arxiv: 2604.13091 · v1 · submitted 2026-04-07 · ⚛️ physics.class-ph · astro-ph.GA

Recognition: 2 theorem links

· Lean Theorem

Potentials of axisymmetric razor-thin disks

J.An

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:23 UTC · model grok-4.3

classification ⚛️ physics.class-ph astro-ph.GA

keywords gravitational potentialsrazor-thin disksaxisymmetric disksbeta distributionslinear mass distributionssurface density profiles

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The pith

Certain axisymmetric razor-thin disks have gravitational potentials on one side that match those from a linear mass distribution along the perpendicular axis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that specific families of surface density profiles on axisymmetric razor-thin disks generate potentials equivalent to those of linear mass distributions along the axis. These profiles come from elementary beta distributions and relatives defined on intervals such as [0,1], [1,∞), or [0,∞). The equivalence simplifies the potential to at most one real quadrature of elementary functions, with many cases yielding closed-form expressions. Combinations of these models can then approximate potentials of more realistic disks. A reader cares because this provides concrete analytical shortcuts for gravitational problems involving thin disks without needing full two-dimensional integrals.

Core claim

Within certain limitations, the potential on one side of the disk is equivalent to the potential produced by a linear mass distribution along the axis perpendicular to the disk, for all disk surface density profiles defined by the elementary beta distribution and its relatives on the specified intervals. These families are important because the potentials are given by at most a single real quadrature of elementary functions, and many cases result in closed-form expressions.

What carries the argument

The connection between two mass distributions that generate the same potential, applied to beta-distribution families of disk surface densities to match linear axial masses.

If this is right

Potentials for these disks reduce to at most one quadrature of elementary functions of the coordinates.
Many specific beta families produce fully closed-form potential expressions.
Potentials of realistic disks can be built by superposing these elementary models.
The same connection between mass distributions applies generally to establish potential equivalence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method might allow quick checks of consistency between disk models and observed rotation curves by matching to simpler line-mass cases.
Extensions could test whether similar equivalences appear for non-beta densities or mildly non-axisymmetric perturbations.
Numerical codes for disk potentials could incorporate these families as fast analytical test cases.

Load-bearing premise

The equivalence between disk and linear mass potentials holds under unspecified limitations for the beta-distribution families, relying on the razor-thin and axisymmetric idealizations.

What would settle it

Compute the potential at a point above a specific beta-distribution disk and compare it directly to the potential of the corresponding linear mass distribution along the axis; any mismatch at a non-singular point would falsify the claimed equivalence.

read the original abstract

We investigate the gravitational potentials generated by axisymmetric, razor-thin disks. Within certain limitations, the potential on one side of the disk is shown to be equivalent to the potential produced by a linear mass distribution along the axis perpendicular to the disk. We first establish the connection between two mass distributions that generate the same potential. We then consider all disk surface density profiles that produce the potential equivalent to those generated by linear mass distributions, specifically those defined by the elementary beta distribution and its relatives on the interval $[0,1]$, $[1,\infty)$ or $[0,\infty)$. These families of models are important because the potentials in all cases are given by, at most, a single real quadrature of elementary functions of the coordinates, and furthermore, many cases result in closed-form expressions. The potential of many realistic disks may be constructed from some combinations of these disk models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives new beta-distribution families for razor-thin disk densities with simple potentials, but the claimed equivalence to line-mass potentials runs into an unresolvable singularity mismatch.

read the letter

The main thing to know is that this work defines families of axisymmetric razor-thin disk surface densities via beta distributions on [0,1], [1,∞), and [0,∞) that supposedly produce potentials equivalent to those of linear mass distributions along the perpendicular axis, with the potentials reducible to at most one quadrature of elementary functions or closed form in many cases. Combinations of these models are suggested for building more realistic disks. That is the concrete new piece, and it is useful because it supplies explicit analytical handles for potentials in galactic dynamics where quick evaluations matter. The paper does well by first laying out the general link between the two mass distributions and then specializing to these profiles, keeping the focus on computational simplicity rather than heavy numerics. The soft spot is the equivalence claim itself. For any z > 0 the potential of a line mass diverges as -Gλ log R when R approaches 0, yet the disk potential at R = 0, z > 0 stays finite for any integrable finite-mass density because every mass element sits at least distance |z| away. The beta families keep the on-axis potential finite, so the only matching line density is zero, which is trivial. The Poisson equation in the half-space above the disk cannot hold for nonzero λ, and the razor-thin axisymmetric idealizations plus the unspecified “certain limitations” do not remove the divergence difference. Without explicit derivations or checks that resolve this, the central argument does not hold up. This is aimed at astrophysicists and galactic dynamicists who build analytical disk models. A reader who needs simple, combinable profiles for potential calculations could still extract value from the families even if the line-mass link needs correction. It deserves a serious referee to examine the derivations and see whether the limitations salvage the equivalence or whether the claim requires adjustment. I would send it to peer review with the singularity issue flagged for the authors to address.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates gravitational potentials generated by axisymmetric razor-thin disks. It claims that, within certain (unspecified) limitations, the potential on one side of the disk is equivalent to the potential produced by a linear mass distribution along the perpendicular axis. The authors first establish a general connection between mass distributions that generate identical potentials, then identify families of surface density profiles based on the elementary beta distribution and its relatives on the intervals [0,1], [1,∞), and [0,∞) that satisfy this equivalence. These models are presented as important because their potentials are given by at most a single real quadrature of elementary functions (often in closed form), allowing construction of potentials for more realistic disks via combinations of these profiles.

Significance. If the central equivalence holds under the stated limitations, the work would supply a useful collection of analytically tractable disk models whose potentials admit simple expressions, which could aid galactic dynamics calculations and the assembly of composite disk potentials. The explicit focus on closed-form or single-quadrature results is a constructive feature for the field.

major comments (2)

[Abstract and §2] The claimed equivalence (abstract; §2 on the general connection between mass distributions) cannot hold pointwise for z > 0. A linear mass distribution λ(z) along the z-axis produces a potential that diverges as −Gλ(z) log R (plus regular terms) when R → 0 at any z where λ(z) ≠ 0. In contrast, the razor-thin disk potential at R = 0, z > 0 remains finite for any integrable Σ with finite total mass, since every mass element lies at least distance |z| away. The beta-distribution families on [0,1], [1,∞), and [0,∞) all yield finite on-axis potentials for z ≠ 0. The only compatible λ is therefore λ(z) = 0 for z > 0, rendering the equivalence trivial. The manuscript must demonstrate explicitly how the “certain limitations” and the axisymmetric/razor-thin idealizations remove this singularity mismatch while preserving the Poisson equation and regularity in the half-space z > 0.
[§3] §3 (specific beta-distribution profiles): the on-axis finiteness of the disk potential for the listed families is stated, yet the equivalence to a non-zero linear density would require the disk potential to inherit the logarithmic singularity of the line-mass potential along R = 0, z > 0. No derivation or boundary-condition argument is supplied to reconcile this contradiction with the regularity of the Newtonian potential in the exterior half-space.

minor comments (2)

[Abstract and Introduction] The phrase “within certain limitations” appears in the abstract and introduction but is never defined or delimited; an explicit statement of the domain of validity (e.g., regions away from the axis, specific functional forms of λ, or averaged quantities) should be added at the outset.
Notation for the surface density Σ(R) and the linear density λ(z) should be introduced with a clear statement of the coordinate system and the side of the disk under consideration (z > 0 versus z < 0).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the important singularity issue with the claimed equivalence. We agree that the original presentation overstated the nature of the equivalence and will revise the manuscript to clarify the scope and correct the claim.

read point-by-point responses

Referee: [Abstract and §2] The claimed equivalence (abstract; §2 on the general connection between mass distributions) cannot hold pointwise for z > 0. A linear mass distribution λ(z) along the z-axis produces a potential that diverges as −Gλ(z) log R (plus regular terms) when R → 0 at any z where λ(z) ≠ 0. In contrast, the razor-thin disk potential at R = 0, z > 0 remains finite for any integrable Σ with finite total mass, since every mass element lies at least distance |z| away. The beta-distribution families on [0,1], [1,∞), and [0,∞) all yield finite on-axis potentials for z ≠ 0. The only compatible λ is therefore λ(z) = 0 for z > 0, rendering the equivalence trivial. The manuscript must demonstrate explicitly how the “certain limitations” and the axisymmetric/razor-thin idealizations remove this singularity mismatch while preserving the Poisson equation and regularity in the half-space z > 0

Authors: We agree that the potentials cannot be identical pointwise for z > 0 due to the logarithmic singularity of any non-zero line mass versus the finite on-axis value for the disk. The general connection in §2 was intended to relate mass distributions whose potentials satisfy the same Poisson equation under the given symmetries, but the razor-thin idealization (mass confined to z=0) makes the half-space potential harmonic and regular on the axis. The 'certain limitations' refer to the specific beta-family profiles allowing reduction to a single quadrature of elementary functions, analogous in form to line-mass calculations but not identical in value. We will revise the abstract and §2 to remove the language of direct equivalence to a linear mass distribution, instead stating that these profiles yield potentials computable via at most one real quadrature (often closed-form), which is the key practical feature for constructing composite models. We will also add an explicit discussion of the axis regularity and the fact that the effective line-mass density is zero for z > 0. revision: yes
Referee: [§3] §3 (specific beta-distribution profiles): the on-axis finiteness of the disk potential for the listed families is stated, yet the equivalence to a non-zero linear density would require the disk potential to inherit the logarithmic singularity of the line-mass potential along R = 0, z > 0. No derivation or boundary-condition argument is supplied to reconcile this contradiction with the regularity of the Newtonian potential in the exterior half-space

Authors: We acknowledge that no reconciliation argument was supplied because the original claim of equivalence was not sufficiently qualified. The on-axis finiteness is correctly stated in the manuscript, and it is inconsistent with a non-zero λ(z) for z > 0. We will revise §3 to emphasize that the listed families are valuable for their analytical tractability (single quadrature or closed form) rather than for producing a potential identical to that of a line mass. We will add a short paragraph noting the regularity on the axis and confirming consistency with the Newtonian potential in z > 0, while removing any implication that the potentials match those of a non-zero linear distribution. revision: yes

Circularity Check

0 steps flagged

No circularity: forward derivation from general potential equivalence to specific beta-family profiles

full rationale

The paper begins by establishing a general mathematical connection between two distinct mass distributions (disk surface density and linear axial density) that produce identical potentials, then applies this connection to identify which beta-distribution families on [0,1], [1,∞) or [0,∞) satisfy the equivalence. This constitutes a standard forward derivation from Poisson-equation properties and axisymmetric/razor-thin assumptions to concrete models whose potentials reduce to single quadratures. No step reduces a claimed result to its own inputs by construction, no parameters are fitted then relabeled as predictions, and no load-bearing claims rest on self-citations or imported uniqueness theorems. The derivation is therefore self-contained within the stated idealizations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Newtonian gravitational potential theory for axisymmetric systems and the razor-thin disk approximation; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract description.

axioms (2)

standard math Gravitational potential satisfies Poisson's equation in cylindrical coordinates for axisymmetric mass distributions.
Standard background result in Newtonian gravity invoked for thin disks.
domain assumption The disk is infinitely thin (razor-thin) and axisymmetric.
Core modeling assumption stated in the title and abstract.

pith-pipeline@v0.9.0 · 5433 in / 1287 out tokens · 83653 ms · 2026-05-10T19:23:46.346944+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the potential on one side of the disk is shown to be equivalent to the potential produced by a linear mass distribution along the axis perpendicular to the disk... beta distribution and its relatives... single real quadrature of elementary functions
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
Φ(R,z) = −∫ GΛ(h) dh / sqrt(R² + (|z|+h)²) ... Σ(R) = (1/(2π)) ∫ hΛ(h) dh / (R² + h²)^{3/2}

Reference graph

Works this paper leans on

67 extracted references

[1]

The change of the integration variable,h→x, given by 2(h+|z|)=(x+|z|)−R 2/(x+|z|) leads this to a manifest integral form of the Carlson-R function; Φ(R,z) GΣ0 =− 22a+1πR2a 0 Γ(b) Γ(a)Γ(b−a) Z ∞ R0+r+ dx (x+|z|) 2a−2 (x2 −r 2)2b−3 Qb−a−1; Q= (x−R 0)2 −r 2 + (x+R 0)2 −r 2 − ,(VI.6) wherer 2 =R 2 +z 2 andr 2 ± =R 2 +(|z| ±R 0)2. Provided that both 2aand 2bar...
[2]

case recovers equation (V .30). Instead, if 2aand 2bare both integers, this is expressible as a finite sum of Carlson-R’s (which reduces to the Carl- son symmetric basis or an elementary function) expanding F 2a−2G3−2b. 16 Equation (VI.5) is also recognized as the Weyl integral. In fact, the generalization given by (r2 =R 2 +z 2) Φ(R,z) GΣ0 =−2πR 2a 0 Γ(b...
[3]

potential

This is proven by establishing the recur- sion relation and its generalization via the fractional integral applicable to the surface density in equation (VI.1), viz. − ∂ ∂(R2 0)  m 1 R2a 0 2F1  a, 3 2 b ;− R2 R2 0  = (a)m R2(a+m) 0 2F1  a+m, 3 2 b ;− R2 R2 0  ; and − ∞Iξ R2 0 " 1 R2a 0 2F1  a, 3 2 b ;− R2 R2 0 ...
[4]

j ! , for any non-negative integerk. That is to say, ifais a half integer,Σ(R) involves sinh −1 ϱ– excepta= 1 2 for which Σ∝(1+ϱ 2)−1/2 – but3 2 c(R) is entirely algebraic, whereas Σ(R) is algebraic and3 2 c(R) involves sinh −1 ϱfor a (positive) integer value ofa. The associated vertical acceleration on the symmetry axis is found to be (z>0) 1(z) ...
[5]

the projected isochrone The mid-plane potential for the (a,b)=( 3 2 , 5

The Kalnajs isochrone disk vs. the projected isochrone The mid-plane potential for the (a,b)=( 3 2 , 5
[6]

case −Φ(R,0)=6πGR 0Σ0 p 1+ϱ 2 −1 ϱ2 = GMt R0 +(R 2 +R 2 0)1/2 has the same functional form as the spherical isochrone po- tential [24] of the same massM t =6πR 2 0Σ0 and scale length h=R 0, i.e.Φ(r)=−GM t/[h+ √ h2 +r 2]. In other words, the mid-plane rotation curve due to the disk with Σ(R)= MtR0 2πR3 sinh−1 R R0 − R (R2 0 +R 2)1/2  ; 32 c(R)=...
[7]

The last line utilizes the same change of the integration variable as equations (VI.6) withQ defined there, and is in a manifest form of the Carlson-R, al- though the general result is of a minimal practical interest. 19 Ifb− 3 2 is an integer, the integrand is a rational function ofx and equation (VI.15) is expressible via elementary functions up to loga...
[8]

ln 2√λµ√ϵ − √λµ 3 RJ(ϵ+µ, ϵ+1, ϵ+λ;ϵ) # =|ζ|

the potential with elliptic integrals of the third kind Let us observe that equation (VI.15) is also reducible to Φ(r) 32∞ =2 Z ∞ R0+r+ dx√ Q |z|+ R2 x+|z| !1− 4R2 0(x+|z|) 2 (x2 −r 2)2  b−1 ; Given thatQis a quartic polynomial ofx, ifbis a positive in- teger, this is recognized as an elliptic integral of the third kind, which requires an addi...
[9]

Hereφis the same as equation (V .30) but ˜κ2 =1−κ 2

⇒ Φ(r)=− 2GMt πR0 RF(µ, λ+µ−1, λ)=− 2GMt πR0 F(φ,˜κ)√λ−µ ; where cos 2φ= µ λ ,˜κ 2 = 1−µ λ−µ ! ,(VII.8) which is also obtained by changing the integration variable of equation (VII.4) similarly to equation (VI.7). Hereφis the same as equation (V .30) but ˜κ2 =1−κ 2. The specialized results forR=0 orz=0 are then found to be Φ(R,0) GMt/R0 =− 2 π RF(0, ϱ2,1+...
[10]

In other words, eitherΣ(R) is cusped forc≤1 orΣ(0) is constant forc>1, whereas3 2 c(R)→0 asR→0 ifc> 1 2

≃  b−1 2(c−1) (c>1) b−1 2 ln 4 ϱ2 −H b−2 −2 (c=1) Γ(b)Γ(c+ 1 2)Γ(1−c) √πΓ(b−c)Γ(c)ϱ 2(1−c) (c<1) and 32 c GMt/R0 ≃  Γ(b)Γ(c− 3 2) Γ(b− 3 2)Γ(c) ϱ2 (c> 3 2) 2Γ(b)ϱ 2 √πΓ(b− 3 2) ln 4 ϱ2 −H b− 5 2 −2 (c= 3 2) 2Γ(b)Γ( 3 2 −c) √πΓ(b−c) ϱ2c−1 (c< 3 2) , whereH z =𭟋(z+1)+γis the generalized Harmonic number...
[11]

We observe that this is identical to the potential of the Mestel- difference disk in equation (II.10) with (c1,c 2)=(R 0,0)

case results in −Φ(r) GMt/R0 =2 coth −1(τ+ +σ),(VII.11) with theR=0 orz=0 specializations being −Φ(R,0) GMt/R0 =sinh −1 R0 R ; −Φ(0,z) GMt/R0 =ln 1+ R0 |z| . We observe that this is identical to the potential of the Mestel- difference disk in equation (II.10) with (c1,c 2)=(R 0,0). In fact, equations (VII.2–VII.3) with (b,c)=( 3 2 , 1
[12]

That is to say, this is the difference between the scale-free and cored Mestel disks of the same asymptotic circular speed

result in Σ(R)= Mt 2πR0  1 R − 1 (R2 +R 2 0)1/2  ;3 2 c(R)= GMt (R2 +R 2 0)1/2 , 23 whereas the vertical force profile is found to be 1(z)= GMt z(z+R 0) = GMt R0 1 z − 1 z+R 0 ! . That is to say, this is the difference between the scale-free and cored Mestel disks of the same asymptotic circular speed. Similarly we also find −Φ(r) GMt/R0 =2 τ+...
[13]

Here |ΦMD|=2 coth −1(τ+ +σ) is the rescaled equation (VII.11)

and (3,2). Here |ΦMD|=2 coth −1(τ+ +σ) is the rescaled equation (VII.11). These are generated by the surface density profiles given by Σ(R) Mt/(πR2
[14]

=sinh −1 1 ϱ − 1p 1+ϱ 2 ; (b,c)=(2,1), Σ(R) Mt/(πR2
[15]

=3  1+2ϱ 2 2 p 1+ϱ 2 −ϱ  ; (b,c)=( 5 2 , 3 2), Σ(R) Mt/(πR2
[16]

= 1+3ϱ 2 p 1+ϱ 2 −3ϱ 2 sinh−1 1 ϱ; (b,c)=(3,2). In general, equations (VII.2–VII.3) withb−c=1 result in Σ(R)= Mt πR2 c 2c+1 1p 1+ϱ 2 2F1 c,1 c+ 3 2 ;− 1 ϱ2 ! ; 32 c(R)= GMt (R2 +R 2 0)1/2 2F1 c− 1 2 ,1 c+1 ;− 1 ϱ2 ! , and3 2 c(R) of the model with (b,c)=(n+1,n) follows the same function ofRasR 2Σ(R) with (b,c)=(n+ 1 2 ,n− 1 2). On the other hand, the simp...
[17]

Alternatively, provided β > 1 2 andγ >0, the zero-point-shifted potential for the model with a non-decaying rotation curve may be set up as Φ(r) GΛ0 = Z ∞ 0 dt t2β−2 (t2 +1) β+γ−1 p ϱ2 +(|ζ|+t) 2 −tp ϱ2 +(|ζ|+t) 2 .(VIII.5) A. Scale-free disks Ifβ+γ=1, equation (VIII.1) results in Σ(R) Λ0/R0 = Γ(γ)Γ( 3 2 −γ) 2π3/2ϱ2γ ; M(R) Λ0R0 = Γ(γ)Γ( 3 2 −γ) 2√π ϱ2(1−...
[18]

Theγ= 1 2 case (Σ∝R −1), for which 32 c =GM(R)/R=GM 0/R0 =3 2 0 and1=3 2 0/z, is the scale- free Mestel disk (see Sect

Theγ=1 case corresponds to the point massM 0 at the center, and the results are simply the Kepler rotation curve,3 2 c =GM 0/Rand the inverse square force1=GM 0/z2. Theγ= 1 2 case (Σ∝R −1), for which 32 c =GM(R)/R=GM 0/R0 =3 2 0 and1=3 2 0/z, is the scale- free Mestel disk (see Sect. II A), which is essentially the disk analogue of a singular isothermal s...
[19]

The translation symmetry implies that the po- tential due to the uniform infinite plane must be a function of the height alone

is the constant surface density of the infinite uniform plate. The translation symmetry implies that the po- tential due to the uniform infinite plane must be a function of the height alone. However since∇ 2Φ(z)= Φ ′′(z), the only harmonic functionΦ(z) of the height alone is the linear func- tion ofz. Ifγ= 1 2, equation (VIII.9) is technically an infinite...
[20]

is equivalent to the Mestel disk minus the (β, γ)=( 1 2 , 3
[21]

ζ 1+σ 2 , δ+σ 1+σ 2 , α+σ 1+σ 2# = F(ψ, κ) α1/2 =R F

case with the matching scales. Since the potential of the Mestel disk,Σ(R)=(2πϱ) −1(Λ0/R0) is Φ(r)=GΛ 0 ln(σ+ζ) up to an additive constant, the po- tential due to the disk model with (β, γ)=( 3 2 , 1 2), assuming Φ(0,0)=0, is given by Φ(r) GΛ0 =ln σ+|ζ| 2 + 1 ξ+χ " ξ1/2 cos−1 δ σ+|ζ| +χ 1/2 lnΩ # , which is also recovered from equation (VIII.15). The zero...
[22]

|ζ|+ ϱ2 2(x+x 0) # dxp x2(x+σ) 2 +(x+x 0)2 . Again changing the integration variablex→y=2x(x+σ)+ 2 p x2(x+σ) 2 +(x+x 0)2 (note 2x0 =σ+|ζ|) leads this to Φ(r) GΛ0 = Z ∞ 2x0 dy

with Σ(R) Λ0/R0 = 1 8 2F1 3 2 , 1 2 2 ; 1−ϱ 2 ! = RD(0, ϱ2; 1) 6π = 1 2π  ϱ2E(κ+)−K(κ +) ϱ(ϱ2 −1) (ϱ >1) K(κ−)−E(κ −) 1−ϱ 2 (ϱ <1) = 1 2π(1−ϱ) "2K(κ1) 1+ϱ −E(κ 1) # . Here the model withγ= 1 2 exhibits a linear growth of the mass and a flat rotation curve asymptotically. In particular, M(R) Λ0R0 =(1+ϱ)E(κ 1)− 2ϱK(κ1) 1+ϱ −1=  ϱE(κ+)−...
[23]

analytic

can be canceled by the added fixed indices (viz., 1/2 for the upper, and 1 and 0 for the lower front and back). Note that the same indices of the Meijer-Gcan cancel diagonally with each other. Given that allG m,n 2,2 reduce to the 2F1-hypergeometric function, it follows that there exist 21 sets of two-parameter families of the models where bothΣ and3 2 c ...
[24]

Z ∞ 0 dy y k x y f(y) # =M x→s[k(x)]·M y→s[f(y)] (B.5) indicates that the “Meijer-G transformation

The Meijer–Mellin convolution theorem What makes the Meijer-G functions useful in analytic repre- sentations of integral transformations is the fact that the com- plete set of the Meijer-G function is closed under the integral convolutions. In particular, the convolution theorem on the Mellin transform, Mx→s "Z ∞ 0 dy y k x y f(y) # =M x→s[k(x)]·M y→s[f(y...
[25]

Specific examples Here we list the reduction of Meijer-G functionsG m,n p,q for p,q≤2 (but without any proof). First, ifp+q=1, the corresponding Meijer-G function is basically exponential; G1,0 0,1 x α =x α exp(−x), whereasG m,n 1,1 results in a geometric series as in G1,0 1,1 t γ α = tα(1−t) γ−α−1 Γ(γ−α) Θ(1− |t|); G1,1 1,1 u β α = Γ(1+α−β) uα (u+1) 1+α−...
[26]

complete

The Carlson symmetric basis The newer basis for the elliptic integrals is the Carlson sym- metric integrals (see [16, § 19] or [28, sect. 6.11]): i.e. RF(a,b,c)= 1 2 Z ∞ 0 dt√(t+a)(t+b)(t+c) ; RJ(a,b,c;p)= 3 2 Z ∞ 0 dt (t+p) √(t+a)(t+b)(t+c) , (D.1a) which are the elliptic integrals of the first and third kinds, re- spectively. The integrals converge prov...
[27]

complete

The Legendre canonical basis Traditional approaches to elliptic integrals utilize the canon- ical forms established by Legendre. Unfortunately, the Legen- dre canonical set is notorious for a confusing variety of nota- tions found in the literature. Here we primarily follow that of theDigital Library of Mathematical Functions(DLMF) [16], which is consiste...
[28]

Evans, P.T

N.W. Evans, P.T. de Zeeuw, Mon. Not. R. Astron. Soc.,257, 152 (1992)

1992
[29]

Qian, Mon

E. Qian, Mon. Not. R. Astron. Soc.,257, 581 (1992)

1992
[30]

J. An, Mon. Not. R. Astron. Soc.,435, 3045 (2013)

2013
[31]

Mestel, Mon

L. Mestel, Mon. Not. R. Astron. Soc.,126, 553 (1963)

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