, n=0,1,2,...). (57)
DENSITIES − dwn ≥
Inthefollowing,welimitourconcerntothecasesforwhich Herethelastisalsoequivalenttosayingthatthefunction (w)
thepotentialandtheradiusdependenciesoftheADaremul- defined in equation (55) is a cm function of w. The BRern-
tiplicativelyseparablesuchthat steintheoremthenindicatesthat (w)isrepresentableasthe
R
Laplacetransformationofanon-negativefunction. Thatisto
N(Ψ,r2)= P(Ψ)R(r2). (48)
say,thereexistsanon-negativefunctionφ(t) 0oft>0such
In addition to mathematical expediency, this assumption is that (w) = t w[φ(t)]. TheinverseLaplac≥etransformation
aelqsuoatnioontab(4le8)b,etchaeusreaduiunsdepratrhteRs(re2p)aroafbitlhitey AasDsumalopntieoncainn φm(ut)laR=(1L9−w),→1wt[hLRic(→wh,)]thmanayksbteofeoquunadtiuosnin(5g4t)h,erePdousct–esWtoidderfor-
uniquelyspecifytheso-calledBinneyanisotropyparameter, 1 t
φ(t)= lim R . (58)
(n)
β(r)=1 v2t =1 m0,1[Ψ(r),r2] n→∞n! (cid:16)n(cid:17)
Finallywefindanotherequivalentnecessarycondition,
− 2v2 − 2m1,0[Ψ(r),r2]
r (49)
1 dn[xnR(x)]
=1− m11,0∂(r∂2mr21,0) =−∂∂lologgmr12,0(cid:12)(cid:12)(cid:12)(cid:12)Ψ(r),r2 It is obviounl→ism∞thna!t equdaxtinon (5(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x3=)t/nim≥p0lies eq(uta>tio0n).(59), p(5ro9-)
such that (Dejonghe 1986; Qian&H(cid:12) unter 1995; videdthatit converges. The converseon the other handfol-
Baes&VanHese 2007; An 2011b; see also vanderMarel lows the Bernstein theorem and the Post–Widder formula.
1994asR−1beingtheintegratingfactoroftheJeansequation, However,the conditionalequivalencegiventhe convergence
i.e,eq.47) of equation (58) may also be inferred from equation (27).
Bydefinition,equation(59)indicatesthatthereexistsasuffi-
dlogR(r2) R(r2) r02β(s) cientlylargeintegerm such thatR (x) 0 forall n m
β(r)= ; =exp ds . (50) (n) ∀ ∃
≥ ≥
Someap−plicdaltoiognrs2arefoundRe(.rg02.),inBaesZ&r VansHese(2007) adfonerdscxex>n>d0in,0ga.nsCduboesrqcourlailpatirtosyno2f(.51R34)tf(hoxel)nl.owsusgsguecsctsestshiavteRa(rmg−u1m)(exn)ts≥o0n
(n)
whileAn(2011b)discussesfurtherimplicationsofthesepa-
rabilityassumption. 5.2. Thepotentialpart
5.1. Theradiuspart VanHeseetal.(2011)provedthat,givenequation(48),
WithaseparableADgivenbyequation(48),equation(39) P(k)(Ψ) 0 (k=0,..., 3 β ) (60)
indicatesthat(hereafterx r2), ≥ ⌊2 − 0⌋
≡ whereβ isthelimitoftheanisotropyparameteratthecenter,
0
+0∂xµ xµ+E0>Ψξ−21N =+E0>Ψξ−12P(Ψ)·+0∂xµ[xµR(x)]≥0 (51) itshinsegceensseararylizfeosritnhceodrfpotorabteinngofnra-ncetigoantaivled.eWriveastihvaelsl.showthat
forµ (cid:0)ξwhereas+(cid:1) ξ−21P>0forξ 1. Therefore, First,wegeneralizetheresultofAn(2011a)toincludear-
≤ E0>Ψ ≥ 2 bitraryrealorderderivatives. Thisistrivialsincetheinverse
+∂µ(xµR) 0 (x>0, µ 0). (52) Abeltransformisjustaparticularfractionalderivativeasde-
0 x ≥ ≥ fined in equation (4). If the AD is given as equation (48),
Thisisactuallyequivalenttothecondition, equation(36)reducesto
R(n)(x)≡ dn[xdnxRn(x)] ≥0 (x>0, n=0,1,2,...), (53) +E0∂Ψµ+0>xξ−21(cid:16)xξN−1/2(cid:17)=+E0∂ΨµP·+0>xξ−12(cid:16)xξ−R1/2(cid:17)≥0, (61)
10 J.An
forµ ξ. SinceR(x) 0istriviallynecessary,Iλ (x λR)>0 Therefore,equation(64c)indicatesthat
for x≤> 0andanyλ ≥0unlessR(x) = 0almosxt|0eve−rywhere
inx≡r2 ∈[0,∞)(Le≥mma2.5),whichwillnotbeconsidered 0<R¯ <∞ =⇒ +E0∂ΨµP≥0 (µ≤ 23 −η). (67)
here.Consequently,equation(61)impliesthat
That is to say, if there exists η < 1 such that R¯ is a (non-
∃ η
0<+0>xλ(cid:16)xRλ(cid:17)<∞ =⇒ +E0∂ΨµP≥0 (µ≤λ+ 12). (62) z32e−ro∃η).pTohsiitsivaectfiunaliltyeecnocnosmtanpta,sstheesneq+Eu0∂aΨtiµoPn(≥620),fwohriacnhyis∀sµee≤n
cWPo(inΨtdh)iλtio=0n.0fFo,rothrµiλs≤i>n0d0iiscoatntreivtshitaehloabtteh+Ece0ar∂uΨhsµaePn+Ed≥0,∂eΨ0q−ufλoaPrtia=onny+E(0>µ6Ψ2≤)λPi12mw–phltiihelees afiλsn<fdo1tlhloawtηRs,:a∼InfdRx¯s−ηoηisiafnsµoxn-→zλer0+o.fi1Hn,eitthneecfneoµr+0>η<xλ<(3x1−,λtηRh.e)ncownevberagseicsaflolyr
that, i≥f x λR(x)dxisintegrableover x = 0, then+ ∂ µP 0 For−example,witha≤consta2ntanisotrop2y−systemgivenby
− Ψ
for any µ λ+ 1 and all accessible Ψ is necessEa0ry for≥the R(x)= x β; R¯ =1 (β 1) (68)
existenceo≤fanon2-negativedf. Alternatively,forafixedµ > − β ≤
1, equation (62) suggests that + ∂ µP 0 is necessary for theconvergenceconditionreducesto
t2hedftobenon-negativeiftherEe0eΨxists≥∃λ ≥ µ− 21 suchthat + λ R = 1 x(x−s)λ−1ds = Γ(1−β−λ) < ,
+0 xλ(x−λR)iswell-defined. 0>x (cid:16)xλ(cid:17) Γ(λ)Z0 sλ+β xβΓ(1−β) ∞
β0>i<Esq1nueaactsieosxnsa(r6y20)fohwrohawilenevoetnhr-iinsseiigsnacntoievnceecsldusfasirgvyieviewfnwheRet(hwxe)err+E∼e0∂tΨox−23eβ−xβwtPeint≥hd fiwnohdriitcchhaetcedosfntthvoaebtrg+Eee0n∂sΨoifnµP0-n(≤eΨg)λat≥<iv0e1wf−ohrβe.µrIe≤tafsoλel+lqou12wa<stitoh23na−t(β6e4qiscu)natseiucogensgs(e6asr2tys)
theresultof→VanHeseetal.(2011).Forthis,wefirstnotethat thesameforµ 3 β.
≤ 2 −
if f(t)isright-continuousatt=a,
6. SUFFICIENTCONDITIONSFORPHASE-SPACECONSISTENCY
b h(t)dt INTERMSOFSEPARABLEAUGMENTEDDENSITIES
lim ǫ = lim h(t)=h(a) (a<b). (63)
ǫ→0+ Za (t−a)1−ǫ t→a+ suffiReccieennttlcyo,nVdaintioHnesfoeretthael.d(f2w01it2h)der=iv0edtothbeenneocne-snseagryataivned,
0
Thisappliedtotheleft-handsideofequation(36)reducesto expressed in terms of the integro-Edifferential constraints on
lim 3 η ξ + ξ−12 N = Pˆη(Ψ) (64a) tHhaeuAsdDo.rffTmheoymaecnhtiepvroebdlethmis,abcycorreddiuncgintogwthheicphrothbeledmfistonothne-
ξ→(32−η)−(cid:0)2 − − (cid:1)0>x (cid:16)xξ−1/2(cid:17) xηΓ(1−η) negativeifandonlyifthemomentsequenceinequation(41a)
is a completely monotone sequence.23 Since the moment
whereη<1and sequence are generated by the AD using equation (42), the
monotonesequenceconditionisexpressibleintermsoffinite
Pˆη(Ψ)= lim xηN(Ψ,x). (64b) differencesofintegro-differentialoperationsontheAD.
x 0+
→ WithaseparableAD,theyhavederivedasimplersufficient
Equation(36)overallthenresultsintheformula, (butnotnecessary)conditiongivenasa unionofconditions,
each of which only involves the potential or the radius part
+E0∂ΨµPˆη(Ψ)=223−ηπ32Γ(1−η)+E0>Ψ23−η−µg˜η(Ψ)≥0, (64c) sseupffiacraietenltycbountdintiootntofogretahesre.paHraebrleewAeDdteorivbee arensualltteedrnfartoivme
where
anon-negativedf,whichturnsouttobeequivalenttothatof
g˜ ( )= lim L2η ( ,L2). (64d)
η E L2 0+ F E VanHeseetal. (2012). The derivation here is based on the
→ properties of cm functions and also uses the Laplace trans-
For µ < 32 −η, this is derived with the limit ξ → (32 −η)− form extensively. In this section, we only consider the case
whilemaintaintingµ<ξ < 3 η. Forµ= 3 ηontheother that = 0 and L¯2 = 2r2Ψ, that is, the df has a compact
hand, the same limit is take2n−with µ = ξ. 2T−herefore, this is suppEor0tand ( <0,L2)=0.
F E
validforµ≤ 32 −ηandη<1,providedthat+0 xξ−21(x21−ξN)is 6.1. Inversionofaseparableaugmenteddensityforthe
well-definedforξ < 3 η(n.b.,theintegrabi>lityofthesame distributionfunction
2 −
forξ= 3 ηisactuallynotrequiredforitsvalidity).Herethe As it has been shown by (Hunter&Qian 1993, see also
2−
non-negativityofequation(64c)followsthenon-negativityof An 2011a), inverting equation (2b) for ( ,L2) is formally
( ,L2). Ofparticularinterestsareequation(64c)forµ = 0 equivalentto recoveringthe two-integraFlevEen df, +( ,J2)
aFndE3 η, from the axisymmetric density ν[Ψ(R2,z2),R2]. FTheEfindz-
2 −
ings of the preceding section together with the inversion of
Pˆη(Ψ)=232−ηπ23Γ(1−η)+E0>Ψ23−ηg˜η(Ψ); (65a) Lthyendlaetnte-rBpelrlob(1le9m62)suwggheostuttihliaztetdhethfeunLcatpiolanceφ(ttr)andseffionremdfboyr
+ ∂Ψ23−ηPˆη(Ψ) equation (58) must be directly related to the underlying df,
g˜η(Ψ)= 2E320−ηπ23Γ(1 η), (65b) F(E,L2). Weinvestigatethisconnectioninthefollowing.
−
thatis,explicitformulaeforPˆ (Ψ)andg˜ (Ψ)fromeachother. 23 A sequence (a0,a1,a2,...) is completely monotone if and only if
ForaseparableADgivenasηinequatioηn(48),wehave (f−or1w)ka∆rdkafijni≥ted0ifffoerreanncyenoopner-anteogradtievfieniendtesguecrhptahiarts∆kk+a1nadj j=. ∆Hkearje+1∆is∆kthaej
−
Pˆη(Ψ)=R¯ηP(Ψ); R¯η = lim xηR(x), (66) and∆0aj=aj.
x 0+
→