www.functions.wolfram.com—The web’s most comprehensive site about special functions

Oleg Marichev and Michael Trott
Wolfram Research Inc.

Abstract

At the Mathematical Functions website, we have a collection of most of the known formulas for practically all mathematical functions. It is organized in such a way as to allow for expansion, citation, searching, conversion of formulas, and substantial computer support from the Mathematica system.

Currently (June 2002) the site includes more than 37,300 formulas for the approximately 250 elementary and special functions that are available in Mathematica. It is already the biggest online collection of formulas presented in different format types (Mathematica notebooks, HTML, and PDF). The site also has a well-developed, unified structure; in particular, the organization of mathematical files and programming solutions allows for semiautomatic updating and citation of formulas. The site contains thousands of new or adapted-to Mathematica formulas for elementary and classical special functions.

In this talk we will outline some of the material at the Mathematical Functions site and also discuss how it was put together.

Why make such a web site?

Existing handbooks are old and static

Many people around the world use mathematical formulas. They always have difficulties in finding, adapting and verifying them for specific applications. Very often people use handbooks; among the most popular are those of Abramowitz and Stegun (1964), Bateman and Erdelyi (1950-54), Gradzhteyn and Ryzhik (1971), and Prudnikov, Brychkov, and Marichev (1981-92). But these handbooks were published 20-50 years ago and many new results have been found since that time (especially for functions like PolyLog, MeijerG, FoxH and etc.). The best known, Abramowitz and Stegun handbook is not uniform in accuracy or coverage; its various chapters were compiled by different authors with different criteria for inclusion of material. It also contains large tables of numerical values, which are now obsolete thanks to inexpensive computing power and the availability of high-precision numeric capabilities for all values of all special functions in modern computer programs such as Mathematica.

Computer systems and Internet opened new opportunities

The appearance of computer systems like Mathematica and the Internet made a revolution in the opportunities to collect, derive, represent and search for mathematical formulas, and produce graphics and related mathematical information in an interactive electronic fashion. Now it is possible to work with functions which do not have simple closed forms but can be described by algorithms that produce as many terms as desired. (Example: series expansions for zeros of Bessel functions \(TraditionalForm\`ξ\_n\%\((ν)\) = ∑\+\(k = 0\)\%∞\( c\_k\)(n, ν)\), (Subscript[J, n])((Subscript[ξ, n]^((ν)))) = 0.) This allows for the possibility to reorganize and redescribe the classical world of elementary and special functions. It can moreover be made more internally consistent and appropriately developed as to branch cuts of mathematical functions. (For example, in all existing handbooks the correct formula (cosh^(-1))(
          z) == (Sqrt[z - 1])/(Sqrt[1 - z]) (cos^(-1))(z) is presented by a "misleading" relation (cosh^(-1))(
          z) == ±i (cos^(-1))(
            z) with some comments about ±.)

Currently the Internet includes many different sites with mathematical formulas. But users of these formulas still can not copy or paste them and should act like Euclid 3000 years ago—they should use pen or pencil (or perhaps a stick, like Euclid) when they want to rewrite formulas from a computer display. This old fashioned situation should be fixed. We have done exactly that, at http://functions.wolfram.com. The engine behind this site is the powerful system Mathematica.

Resources for making such web site

Existing and regularly upgraded system Mathematica

General remarks about system Mathematica

Mathematica already has extraordinary capabilities in calculation, plotting, programming, publication, data manipulation, etc. New functions and functionality are always being added to this already very general system. And this is true not only for the world of mathematics, but also other uses of the computer, such as electronic publishing by Mathematica-generated web site.

Currently Mathematica supports over 250 elementary and special functions (more than exist in any other system). For example,

Select[Names["*"],Function[f,MemberQ[Attributes[f],NumericFunction]]]//Short[#,10]&

These functions have a “natural” naming (for example, LegendreP, HypergeometricPFQ,…). They can be manipulated algebraically, differentiated, integrated, plotted, simplified and expanded. Their numerical values can be calculated to arbitrary precision for any complex argument and they can be presented in different formats and forms (based on Mathematica notebook technology that enables the automatic generation of different presentation media). Current formats include HTML, PDF, and Mathematica notebooks. All formulas are available as MathML Future formats will include XML as it becomes more prevalent.

The large potential of the computer system Mathematica allows for the derivation of many new formulas, from simple to extrememly complicated, as well as to revise existing formulas and verify them not only theoretically but numerically.

Branch cuts are more consistent in Mathematica than in textbook-mathematics itself!

The system Mathematica has about 90 mathematical functions with branch cuts. The behavior of these functions near their branch cuts uniquely follows from the behavior of logarithmic function Log[z]. These functions are more internally self consistent with one another in Mathematica than in textbook mathematics.

The picture below illustrates some graphical opportunities of Mathematica in description of behavior of function (Sqrt[(((x + i y))^2) - 1]) near its complicated double branch cut lines.

ParametricPlot3D[{x,y,Im[(Sqrt[(((x+I y))^2)-1])],{EdgeForm[], SurfaceColor[Hue[0.22]]}},{x,-2,2},{y,-2,2},
               PlotPoints -> 100, BoxRatios->{1,1,0.48}];

Desire and capability for making such a site

For the first time in the history of mathematical functions, due to hardware and software advances and the permanently increasing opportunities of the Internet (not to mention the system Mathematica), there is the possibility to create a complete listing of all identities, equations, theorems, and inequalities about mathematical functions with maximal modern technological and programming support. The building of this site commenced with the over 250 built-in functions of Mathematica and in the future will cover more additional special functions for which a Mathematica code repository will be also maintained. To provide permanent updating, contributions from researchers are welcome to help expand the scope of the site. For details about communication with Wolfram team who develop this site see Be a Contributor (with Wish List), Sign the Guestbook and Email Comments.

The structure of this site

General description of front page

The first page of the site includes Keyword search, Function Categories (with ElementaryFunctions (9233 formulas), Constants (573 formulas), Bessel, Airy, Struve Function (2193), Integer Functions (1321), Polynomials (1675), Gamma, Beta, Erf (3818), Hypergeometric Functions (11295), Elliptic Integrals (686), Zeta Functions and Polylogarithms (997), Mathieu Functions (135), Complex Components (450), Number Theory Functions (787), Generalized Functions (164)—together 37366 formulas as of June 18, 2002)), Find a citation, Wish List, Alphabetical Index, Notations, General Identities, About This Site, Be a Contributor, Sign the Guestbook, Email Comments.

The site was designed to be similar in nature to existing handbooks that detail special mathematical functions. The main part if this site is the Function categories which currently includes near 270 mathematical functions divided on the following basic groups:

Function categories

1) Elementary Functions: Power Functions: Sqrt, Power; Exponential Function: Exp; Logarithmic Functions: Log (2 types); Trigonometric Functions: Sin, Cos, Tan, Cot, Csc, Sec; Inverse Trigonometric Functions: ArcSin, ArcCos, ArcTan (2 types), ArcCot, ArcCsc, ArcSec; Hyperbolic Functions: Sinh, Cosh, Tanh, Coth, Csch, Sech; Inverse Hyperbolic Functions: ArcSinh, ArcCosh, ArcTanh, ArcCoth, ArcCsch, ArcSech; Product Logarithms: ProductLog (2 types); Roots: Root; Max and Min: Max, Min

2) Constants: Imaginary Unit: I; Classical Constants: GoldenRatio, Pi, Degree, E, EulerGamma, Catalan, Khinchin; Symbols: Indeterminate, Infinity, ComplexInfinity, DirectedInfinity

3) Bessel, Airy, Struve Functions: Bessel Functions: BesselJ, BesselI, BesselY, BesselK; Airy Functions: AiryAi, AiryBi, AiryAiPrime, AiryBiPrime; Struve Functions: StruveH, StruveL

4) Integer Functions: Rounding and Congruence Functions: Floor, Ceiling, Round, IntegerPart, FractionalPart, Mod, Quotient; GCD and LCM: GCD, ExtendedGCD, LCM; Fibonacci Numbers: Fibonacci; Euler and Bernoulli Numbers: EulerE, BernoulliB; Stirling Numbers: StirlingS1, StirlingS2: Partitions: PartitionsP, PartitionsQ; Tensor Functions: DiscreteDelta (2 types), KroneckerDelta, Signature

5) Polynomials: Classical Orthogonal Polynomials: HermiteH, LaguerreL, LegendreP, ChebyshevT, ChebyshevU, JacobiP; Associated Polynomials: LegendreP, LaguerreL, GegenbauerC, SphericalHarmonicY; Other Polynomials: Cyclotomic, Fibonacci, EulerE, BernoulliB

6) Gamma, Beta, Erf: Factorials: Factorial, Factorial2; Binomials: Binomial, Multinomial; Gamma Functions: Gamma (3 types), GammaRegularized (2 types), Pochhammer, LogGamma, InverseGammaRegularized (2 types); Differentiated Gamma Functions: PolyGamma (2 types), HarmonicNumber (2 types); Beta Functions and Inverses: Beta (3 types), BetaRegularized (2 types), InverseBetaRegularized (2 types); Probability Integrals and Inverses: Erf (2 types), Erfc, Erfi, InverseErf (2 types), InverseErfc; Fresnel Integrals: FresnelS, FresnelC; Exponential Integrals: ExpIntegralE, ExpIntegralEi, LogIntegral, SinIntegral, CosIntegral, SinhIntegral, CoshIntegral

7) Hypergeometric Functions: Hermite and Laguerre Functions: HermiteH, LaguerreL (2 types); Chebyshev Functions: ChebyshevT, ChebyshevU, Fibonacci; Legendre Functions: LegendreP (3 types), LegendreQ (3 types); Gegenbauer and Jacobi Functions: GegenbauerC (2 types), JacobiP; Hypergeometric Functions of One Variable: HypergeometricPFQ (0F0 function), Hypergeometric0F1 (0F1), Hypergeometric0F1Regularized, HypergeometricPFQ (1F0), Hypergeometric1F1 (1F1), Hypergeometric1F1Regularized, HypergeometricPFQ (1F2), Hypergeometric2F1 (2F1), Hypergeometric2F1Regularized, HypergeometricPFQ (2F2), HypergeometricPFQ (2F3), HypergeometricPFQ (3F2), HypergeometricPFQ (4F3), HypergeometricPFQ (5F4), HypergeometricPFQ (6F5), HypergeometricPFQ (PFQ), HypergeometricPFQRegularized, HypergeometricU; Meijer G Functions: MeijerG (2 types); Hypergeometric Functions of Several Variables: AppellF1; Angular Momentum Functions: SphericalHarmonicY, ClebschGordan, ThreeJSymbol, SixJSymbol

8) Elliptic Integrals: Complete Elliptic Integrals: EllipticE, EllipticK, EllipticPi; Incomplete Elliptic Integrals: EllipticE, EllipticF, EllipticPi, JacobiZeta

9) Elliptic Functions: Jacobi Theta Functions: EllipticTheta (4 types), EllipticThetaPrime (4 types); Neville Theta Functions: NevilleThetaC, NevilleThetaD, NevilleThetaN, NevilleThetaS; Weierstrass Functions and Inverses: WeierstrassP, WeierstrassPPrime, WeierstrassSigma (2 types), WeierstrassZeta, WeierstrassHalfPeriods, WeierstrassInvariants, WeierstrassPHalfPeriodValues, WeierstrassZetaHalfPeriodValues, InverseWeierstrassP (2 types); Jacobi Functions: JacobiAmplitude, JacobiCD, JacobiCN, JacobiCS, JacobiDC, JacobiDN, JacobiDS, JacobiNC, JacobiND, JacobiNS, JacobiSC, JacobiSD, JacobiSN; Inverse Jacobi Functions: InverseJacobiCD, InverseJacobiCN, InverseJacobiCS, InverseJacobiDC, InverseJacobiDN, InverseJacobiDS, InverseJacobiNC, InverseJacobiND, InverseJacobiNS, InverseJacobiSC, InverseJacobiSD, InverseJacobiSN; Modular Functions and Inverses: DedekindEta, KleinInvariantJ, ModularLambda, InverseEllipticNomeQ, EllipticNomeQ; Arithmetic Geometric Mean: ArithmeticGeometricMean; Elliptic Exp and Elliptic Log: EllipticExp, EllipticExpPrime, EllipticLog

10) Zeta Functions and Polylogarithms: Zeta Functions: Zeta (2 types), RiemannSiegelTheta, RiemannSiegelZ, StieltjesGamma, LerchPhi; Polylogarithms: PolyLog (3 types)

11) Mathieu Functions: Mathieu Functions: MathieuC, MathieuS, MathieuCPrime, MathieuSPrime; Characteristics: MathieuCharacteristicA, MathieuCharacteristicB, MathieuCharacteristicExponent

12) Complex Components: Abs, Arg, Re, Im, Conjugate, Sign

13) Number Theory Functions: Prime Factorization Related Functions: FactorInteger, Divisors, Prime, PrimePi, DivisorSigma, EulerPhi, MoebiusMu; Other Functions: JacobiSymbol, CarmichaelLambda, DigitCount

14) Generalized Functions: Discontinuous Functions: UnitStep (2 types); Functions with Singular Support: DiracDelta (2 types)

Help and searching categories

To improve convenience for site users, special help and searching categories are under development. Currently it supports Keyword search, Find a citation, Alphabetical Index, Notations, General Identities and About This Site.

Keyword search does word searching inside of the site.

Find a citation provides openning of passport-page for each formula.

Alphabetical Index includes the list of all mathematical functions from this site along with the quantity of 
formulas therein. For example, it includes "ArcTan[z] (187 formulas)" with link to this function.

Notations and General Identities are available as a Mathematica Notebooks and as a PDF files. Viewing these guides requires MathReader, our downloadable notebook reader, or Adobe Acrobat Reader.

Notations (more than 350 such) is still under construction and includes the following chapters: Numbers, Variables and Functions (with Domains and Domain Membership; Types of Variables; Set Membership; Types of Functions); Logical Operators and Conditionals (with Logical Operators; Conditionals, Equality and Ordering Operators); Operations (with Domain and Range; Branch Cuts and Points, Singularities and Discontinuities; Asymptotics and Series; Summations; Products; Differentiations; Fractional Integro-Differentiations; Integrations; Integral Transforms; Limits; Continued Fractions; Matrices and Determinants); Symbols; Functions in Alphabetical Order(with A; B;…, Z; Greek; Other).

General Identities is still under construction. It has the same structure and organization of information as for each function of the site. But this structure is applied for a general abstract function f(z) (instead of sin(z), for example).

About This Site includes the following articles: Overview; Mathematica and This Site; The Developers; Our Vision; What's New; How to Cite This Site; FAQs.

Communication (with users) categories

Wish List includes the list of ten problems which we deem to be the most in need of work.

Be a Contributor invites everybody to send interesting formulas and information which can be included
in the site (after corresponding verification).

Sign the Guestbook includes form which allow to get basic information about users of this site.

Email Comments allows everybody send email to developers of site with his comments and suggestions.

Description of front pages for functions

The front page for each function includes the name and title of the function, its Mathematica notations, traditional Notation, the name of its category, standard notation (with a quantity of formulas for this function) and the list of sections (with the same names for all functions).

On the top of front page the user can find a link to related information from The Mathematica Book. There is also the opportunity to download all formulas for this function in the form of a Mathematica notebook or PDF file. Downloading of PDF file can be done when the user fills out the simple registration form.

Some functions have the same names but may have different sets of arguments. 
For example, Mathematica has two kinds of inverse tangent functions: ArcTan[z] and ArcTan[x,y]. For more convenient separation of these cases the user can click on “See also Variants of these function” on the top of the front page which allows one to select the corresponding function.

Front page for the power function

For example, the front page of function Power includes the following: Power function (name), Power (title), Mathematica notation: Power, Traditional Notation: (z^a), Elementary functions (the name of its category), Power[z,a](343 formulas) (Standard notation) and the following list of sections for power function: Primary definition, Specific values, General characteristics, Series representations, Integral representations, Continued fraction representations, Differential equations, Transformations, Identities, Complex characteristics, Differentiation, Integration, Integral transforms, Summation, Products, Operations, Representations through more general functions, Representations through equivalent functions, Inequalities, Zeros, Theorems, Other information, History.

Sections names for each function (all possibilities)

The corresponding formulas and information for each function are distributed among the following sections:

Notations, Primary definition, Specific values, General characteristics, Analytic continuations, Series representations, Integral representations, Product representations, Limit representations, Continued fraction representations, Generating functions, Group representations, Differential equations, Difference equations, Equations involving this function, Transformations, Identities, Identities involving the group of functions, Complex characteristics, Differentiation, Integration, Integral transforms, Summation, Products, Operations, Representations through more general functions, Representations through equivalent functions, Inequalities, Zeros, Theorems, Other information, Applications, History, References

If some sections can not be applied to corresponding function or currently can not be filled out they are omitted.

Working in this site

General statements

For convenience this site supports several ways of finding information. One can use Keyword search, Alphabetical Index or Function categories. For example, after printing cos or Cos in the window of Keyword search and clicking "go" the user gets a link to http://functions.wolfram.com/ElementaryFunctions/Cos/. A click on Alphabetical Index or Elementary Functions (in Function categories) will give the list of functions where one can find Cos.

On the front page of Cos the user can see its full name (Cosine), Mathematica notations (in StandardForm Cos[z] and in TraditionalForm cos (z)), classification (Elementary Functions), title with quantities of formulas in the section (Cos[z] (545 formulas)), list of sections with quantities of formulas there (Primary definition (1 formula), Specific values (159 formulas), etc.). On the top of the front page the user finds links to
corresponding Cos chapter of Mathematica book and also can freely download all formulas for the function.

A user wanting to find formulas that connect Cos and Sin can read the titles of corresponding sections and click on the the section Representations through equivalent functions (54 formulas). This can be used in two ways.
  1. One can click on one of the two  Subsection With related functions (50 formulas) and one of the twelve  Subsubsection Involving sin (15 formulas). As result the user will see 15 formulas. 
  2. One can click on "Show All Below" and on separate page the user will see all 54 formulas from the hole .Section.

If the user clicks on 5th formula on the Subsubsection Involving sin (15 formulas)

cos(z)==(((-1))^⌊)(1/2) - ((Re(z))/π)⌋ (Sqrt[1 - (sin^2)(z)]) (1-(1 + (((-1))^(⌊((Re(z))/π) - (1/2)⌋ + ⌊(1/2) - ((Re(z))/π)⌋))) θ(Im(z)))

it gives the special passport-page which includes a classification of this formula within the site:
Elementary Functions
 /Cos[z]
   /Representations through equivalent functions
    /With related functions
     /Involving sin, 
citation information (http://functions.wolfram.com/01.07.27.0010.01), its traditional form (see above form) with opportunity to enlarge it), InputForm

Cos[
        z] == (-1)^Floor[1/2 - Re[z]/Pi] Sqrt[
          1 - Sin[z]^2] (1 - (1 + (-1)^(Floor[
                          1/2 - Re[z]/Pi] + 
                        Floor[-(1/2) + Re[z]/Pi])) UnitStep[
                Im[z]])

StandardForm (long box form), and Date Added to functions.wolfram.com (modification date) (2001-10-29).

The above formula is a new one. It shows the coefficient in the famous relation between cos(z) and (Sqrt[1 - (sin^2)(z)]) which is valid for all complex argument z. Its citation index http://functions.wolfram.com/01.07.27.0010.01 can be used for direct linking to this formula from other resources. For example, if the user prints in the Find a citation window the number 01.07.27.0010.01 and clicks "go", it will directly open the passport-page of above formula.

Each formula on this web site is given a unique numeric identifier. As this site expands and more information is added, the position of a given formula within its group may change. However,the identifier itself will remain the same. For example, index 01.07.27.0010.01 provides one-to-one identification of position of above formula. Indeed, 01. means the number of Function Category - Elementary Functions; 07. means the number of the function Cos in the list of Elementary Functions); 27. means the number of the section Representations through equivalent functions); 0010. means the number of formula in this section, calculated from the beginning of this section ); 01 means the number of modification of this formula.

Some formulas are placed very deep in the hierarchical structure of the formula pyramid.
For example, the formula http://functions.wolfram.com/06.38.21.0004.01

∫(z^(α - 1)) Ci(
              a z) \[DifferentialD]z == 
      ((z^α)/(2  a)) (Γ(α, -I a z) (((-I a z))^(-α)) + 
            2 Ci(a z) + (((I a z))^(-α)) Γ(α, I a z))

is placed in 
Gamma,Beta,Erf
 /CosIntegral[z]
  /Integration
   /Indefinite integration
    /Involving one direct function and elementary functions
     /Involving power function
      /InvolvingPower
       /Linear argument.

Examples of new formulas

The site includes thousands of new formulas or old “forgotten” formulas that were adopted to Mathematica and fixed. Many such type new formulas were derived and presented in the site at first time (without publication in literature) even for basic elementary functions like Sqrt or Log. The examples below illustrate some of these formulas

Elementary functions

Sqrt

Many people use the power-expansion formula (Sqrt[(z^2)])==(Sqrt[z])(Sqrt[z])==z. But this is correct only for real z>0 or for complex z with -π/2<Arg(z)≤π/2. Instead of this formula we have the completely correct formulas (http://functions.wolfram.com/01.01.16.0024.01 and http://functions.wolfram.com/01.01.16.0025.01)

(Sqrt[(z^2)])==(Sqrt[-I z]) (Sqrt[I z]);
(Sqrt[-(z^2)])==(Sqrt[ z]) (Sqrt[- z])

Also we have the corresponding restricted formula
 (http://functions.wolfram.com/01.01.16.0008.01)

(Sqrt[(Subscript[z, 1]) (Subscript[z, 2])]) == (Sqrt[(Subscript[z, 1])]) (Sqrt[(Subscript[z, 2])]) /;  
      (Subscript[z, 1]) + (Subscript[z, 2]) ≥ 0

In particular, we have \(TraditionalForm\`\@\(1 - z\^2\) == \@\(1 - z\)\ \@\(1 + z\)\); however \(TraditionalForm\`\@\(z\^2 - 1\) == \@\(z - 1\)\ \@\(z + 1\) /; \ Re(z) > 0\).

Above restriction can be removed but the formula will become more complicated one including function Floor (http://functions.wolfram.com/01.01.16.0011.01):

(Sqrt[(Subscript[z, 1]) (Subscript[z, 2])]) == (Sqrt[(Subscript[z, 1])]) (Sqrt[(Subscript[z, 2])]) exp(π I ⌊((π - Arg((Subscript[z, 1])) - Arg((Subscript[z, 2])))/(2 π))⌋)

ArcSin

The following well-known formula (http://functions.wolfram.com/01.12.27.0001.01):

(sin^(-1))(sin(z))==z/;-π/2<Re(z)<π/2∨(Re(z)==-π/2∧Im(z)≥0)∨(Re(z)==π/2∧Im(z)≤0)

was generalized for arbitrary values of argument
 (http://functions.wolfram.com/01.12.27.0005.01):

(sin^(-1))(
        sin(z)) == (((-1))^⌊)((Re(z))/π) - (1/2)⌋ ((1 + (((-1))^(⌊((Re(z))/π) + (1/2)⌋ - ⌈((Re(z))/π) + (1/2)⌉))) θ(-Im(z)) - 
            1) (z - π ⌊((Re(z))/π) - 
                  (1/2)⌋ + (π/2) (1 + (((-1))^(⌊((Re(z))/π) + (1/2)⌋ - ⌈((Re(z))/π) + (1/2)⌉))) θ(-Im(
                    z)) - π)

Special functions

Stokes phenomena for AiryBi

Here is a single formula which correctly describes the asymptotic behavior of AiryBi as \(TraditionalForm\`|z| -> ∞\) in any direction.

The asymptotic behavior of AiryBi as |z|->∞ is described in literature by different formulas for different sectors separated by lines |Arg(z)|==(π/3) or |Arg(z)|==((2π)/3) (see
http://functions.wolfram.com/03.06.06.0005.01 
and http://functions.wolfram.com/03.06.06.0007.01). Using
Mathematica we found the single unified formula which is applied for whole complex plane (http://functions.wolfram.com/03.06.06.0013.01):

Bi(z)∝(1/(2 (Sqrt[π])))(((-(z^3)))^(-5/12)) ((((-1))^(5/12)) (E^(-(( 2 I)/3) (Sqrt[-(z^3)]))) ((1/((-1)^(1/3))) ((-(z^3))^(1/3))+z) (Subscript[, 2])(Subscript[F, 0])((1/6),(5/6);\[Null];((3 I)/(4 (Sqrt[-(z^3)]))))-(((-1))^(7/12)) (E^(((2I)/3)  (Sqrt[-(z^3)]))) (((-1)^(1/3)) ((-(z^3))^(1/3))+z) (Subscript[, 2])(Subscript[F, 0])((1/6),(5/6);\[Null];-((3 I)/(4 (Sqrt[-(z^3)])))))/;(\[LeftBracketingBar]z\[RightBracketingBar]->∞)