Especial Functions

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Introduction

Reading Problems

Computer Algebra Systems

Computer Algebra Systems (CAS) are software packages used in the manipulation of math-ematical formulae in order to automate tedious and sometimes difficult algebraic tasks. Theprincipal difference between a CAS and a traditional calculator is the ability of the CASto deal with equations symbolically rather than numerically. Specific uses and capabilities

of CAS vary greatly from one system to another, yet the purpose remains the same: ma-nipulation of symbolic equations. In addition to performing mathematical operations, CASoften include facilities for graphing equations and programming capabilities for user-definedprocedures.

Computer Algebra Systems were originally conceived in the early 1970s by researchers work-ing in the area of artificial intelligence. The first popular systems were Reduce, Deriveand Macsyma. Commercial versions of these programs are still available. The two mostcommercially successful CAS programs are Maple and Mathematica. Both programshave a rich set of routines for performing a wide range of problems found in engineering

applications associated with research and teaching. Several other software packages, such asMathCAD and MATLAB include a Maple kernal for performing symbolic-based calcu-lation. In addition to these popular commercial CAS tools, a host of other less popular ormore focused software tools are availalbe, including Axiom and MuPAD.

The following is a brief overview of the origins of several of these popular CAS tools alongwith a summary of capabilities and availability.

Axiom

Axiom is a powerful computer algebra package that was originally developed as a researchtool by Richard Jenks Computer Mathematics Group at the IBM Research Laboratory inNew York. The project began in 1978 under the name Scratchpad and the first release ofAxiom was launched in 1991. The research group is widely recognized for some outstandingachievements in the field of computer algebra and their work was augmented by contributionsfrom a multitude of experts around the world. This collaboration continues to play animportant part in the development of Axiom.

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The object oriented design of Axiom is unique in its recognition of a universe of mathemat-ical components and the relationships amongst them. The strong typing and hierarchicalstructure of the system leads to it being both consistent and robust. Many users are waryof strongly typed languages but Axiom employs comprehensive type-inferencing techniquesto overcome problems encountered in some older packages. It is a very open system with all

library source code held on-line. This means that all the algorithms employed and the type-inferencing performed on behalf of the user are readily visible - making Axiom particularlyattractive for teaching purposes.

The package includes an interface that allows users to develop their own object orientedprogramming code and add it in compiled form to the Axiom library which leads to significantperformance improvements. Axiom is the only computer algebra package providing such afacility. Code can also be lifted out of the Axiom system to form stand-alone programswhich are independent of the Axiom environment.

Axiom was available commercially for several years through NAG (Numerical AlgorithmGroup) but their support of the product terminated in December 2001. It is now distributedas an open license through several ftp sites worldwide.

Web Site: http://home.earthlink.net/ jgg964/axiom.html

Derive

Derive had its roots in the Mu-Math program first released in 1979, with the first availableversion appearing in 1988. Its main competitive advantage to programs such as Maple

and Mathematica was its limited use of computer resourses including RAM and hard diskspace. While Derive does not have the same power and breadth of features as Maple orMathematica, it is a more suitable package for the casual or occasional user and probablyfor many professional users whose needs are for standard rather than esoteric features.

Derive is the ideal program for demonstration, research, and mathematical exploration be-cause of its limited use of computer resources. Users appreciate the easy to use, menu-driveninterface. Just enter a formula, using standard mathematical operators and functions. De-rive will display it in an easy-to-read format using raised exponents and built up fractions.Select from the menu to simplify, plot, expand, factor, place over a common denominator,

integrate, or differentiate. Derive intelligently applies the rules of algebra, trigonometry,calculus, and matrix algebra to solve a wide range of mathematical problems. This non-numeric approach goes far beyond the capabilities of mere statistics packages and equationsolvers that use only approximate numerical techniques. Powerful capabilities exist for 2D(Cartesian, polar, and parametric) and 3D graphing.

The current version of Derive for PCs is Derive 5, available from Texas Instruments for$199US or $99US for an Educational version. System Requirements include, Windows 95,98, ME, NT, 2000 or XP compatible PC (minimum RAM and processor requirements are

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the same as the operating system requirements), CD ROM Drive, and less than 4 MB ofdisk space.

Web Site: http://education.ti.com/us/product/software/derive/features/features.html

Macsyma

Macsyma evolved from research projects funded by the U.S. Defense Advanced ResearchProjects Agency at the Massachusetts Institute of Technology around 1968. By the early1970s, Macsyma was being widely used for symbolic computation in research projects atM.I.T and the National Labs. In the late 1970s the U.S. government drastically reducedthe funding for symbolic mathematics software development. Authorities reasoned then thatfaster vectorized supercomputers and better numerical mathematical software could solve allU.S. mathematical analysis needs. Around 1980 M.I.T. began seeking a commercial licensee

for Macsyma and in 1982, licensed Macsyma to Symbolics, Inc., an early workstation spin-offfrom M.I.T.

In April of 1992, Macsyma Inc. acquired the Macsyma software business from Symbolics,Inc. Bolstered by private investors from across the U.S. who understand the potential of thesoftware, Macsyma Inc. has been reinvesting in Macsyma to make the software a suitabletool for a wide range of users and has already brought the PC user interface and scientificgraphics up to modern windows standards.

Macsymas strength lies in the areas of basic algebra and calculus, O.D.E.s, symbolic-numerical linear algebra and, when combined with PDEase, in symbolic and numerical

treatment of P.D.E.s. The current version of PC Macsyma 2.4 is available for MS-Windows.

MAXIMA is a COMMON LISP implementation due to William F. Schelter, and is basedon the original implementation of Macsyma at MIT. This particular variant of Macsymawas maintained by William Schelter from 1982 until he passed away in 2001. In 1998 heobtained permission to release the source code under GPL. Since his passing a group ofusers and developers has been formed to keep Maxima alive and kicking. Maxima itself isreasonably feature complete at this stage, with abilities such as symbolic integration, 3Dplotting, and an ODE solver, but there is a lot of work yet to be done in terms of bug fixing,cleanup, and documentation.

Web Site: http://www.scientek.com/macsyma/main.htm

Maple

The MAPLE project was conceived in November 1980 with the primary goal to designa computer algebra system which would be accessible to large numbers of researchers inmathematics, engineering, and science, and to large numbers of students for educational

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purposes. One of the key ideas was to make space efficiency, as well as time efficiency, a fun-damental criterion. The vehicle for achieving this goal was to use a systems implementationlanguage from the BCPL family rather than LISP. Another aspect of making the systemwidely accessible was to design for portability, so that the system could be ported to thevarious microcomputers which were appearing in the marketplace. A very important aspect

of achieving the efficiency goal was to carry out research into the design of algorithms forthe various mathematical operations.

Maple is a comprehensive general purpose computer algebra system that can do both sym-bolic and numerical calculations and has facilities for 2 and 3-dimensional graphical output.Maple is also a programming language. In fact almost all of the mathematical and graphi-cal facilities are written in Maple and not in a systems implementation language like othercomputer algebra systems. These Maple programs reside on disk in the Maple library andare loaded on demand. The programming language supports procedural and functional pro-gramming. Because of the clean separation of the user interface from the kernel and library,

Maple has been incorporated into other software packages, such as Mathcad and MatLAB,to allow the symbolic functionality of the program to be accessable to as wide and audienceas possible.

At the University of Waterloo, Maple 8 is available to all students through the Nexus systemor through the University dial up system using an X-Windows package.(see http://ist.uwaterloo.ca/cs/chip/gs/newgs.html for further details)

Web Site: http://www.maplesoft.com/

Mathematica

Mathematica is a product of Wolfram Research Inc. founded by the architect of the system,Stephen Wolfram. Stephen Wolfram was a MacArthur Prize recipient in 1981. During thisperiod (in the early 1980s) Stephen Wolfram developed a language called SMP (SymbolicManipulation Processor) in C. This evolved into another program, Mathematica.

Mathematica, like Maple, offers capabilities for symbolic and numerical computations. Nu-meric computations can be carried out to arbitrary precision, though obviously the higherthe precision, the more time required to complete the calculation. There is a full suite of

functions supporting 2- and 3-dimensional plotting of data and functions. Mathematica in-corporates a graphics language capability which can be used to produce visualisations ofcomplex objects.

There is an Applications Library available which includes a range of application-tailored toolswritten in Mathematicas language. These include: a 3-D real-time graphics tool; a control-system tool; an experimental data analyser; mechanical, electrical, and signal analysers.

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Web Site: http://www.wolfram.com/

MuPAD

MuPAD is a general purpose (parallel) computer algebra system, developed at the Universityof Paderborn in Germany. MuPAD is available via FTP for several operating systems. Thenet version is limited in its memory access and cannot be used to solve real hard problems.But all non-commercial users can get the full-version for free by obtaining a MuPAD license(key) that unlocks all memory.

Web Site: http://www.mupad.de/

Reduce

The first version of REDUCE was developed and published by Anthony C. Hearn morethan 25 years ago. The starting point was a class of formal computations for problemsin high energy physics (Feynman diagrams, cross sections etc.), which are hard and timeconsuming if done by hand. Although the facilities of the current REDUCE are much moreadvanced than those of the early versions, the direction towards big formal computationsin applied mathematics, physics and engineering has been stable over the years, but with amuch broader set of applications.

Like symbolic computation in general, REDUCE has profited by the increasing power ofcomputer architectures and by the information exchange made available by recent network

developments. Spearheaded by A.C. Hearn, several groups in different countries take partin the REDUCE development, and the contributions of users have significantly widened theapplication field.

Today REDUCE can be used with a variety of hardware platforms from the Windows-basedpersonal computer up to the Cray supercomputer. However, the primary vehicle is the classof advanced UNIX workstations.

Although REDUCE is a mature program system, it is extended and updated on a continuousbasis. Since the establishment of the REDUCE Network Library in 1989, users take part in

the development, thus reducing the incompatibilities encountered with new system releases.

REDUCE is based on a dialect of Lisp called Standard Lisp, and the differences betweenversions are the result of different implementations of this Lisp; in each case the source codefor REDUCE itself remains the same. The complete source code for REDUCE is availablethrough ftp sites worldwide.

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Web Site: http://www.uni-koeln.de/REDUCE/ or http://www.zib.de/Symbolik/reduce/

Others

Mathcad is an easy to use tool for basic matematical calculations that has a Maple enginefor doing symbolic computation. It is fully WYSYWIG.

Web Site: http://www.mathsoft.com/products/mathcad/

MatLab is a good number crucher. It also has a Maple engine for doing Symbolic operations.A Student version is available.

Web Site: http://www.mathworks.com/

GNU-calc runs inside GNU Emacs and is written entirely in Emac Lisp. It does the usual

things: arbitrary precision integer, real, and complex arithmetic (all written in Lisp), sci-entific functions, symbolic algebra and calculus, matrices, graphics, etc. and can displayexpressions with square root signs and integrals by drawing them on the screen with ASCIIcharacters. It comes with a well written 600 page on-line manual. You can FTP it from anyGNU site.

Web Site: http://www.gnu.org/home.html

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Glossary of Symbols

Symbol Quantity

arg(z) argument or phase of the complex number z

bei(z), ber(z) Kelvin functions of order zero

bei(z), ber(z) Kelvin functions of order

B(a, b) Beta function

Bn(a, b) Incomplete Beta function

Bn Bernoulli number

Bn(x) Bernoulli polynomial

C(x) Fresnel integral

Ci(x) Cosine integral

Dn(x) Debye functions

D(x) Parabolic cylinder (or Webers) function

E(k, ) Elliptic integral of the second kind

E(k) Complete elliptic integral of the second kind

Ei(x), E1(x) Exponential integrals

En(k) Generalized error function

F(k, ) Elliptic integral of the first kind

F(a,c,x),F1(a,c,x) Kummers function (or confluent hypergeometric function)

F1(a,b,c,x) Gaussian hypergeometric function

Hn(x) Hermite polynomial

H(1)n

(x), H(2)n

(x) Hankel functions (or Bessel functions) of the third kind

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Symbol Quantity

h(1)n

(x), h(2)n

(x) Spherical Hankel functions

I(x) Modified Bessel Functions of order

Im(z) Imaginary part of the complex number z

J(x) Bessel function of the first kind of order

jn(x) Spherical Bessel function

K(x) Complete elliptic integral of the first kind

K(x) Modified Bessel function of the second kind or order

kei(z), ker(z) Kelvin functions or order zero

kei(z), ker(z) Kelvin functions or order

Ln(x) Laguerre polynomial

Lkn

(x) Associated Laguerre polynomial

i(x) Logarithmic integral

M(a,c,x) Confluent hypergeometric function

Pn(x) Legendre (or spherical) polynomial

Pmn

(x) Associated Legendre function of the first kind

Qn(x) Legendre polynomial

Qmn

(x) Associated Legendre function of the second kind

q Jacobi nome

Re(z) Real part of the complex number z

S(x) Fresnel integral

Si(x), si(x) Sine integrals

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Symbol Quantity

Tn(x) Chebyshev polynomial of the first kind

Un(x) Chebyshev polynomial of the second kind

Y(x) Bessel function of the second kind of order

Yn(x) Spherical Bessel function

Greek Characters

Euler-Mascheroni constant

(x) Gamma function

(x, a) Incomplete Gamma function

(x, a) Incomplete Gamma function

()n Pochhammer symbol

(

n) Binomial coefficient

(x) Probability integral

(x) Riemanns zeta function

(k,,a) Elliptic integral of the third kind

(k, a) Complete elliptic integral of the third kind

n(x) Weber-Hermite function

(x) Psi function (or the logarithmic derivative of the Gamma function)

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Selected References1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover,

New York, 1965.

2. Artin, E., The Gamma Function, Holt, Rinehart and Winston, New York, 1964.

3. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher Tran-scendental Functions, Bateman Manuscript Project, Vols. 1-3, McGraw-Hill, NewYork, 1953.

4. Fletcher, A., Miller, J.C.P., Rosehead, L. and Comrie, L.J., An Index ofMathematical Tables, Vols. 1 and 2, 2 edition, Addison-Wesley, Reading, Mass., 1962.

5. Hobson, E.W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge Uni-versity Press, London, 1931.

6. Hochsadt, H., Special Functions of Mathematical Physics, Holt, Rinehart and Win-ston, New York, 1961.

7. Jahnke, E., Emdw, F. and Losch, F., Tables of Higher Functions, 6th Edition,McGraw-Hill, New York, 1960.

8. Lebedev, A.V. and Fedorova, R.M., A Guide to Mathematical Tables, PergamonPress, Oxford, 1960.

9. Lebedev, N.N., Special Functions and Their Applications, Prentice-Hall, EnglewoodCliffs, NJ, 1965.

10. MacRobert, T.M., Spherical Harmonics, 2nd Edition, Mathuen, London, 1947.

11. Magnus, W., Oberhettinger, F. and Soni, R.P., Formulas and Theorems for theFunctions of Mathematical Physics, 3rd Edition, Springer-Verlag, New York, 1966.

12. McLachlan, N.W., Bessel Functions for Engineers, 2nd Edition, Oxford UniversityPress, London, 1955.

13. Rainville, E.D., Special Functions, MacMillan, New York, 1960.

14. Relton, F.E., Applied Bessel Functions, Blackie, London, 1946.

15. Slater, L.J., Confluent Hypergeometric Functions, cambridge University Press, Lon-don, 1960.

16. Sneddon, I.N., Special Functions of Mathematical Physics and Chemistry, 2nd Edi-tion, Oliver and Boyd, Edinburgh, 1961.

17. Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd Edition, CambridgeUniversity Press, London, 1931.

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18. National Bureau of Standards, Applied Mathematics Series 41, Tables of ErrorFunction and Its Derivatives, 2nd Edition, Washington, DC, US Government PrintingOffice, 1954.

19. Franklin P., A Treatise on Advance Calculus, Chapter 16, Dover Publications, New

York, 1940.

20. Hancock H., Elliptic Integralsd, Dover Publications, New York, 1917, p69 and p81.

21. Tranter, C.J., Integral Transforms in Mathematical Physics, 2nd Edition, Methuen,London, 1956, pp 67-72.

22. Stroud, A.H. and Secrest, D., Gaussian Quadrature Formulas, Prentice-Hall Inc.,Englewood Cliffs, NJ, 1966.

23. Bauer, F.L., Rutishauser, H. and Stiefel, E., New Aspects in Numerical Quadra-ture, Symposia in Applied Mathematics, 15, Providence, RI, American Mathematical

Society, 1963, pp. 199-218.

24. Ralston, A., A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.

25. Gerald, C.F., Applied Numerical Analysis, 2nd Edition, Addison-Wesley, Reading,Mass., 1978.

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Factorial, Gamma and Beta Functions

Reading Problems

OutlineBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Factorial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Digamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8

Incomplete Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

Incomplete Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

Assigned Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

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Background

Louis Franois Antoine Arbogast (1759 - 1803) a French mathematician, is generally creditedwith being the first to introduce the concept of the factorial as a product of a fixed numberof terms in arithmetic progression. In an effort to generalize the factorial function to non-

integer values, the Gamma function was later presented in its traditional integral form bySwiss mathematician Leonhard Euler (1707-1783). In fact, the integral form of the Gammafunction is referred to as the second Eulerian integral. Later, because of its great importance,it was studied by other eminent mathematicians like Adrien-Marie Legendre (1752-1833),Carl Friedrich Gauss (1777-1855), Cristoph Gudermann (1798-1852), Joseph Liouville (1809-1882), Karl Weierstrass (1815-1897), Charles Hermite (1822 - 1901), as well as many others.1

The first reported use of the gamma symbol for this function was by Legendre in 1839. 2

The first Eulerian integral was introduced by Euler and is typically referred to by its morecommon name, the Beta function. The use of the Beta symbol for this function was first

used in 1839 by Jacques P.M. Binet (1786 - 1856).

At the same time as Legendre and Gauss, Cristian Kramp (1760 - 1826) worked on thegeneralized factorial function as it applied to non-integers. His work on factorials was in-dependent to that of Stirling, although Sterling often receives credit for this effort. He didachieve one first in that he was the first to use the notation n! although he seems not tobe remembered today for this widely used mathematical notation3.

A complete historical perspective of the Gamma function is given in the work of Godefroy4

as well as other associated authors given in the references at the end of this chapter.

1http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html2Cajori, Vol.2, p. 2713Elements darithmtique universelle , 18084M. Godefroy, La fonction Gamma; Theorie, Histoire, Bibliographie, Gauthier-Villars, Paris (1901)

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Definitions

1. Factorial

n! = n(n 1)(n 2) . . . 3 2 1 for all integers, n > 0

2. Gamma

also known as: generalized factorial, Eulers second integral

The factorial function can be extended to include all real valued argumentsexcluding the negative integers as follows:

z! =

0

et tz dt z = 1, 2, 3, . . .

or as the Gamma function:

(z) =

0

et tz1 dt = (z 1)! z = 1, 2, 3, . . .

3. Digamma

also known as: psi function, logarithmic derivative of the gamma function

(z) =d ln(z)

dz=

(z)

(z)z = 1, 2, 3, . . .

4. Incomplete Gamma

The gamma function can be written in terms of two components as follows:

(z) = (z, x) + (z, x)

where the incomplete gamma function, (z, x), is given as

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(z, x) =

x0

et tz1 dt x > 0

and its compliment, (z, x), as

(z, x) =

x

et tz1 dt x > 0

5. Beta

also known as: Eulers first integral

B(y, z) =

10

ty1 (1 t)z1 dt

=(y) (z)

(y + z)

6. Incomplete Beta

Bx(y, z) =

x0

ty1 (1 t)z1 dt 0 x 1

and the regularized (normalized) form of the incomplete Beta function

Ix(y, z) =Bx(y, z)

B(y, z)

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Theory

Factorial Function

The classical case of the integer form of the factorial function, n!, consists of the product of

n and all integers less than n, down to 1, as follows

n! =

n(n 1)(n 2) . . . 3 2 1 n = 1, 2, 3, . . .

1 n = 0(1.1)

where by definition, 0! = 1.

The integer form of the factorial function can be considered as a special case of two widely

used functions for computing factorials of non-integer arguments, namely the Pochham-mers polynomial, given as

(z)n =

z(z + 1)(z + 2) . . . (z + n 1) = (z + n)(z)

n > 0

=(z + n 1)!

(z 1)!

1 = 0! n = 0

(1.2)

and the gamma function (Eulers integral of the second kind).

(z) = (z 1)! (1.3)

While it is relatively easy to compute the factorial function for small integers, it is easy to seehow manually computing the factorial of larger numbers can be very tedious. Fortunately

given the recursive nature of the factorial function, it is very well suited to a computer andcan be easily programmed into a function or subroutine. The two most common methodsused to compute the integer form of the factorial are

direct computation: use iteration to produce the product of all of the counting numbersbetween n and 1, as in Eq. 1.1

recursive computation: define a function in terms of itself, where values of the factorialare stored and simply multiplied by the next integer value in the sequence

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Another form of the factorial function is the double factorial, defined as

n!! =

n(n 2) . . . 5 3 1 n > 0 odd

n(n 2) . . . 6 4 2 n > 0 even

1 n = 1, 0

(1.4)

The first few values of the double factorial are given as

0!! = 1 5!! = 151!! = 1 6!! = 482!! = 2 7!! = 105

3!! = 3 8!! = 3844!! = 8 9!! = 945

While there are several identities linking the factorial function to the double factorial, perhapsthe most convenient is

n! = n!!(n 1)!! (1.5)

Potential Applications

1. Permutations and Combinations: The combinatory function C(n, k) (n choose k)allows a concise statement of the Binomial Theorem using symbolic notation and inturn allows one to determine the number of ways to choose k items from n items,regardless of order.

The combinatory function provides the binomial coefficients and can be defined as

C(n, k) =n!

k!(n k)! (1.6)

It has uses in modeling of noise, the estimation of reliability in complex systems aswell as many other engineering applications.

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Gamma Function

The factorial function can be extended to include non-integer arguments through the use ofEulers second integral given as

z! =

0

et tz dt (1.7)

Equation 1.7 is often referred to as the generalized factorial function.

Through a simple translation of the z variable we can obtain the familiar gamma functionas follows

(z) =0

et tz1 dt = (z 1)! (1.8)

The gamma function is one of the most widely used special functions encountered in advancedmathematics because it appears in almost every integral or series representation of otheradvanced mathematical functions.

Lets first establish a direct relationship between the gamma function given in Eq. 1.8 andthe integer form of the factorial function given in Eq. 1.1. Given the gamma function(z + 1) = z! use integration by parts as follows:

u dv = uv

v du

where from Eq. 1.7 we see

u = tz du = ztz1 dt

dv = et dt v = et

which leads to

(z + 1) =

0

et tz dt =

et tz

0

+ z

0

et tz1 dt

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Given the restriction of z > 0 for the integer form of the factorial function, it can be seenthat the first term in the above expression goes to zero since, when

t = 0 tn 0

t = et 0

Therefore

(z + 1) = z

0

et tz1 dt

(z)= z (z), z > 0 (1.9)

When z = 1 tz1 = t0 = 1, and

(1) = 0! =

0

et dt =et

0= 1

and in turn

(2) = 1 (1) = 1 1 = 1!

(3) = 2 (2) = 2 1 = 2!

(4) = 3 (3) = 3 2 = 3!

In general we can write

(n + 1) = n! n = 1, 2, 3, . . . (1.10)

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The gamma function constitutes an essential extension of the idea of a factorial, since theargument z is not restricted to positive integer values, but can vary continuously.

From Eq. 1.9, the gamma function can be written as

(z) =(z + 1)

z

From the above expression it is easy to see that when z = 0, the gamma function approaches or in other words (0) is undefined.

Given the recursive nature of the gamma function, it is readily apparent that the gammafunction approaches a singularity at each negative integer.

However, for all other values of z, (z) is defined and the use of the recurrence relationshipfor factorials, i.e.

(z + 1) = z (z)

effectively removes the restriction that x be positive, which the integral definition of thefactorial requires. Therefore,

(z) =

(z + 1)

z , z = 0, 1, 2, 3, . . . (1.11)

A plot of (z) is shown in Figure 1.1.

Several other definitions of the -function are available that can be attributed to the pio-neering mathematicians in this area

Gauss

(z) = limn

n! nz

z(z + 1)(z + 2) . . . (z + n), z = 0, 1, 2, 3, . . . (1.12)

Weierstrass

1

(z)= z ez

n=1

1 +

z

n

ez/n (1.13)

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4 2 0 2 4z

15

10

5

0

5

10

15

z

Figure 1.1: Plot of Gamma Function

where is the Euler-Mascheroni constant, defined by

= limn

nk=1

1

k ln(n) = 0.57721 56649 0 . . . (1.14)

An excellent approximation of is given by the very simple formula

=1

2

3

10 1

= 0.57721 73 . . .

Other forms of the gamma function are obtained through a simple change of variables, asfollows

(z) = 2

0

y2z1 ey2

dy by letting t = y2 (1.15)

(z) =

10

ln

1

y

z1dy by letting et = y (1.16)

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Relations Satisfied by the -Function

Recurrence Formula

(z + 1) = z (z) (1.17)

Duplication Formula

22z1 (z)

z +

1

2

=

(2z) (1.18)

Reflection Formula

(z) (1 z) = sin z

(1.19)

Some Special Values of the Gamma Function

Using Eq. 1.15 or Eq. 1.19 we have

(1/2) = (1/2)! = 2

0

ey2

dy I

=

(1.20)

where the solution to I is obtained from Schaums Handbook of Mathematical Functions(Eq. 18.72).

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Combining the results of Eq. 1.20 with the recurrence formula, we see

(1/2) =

(3/2) =1

2(1/2) =

2

(5/2) =3

2(3/2) =

3

2

2=

3

4

...

n +

1

2

=

1 3 5 (2n 1)2n

n = 1, 2, 3, . . .

For z > 0, (z) has a single minimum within the range 1 z 2 at 1.46163 21450where (z) = 0.88560 31944. Some selected 10 decimal place values of (z) are found inTable 1.1.

Table 1.1: 10 Decimal Place Values of (z) for 1 z 2

z (z)

1.0 1.00000 000001.1 0.95135 076991.2 0.91816 874241.3 0.89747 069631.4 0.88726 381751.5 0.88622 692551.6 0.89351 534931.7 0.90863 873291.8 0.93138 377101.9 0.96176 58319

2.0 1.00000 00000

For other values of z (z = 0, 1, 2. . . . ), (z) can be computed by means of therecurrence formula.

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Approximations

Asymptotic Representation of the Factorial and Gamma Functions

Asymptotic expansions of the factorial and gamma functions have been developed for

z >> 1. The expansion for the factorial function is

z! = (z + 1) =

2z zz ez A(z) (1.21)

where

A(z) = 1 +1

12z+

1

288z2 139

51840z3 571

2488320z4+ (1.22)

The expansion for the natural logarithm of the gamma function is

ln(z) =

z 1

2

ln z z + 1

2ln(2) +

1

12z 1

360z3+

1

1260z5

11680z7

+ (1.23)

The absolute value of the error is less than the absolute value of the first term neglected.

For large values of z, i.e. as z , both expansions lead to Stirlings Formula, given as

z! =

2 zz+1/2 ez (1.24)

Even though the asymptotic expansions in Eqs. 1.21 and 1.23 were developed for very largevalues ofz, they give remarkably accurate values of z! and (z) for small values ofz. Table

1.2 shows the relative error between the asymptotic expansion and known accurate valuesfor arguments between 1 z 7, where the relative error is defined as

relative error =approximate value accurate value

accurate value

13


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Table 1.2: Comparison of Approximate value of z! by Eq. 1.21 and (z) by Eq. 1.23 withthe Accurate values of Mathematica 5.0

zz! Eq.1.21

z! Mathematica

error(z) Eq.1.23

(z) Mathematica

error

1 0.99949 9469 5.0 104 0.99969 2549 3.1 104

2 0.99997 8981 2.1 105 0.99999 8900 1.1 106

3 0.99999 7005 3.0 106 0.99999 9965 3.5 108

4 0.99999 9267 7.3 107 0.99999 9997 2.8 109

5 0.99999 9756 2.4 107 0.99999 9999 4.0 1010

6 0.99999 9901 9.9 108 0.99999 9999 7.9 1011

7 0.99999 9954 4.6 108 0.99999 9999 2.0 1011

The asymptotic expansion for (z) converges very quickly to give accurate values for rela-tively small values of z. The asymptotic expansion for z! converges less quickly and doesnot yield 9 decimal place accuracy even when z = 7.

More accurate values of(z) for small z can be obtained by means of the recurrence formula.For example, if we want (1+z) where 0 z 1, then by means of the recurrence formulawe can write

(1 + z) =(n + z)

(1 + z)(2 + z)(3 + z) . . . (n 1 + z) (1.25)

where n is an integer greater that 4. For n = 5 and z = 0.3, we have

(1 + 0.3) =(5.3)

(1.3)(2.3)(3.3)(4.3)= 0.89747 0699

This value can be compared with the 10 decimal place value given previously in Table 1.1.We observe that the absolute error is approximately 3 109. Comparable accuracy canbe obtained by means of the above equation with n = 6 and 0 z 1.

14


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Polynomial Approximation of (z + 1) within 0 z 1

Numerous polynomial approximations which are based upon the use of Chebyshev polyno-mials and the minimization of the maximum absolute error have been developed for varyingdegrees of accuracy. One such approximation developed for 0

z

1 due to Hastings8 is

(z + 1) = z!

= 1 + z(a1 + z(a2 + z(a3 + z(a4 + z(a5 +

z(a6 + z(a7 + a8z))))))) + (z) (1.26)

where

|(z)| 3 107

and the coefficients in the polynomial are given as

Table 1.3: Coefficients of Polynomial of Eq. 1.26

a1 = 0.57719 1652 a5 = 0.75670 4078a2 = 0.98820 5891 a6 = 0.48219 9394a3 = 0.89705 6937 a7 = 0.19352 7818a4 = 0.91820 6857 a8 = 0.03586 8343

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Series Expansion of 1/(z) for |z|

The function 1/(z) is an entire function defined for all values ofz. It can be expressed asa series expansion according to the relationship

1

(z)=

k=1

Ckzk, |z| (1.27)

where the coefficients Ck for 0 k 26, accurate to 16 decimal places are tabulated inAbramowitz and Stegun1. For 10 decimal place accuracy one can write

1(z)

=

19k=1

Ckzk (1.28)

where the coefficients are listed below

Table 1.4: Coefficients of Expansion of 1/(z) of Eq. 1.28

k Ck k Ck

1 1.00000 00000 11 0.00012 805022 0.57721 56649 12 0.00002 013483 0.65587 80715 13 0.00000 125044 0.04200 26350 14 0.00000 113305 0.16653 86113 15 0.00000 020566 0.04219 77345 16 0.00000 000617 0.00962 19715 17 0.00000 000508 0.00721 89432 18 0.00000 000119 0.00116 51675 19 0.00000 00001

10 0.00021 52416

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1. Gamma Distribution: The probability density function can be defined based on theGamma function as follows:

f(x,,) =1

()x1ex/

This function is used to determine time based occurrences, such as:

life length of an electronic component

remaining life of a component

waiting time between any two consecutive events

waiting time to see the next event

hypothesis tests

confidence intervals

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Digamma Function

The digamma function is the regularized (normalized) form of the logarithmic derivative ofthe gamma function and is sometimes referred to as the psi function.

(z) =d ln(z)

dz=

(z)

(z)(1.29)

The digamma function is shown in Figure 1.2 for a range of arguments between 4 z 4.

4 2 0 2 4

z

20

10

0

10

20

z

Figure 1.2: Plot of the Digamma Function

The -function satisfies relationships which are obtained by taking the logarithmic derivativeof the recurrence, reflection and duplication formulas of the -function. Thus

(z + 1) =1

z+ (z) (1.30)

(1 z) (z) = cot( z) (1.31)(z) + (z + 1/2) + 2 ln 2 = 2(2z) (1.32)

These formulas may be used to obtain the following special values of the -function:

(1) = (1) = (1.33)

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where is the Euler-Mascheroni constant defined in Eq. (1.14). Using Eq. (1.30)

(n + 1) =

+

n

k=11

k

n = 1, 2, 3, . . . (1.34)

Substitution of z = 1/2 into Eq. (1.32) gives

(1/2) = 2 ln 2 = 1.96351 00260 (1.35)

and with Eq. (1.30) we obtain

(n + 1/2) = 2 ln 2 + 2n

k=1

1

2k 1 , n = 1, 2, 3, . . . (1.36)

Integral Representation of (z)

The -function has simple representations in the form of definite integrals involving thevariable z as a parameter. Some of these are listed below.

(z) = +10

(1 t)1(1 tz1) dt, z > 0 (1.37)

(z) = cot( z) +10

(1 t)1(1 tz) dt, z < 1 (1.38)

(z) =

0

et

t e

zt

1 et

dt, z > 0 (1.39)

(z) =

0 et (1 + t)zdt

t, z > 0

= +

0

(1 + t)1 (1 + t)z dt

t, z > 0 (1.40)

(z) = ln z +

0

1

t 1

1 et

ezt dt, z > 0

= ln z 12z

0

1

1 et 1

t 1

2

ezt dt, z > 0 (1.41)

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Series Representation of (z)

The -function can be represented by means of several series

(z) = k=0

1z + k

1

1 + k z = 1, 2, 3, . . . (1.42)

(x) = 1x

+ xk=1

1

k(z + k)z = 1, 2, 3, . . . (1.43)

(z) = ln z k=0

1

z + k ln

1 +

1

z + k

z = 1, 2, 3, . . . (1.44)

Asymptotic Expansion of (z) for Large z

The asymptotic expansion of the -function developed for large z is

(z) = ln z 12z

n=1B2n

2nz2nz (1.45)

where B2n are the Bernoulli numbers

B0 = 1 B6 = 1/42B2 = 1/6 B8 = 1/30B4 = 1/30 B10 = 5/66

(1.46)

The expansion can be expressed as

(z) = ln z 12z

112z2

+1

120z4 1

252z6+ z (1.47)

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The Incomplete Gamma Function (z, x), (z, x)

We can generalize the Euler definition of the gamma function by defining the incompletegamma function (z, x) and its compliment (z, x) by the following variable limit integrals

(z, x) =

x0

et tz1 dt z > 0 (1.48)

and

(z, x) =

x

et tz1 dt z > 0 (1.49)

so that

(z, x) + (z, x) = (z) (1.50)

Figure 1.3 shows plots of (z, x), (z, x) and (z) all regularized with respect to (z).We can clearly see that the addition of (z, x)/(z) and (z, x)/(z) leads to a value ofunity or (z)/(z) for each value of z.

The choice of employing (z, x) or (z, x) is simply a matter of analytical or computationalconvenience.

Some special values, integrals and series are listed below for convenience

Special Values of (z, x) and (z, x) for z Integer (let z = n)

(1 + n, x) = n!

1 ex

nk=0

xk

k!

n = 0, 1, 2, . . . (1.51)

(1 + n, x) = n! exk=0

xk

k!n = 0, 1, 2, . . . (1.52)

(n, x) = (1)n

n!

(0, x) ex

n1k=0

(1)k k!xk+1

n = 1, 2, 3 . . . (1.53)

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0 2 4 6 8 10x

0.2

0.4

0.60.8

1

z,x

z

z,xz, z 1, 2, 3, 4

0 2 4 6 8 10

x

0.2

0.4

0.6

0.8

1

z,x

z

z,xz, a 1, 2, 3, 4

0 2 4 6 8 10x

0.2

0.4

0.6

0.8

1

xa

zz, a 1, 2, 3, 4

Figure 1.3: Plot of the Incomplete Gamma Function where(z, x)

(z)+

(z, x)

(z)=

(z)

(z)

22


34/171

Integral Representations of the Incomplete Gamma Functions

(z, x) = xz cosec( z)

0

ex cos cos(z + x sin ) d

x = 0, z > 0, z = 1, 2, . . . (1.54)

(z, x) =exxz

(1 z)

0

et tz

x + tdt z < 1, x > 0 (1.55)

(z,xy) = yzexy

0

ety (t + x)z1 dt y > 0, x > 0, z > 1 (1.56)

Series Representations of the Incomplete Gamma Functions

(z, x) =n=0

(1)n xz+nn! (z + n)

(1.57)

(z, x) = (z) n=0

(1)n xz+nn! (z + n)

(1.58)

(z + x) = exxz

n=0

Lzn

(x)

n + 1 x > 0 (1.59)

where Lzn(x) is the associated Laguerre polynomial.

Functional Representations of the Incomplete Gamma Functions

(z + 1, x) = z(z, x) xzex (1.60)

(z + 1, x) = z(z, x) + xzex (1.61)

(z + n, x)

(z + n)=

(z, x)

(z)+ ex

n1k=0

xz+k

(z + k + 1)(1.62)

d(z, x)

dx= d(z, x)

dx= xz1 ex (1.63)

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Asymptotic Expansion of (z, x) for Large x

(z, x) = xz1 ex

1 +(z 1)

x+

(z 1)(z 2)x2

+

x (1.64)

Continued Fraction Representation of (z, x)

(z, x) =ex xz

z +1 z

1 +1

x +2 z

1 +2

x +3

z

1 + . . .

(1.65)

for x > 0 and |z| < .

Relationships with Other Special Functions

(0, x) =

Ei(

x) (1.66)

(0, ln 1/x) = i(x) (1.67)

(1/2, x2) =

(1 erf(x)) = erfc(x) (1.68)

(1/2, x2) =

erf(x) (1.69)

(z, x) = z1 xz ex M(1, 1 + z, x) (1.70)

(z, x) = z1 xz M(z, 1 + z, x) (1.71)

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Beta Function B(a, b)

Another definite integral which is related to the -function is the Beta function B(a, b)which is defined as

B(a, b) =

10

ta1 (1 t)b1 dt, a > 0, b > 0 (1.72)

The relationship between the B-function and the -function can be demonstrated easily. Bymeans of the new variable

u =t

(1 t)

Therefore Eq. 1.72 becomes

B(a, b) =

0

ua1

(1 + u)a+bdu a > 0, b > 0 (1.73)

Now it can be shown that

0

ept tz1 dt = (z)pz

(1.74)

which is obtained from the definition of the -function with the change of variable s = pt.Setting p = 1 + u and z = a + b, we get

1

(1 + u)a+b=

1

(a + b)

0

e(1+u)t ta+b1 dt (1.75)

and substituting this result into the Beta function in Eq. 1.73 gives

B(a, b) =1

(a + b)

0

et ta+b1 dt

0

eut ua1 du

=(a)

(a + b)

0

et tb1 dt

=(a) (b)

(a + b)(1.76)

25


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4 2 0 2 4y

10

5

0

5

10

15

By,z

Betay,.5

Figure 1.4: Plot of Beta Function

All the properties of the Beta function can be derived from the relationships linking the-function and the Beta function.

Other forms of the beta function are obtained by changes of variables. Thus

B(a, b) =

0

ua1 du

(1 + u)a+bby t =

u

1

u(1.77)

B(a, b) = 2

/20

sin2a1 cos2a1 d by t = sin2 (1.78)


1. Beta Distribution: The Beta distribution is the integrand of the Beta function. It canbe used to estimate the average time of completing selected tasks in time management

problems.

Incomplete Beta Function Bx(a, b)

Just as one can define an incomplete gamma function, so can one define the incomplete betafunction by the variable limit integral

26


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Bx(a, b) =

x0

ta1 (1 t)b1 dt 0 x 1 (1.79)

with a > 0 and b > 0 if x = 1. One can also define

Ix(a, b) =Bx(a, b)

B(a, b)(1.80)

Clearly when x = 1, Bx(a, b) becomes the complete beta function and

I1(a, b) = 1

The incomplete beta function and Ix(a, b) satisfies the following relationships:

Symmetry

Ix(a, b) = 1 I1x(b, a) (1.81)

Recurrence Formulas

Ix(a, b) = xIx(a 1, b) + (1 x)Ix(a, b 1) (1.82)

(a + b ax)Ix(a, b) = a(1 x)Ix(a + 1, b 1) + bIx(a, b + 1) (1.83)

(a + b)Ix(a, b) = aIx(a + 1, b) + bIx(a, b + 1) (1.84)

Relation to the Hypergeometric Function

Bx(a, b) = a1xa F(a, 1 b; a + 1; x) (1.85)

27


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4 2 0 2 4y

80

60

40

20

020

40

Bx,y,z

Beta.25, y, .5

4 2 0 2 4y

10

5

0

5

10

Bx,y,z

Beta.75, y, .5

4 2 0 2 4y

10

5

0

5

10

15

Bx,y,z

Beta1, y, .5

Figure 1.5: Plot of the Incomplete Beta Function

28


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Assigned Problems

Problem Set for Gamma and Beta Functions

1. Use the definition of the gamma function with a suitable change of variable to provethat

i)

0

eaxxn dx =1

an+1(n + 1) with n > 1, a > 0

ii)

a

exp(2ax x2) dx =

2exp(a2)

2. Prove that

/20

sinn d =

/20

cosn d =

2

([1 + n]/2)

([2 + n]/2)

3. Show that

12

+ x 12

x = cos x

Plot your results over the range 10 x 10.

4. Evaluate

1

2

and

7

2

.

5. Show that the area enclosed by the axes x = 0, y = 0 and the curve x4 + y4 = 1 is

1

4

28

Use both the Dirichlet integral and a conventional integration procedure to substantiatethis result.

29


41/171

6. Express each of the following integrals in terms of the gamma and beta functions andsimplify when possible.

i) 1

0 1

x 1

1/4

dx

ii)

ba

(b x)m1(x a)n1 dx, with b > a, m > 0, n > 0

iii)

0

dt

(1 + t)

t

Note: Validate your results using various solution procedures where possible.

7. Compute to 5 decimal places

A

4ab=

1

2n

1

n

2

2

n

for n = 0.2, 0.4, 0.8, 1.0, 2.0, 4.0, 8.0, 16.0, 32.0, 64.0, 100.0

8. Sketch x3 + y3 = 8. Derive expressions of the integrals and evaluate them in termsof Beta functions for the following quantities:

a) the first quadrant area bounded by the curve and two axesb) the centroid (x, y) of this areac) the volume generated when the area is revolved about the yaxisd) the moment of inertia of this volume about its axis

Note: Validate your results using various solution procedures where possible.

9. Starting with

1

2

=

0

et dtt

and the transformation y2 = t or x2 = t, show that

30


42/171

1

2

2= 4

0

0

exp(x2 + y2) dx dy

Further prove that the above double integral over the first quadrant when evaluatedusing polar coordinates (r, ) yields

1

2

=

31


43/171

References1. Abramowitz, M. and Stegun, I.A., (Eds.), Gamma (Factorial) Function and

Incomplete Gamma Function. 6.1 and 6.5 in Handbook of Mathematical Functionsand Formulas, Graphs and Mathematical Tables, 9th printing, Dover, New York, 1972,pp. 255-258 and pp 260-263.

2. Andrews, G.E., Askey, R. and Roy, R., Special Functions, Cambridge UniversityPress, Cambridge, 1999.

3. Artin, E. The Gamma Function, Holt, Rinehart, and Winston, New York, 1964.

4. Barnew, E.W., The Theory of the Gamma Function, Messenger Math., (2), Vol.29, 1900, pp.64-128..

5. Borwein, J.M. and Zucker, I.J., Elliptic Integral Evaluation of the Gamma Func-tion at Rational Values and Small Denominator, IMA J. Numerical Analysis, Vol.12, 1992, pp. 519-526.

6. Davis, P.J., Leonhard Eulers Integral: Historical profile of the Gamma Function,Amer. Math. Monthly, Vol. 66, 1959, pp. 849-869.

7. Erdelyl, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., The GammaFunction, Ch. 1 in Higher Transcendental Functions, Vol. 1, Krieger, New York,1981, pp. 1-55.

8. Hastings, C., Approximations for Digital Computers, Princeton University Press,Princeton, NJ, 1955.

9.Hochstadt, H.

, Special Functions of Mathematical Physics, Holt, Rinehart and Win-ston, New York, 1961.

10. Koepf, W.. The Gamma Function, Ch. 1 in Hypergeometric Summation: AnAlgorithmic Approach to Summation and Special Identities, Vieweg, Braunschweig,Germany, 1998, pp. 4-10.

11. Krantz, S.C., The Gamma and Beta Functions, 13.1 in Handbook of ComplexAnalysis, Birkhauser, Boston, MA, 1999, pp. 155-158.

12. Legendre, A.M., Memoires de la classe des sciences mathematiques et physiques delInstitut de France, Paris, 1809, p. 477, 485, 490.

13. Magnus, W. and Oberhettinger, F., Formulas and Theorems for the Special Func-tions of Mathematical Physics, Chelsea, New York, 1949.

14. Saibagki, W., Theory and Applications of the Gamma Function, Iwanami Syoten,Tokyo, Japan, 1952.

15. Spanier, J. and Oldham, K.B., The Gamma Function (x) and The IncompleteGamma (, x) and Related Functions, Chs. 43 and 45 in An Atlas of Functions,Hemisphere, Washington, DC, 1987, pp. 411-421 and pp. 435-443.

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Error and Complimentary Error Functions

Reading Problems


Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Gaussian function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Complementary Error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Relations and Selected Values of Error Functions . . . . . . . . . . . . . . . . . . . . . . . . 1 2

Numerical Computation of Error Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Rationale Approximations of Error Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

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Background

The error function and the complementary error function are important special functionswhich appear in the solutions of diffusion problems in heat, mass and momentum transfer,probability theory, the theory of errors and various branches of mathematical physics. It

is interesting to note that there is a direct connection between the error function and theGaussian function and the normalized Gaussian function that we know as the bell curve.The Gaussian function is given as

G(x) = Aex2/(22)

where is the standard deviation and A is a constant.

The Gaussian function can be normalized so that the accumulated area under the curve isunity, i.e. the integral from to + equals 1. If we note that the definite integral

eax2

dx =

a

then the normalized Gaussian function takes the form

G(x) =1

2ex

2/(22)

If we let

t2 =x2

22and dt =

12

dx

then the normalized Gaussian integrated between x and +x can be written as

xx

G(x) dx =1

et

2

dt

or recognizing that the normalized Gaussian is symmetric about the yaxis, we can write

2


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x0

G(x) dx =2

x0

et2

dt = erf x = erf

x2

and the complementary error function can be written as

erfc x = 1 erf x = 2

x

et2

dt

Historical Perspective

The normal distribution was first introduced by de Moivre in an article in 1733 (reprinted inthe second edition of his Doctrine of Chances, 1738 ) in the context of approximating certainbinomial distributions for large n. His result was extended by Laplace in his book AnalyticalTheory of Probabilities (1812 ), and is now called the Theorem of de Moivre-Laplace.

Laplace used the normal distribution in the analysis of errors of experiments. The importantmethod of least squares was introduced by Legendre in 1805. Gauss, who claimed to haveused the method since 1794, justified it in 1809 by assuming a normal distribution of theerrors.

The name bell curve goes back to Jouffret who used the term bell surface in 1872 for a

bivariate normal with independent components. The name normal distribution was coinedindependently by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875 [Stigler].This terminology is unfortunate, since it reflects and encourages the fallacy that everythingis Gaussian.

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Definitions

1. Gaussian Function

The normalized Gaussian curve represents the probability distribution with standard

distribution and mean relative to the average of a random distribution.

G(x) =1

2e(x)

2/(22)

This is the curve we typically refer to as the bell curve where the mean is zero andthe standard distribution is unity.

2. Error Function

The error function equals twice the integral of a normalized Gaussian function between0 and x/

2.

y = erf x =2

x0

et2

dt for x 0, y [0, 1]

where

t =x2

3. Complementary Error Function

The complementary error function equals one minus the error function

1 y = erfc x = 1 erf x = 2

x

et2

dt for x 0, y [0, 1]

4. Inverse Error Function

x = inerf y

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inerf y exists for y in the range 1 < y < 1 and is an odd function of y with aMaclaurin expansion of the form

inverf y =

n=1

cn y2n1

5. Inverse Complementary Error Function

x = inerfc (1 y)

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Theory

Gaussian Function

The Gaussian function or the Gaussian probability distribution is one of the most fundamen-

tal functions. The Gaussian probability distribution with mean and standard deviation is a normalized Gaussian function of the form

G(x) =1

2e(x)

2/(22) (2.1)

where G(x), as shown in the plot below, gives the probability that a variate with a Gaussiandistribution takes on a value in the range [x, x + dx]. Statisticians commonly call thisdistribution the normal distribution and, because of its shape, social scientists refer to it as

the bell curve. G(x) has been normalized so that the accumulated area under the curvebetween x + totals to unity. A cumulative distribution function, which totalsthe area under the normalized distribution curve is available and can be plotted as shownbelow.

4 2 2 4x

Gx

4 2 2 4x

Dx

Figure 2.1: Plot of Gaussian Function and Cumulative Distribution Function

When the mean is set to zero ( = 0) and the standard deviation or variance is set to unity( = 1), we get the familiar normal distribution

G(x) = 12

ex2/2dx (2.2)

which is shown in the curve below. The normal distribution function N(x) gives the prob-ability that a variate assumes a value in the interval [0, x]

N(x) =12

x0

et2/2 dt (2.3)

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4 2 2 4x

0.1

0.2

0.3

0.4

Nx

Figure 2.2: Plot of the Normalized Gaussian Function

Gaussian distributions have many convenient properties, so random variates with unknown

distributions are often assumed to be Gaussian, especially in physics, astronomy and variousaspects of engineering. Many common attributes such as test scores, height, etc., followroughly Gaussian distributions, with few members at the high and low ends and many inthe middle.


Function Maple Mathematica

Probability Density Function statevalf[pdf,dist](x) PDF[dist, x]- frequency of occurrence at x

Cumulative Distribution Function statevalf[cdf,dist](x) CDF[dist, x]- integral of probability

density function up to x dist = normald[, ] dist = NormalDistribution[ = 0 (mean) = 0 (mean) = 1 (std. dev.) = 1 (std. dev.)


1. Statistical Averaging:

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Error Function

The error function is obtained by integrating the normalized Gaussian distribution.

erf x =2

x

0

et2

dt (2.4)

where the coefficient in front of the integral normalizes erf() = 1. A plot of erf x overthe range 3 x 3 is shown as follows.

3 2 1 0 1 2 3

x

1

0.5

0

0.5

1

erfx

Figure 2.3: Plot of the Error Function

The error function is defined for all values of x and is considered an odd function in x sinceerf x = erf (x).

The error function can be conveniently expressed in terms of other functions and series asfollows:

erf x =1

1

2, x2

(2.5)

=2x

M

1

2,

3

2,x2 =

2x

ex2

M1,3

2, x2 (2.6)

=2

n=0

(1)nx2n+1n!(2n + 1)

(2.7)

where () is the incomplete gamma function, M() is the confluent hypergeometric functionof the first kind and the series solution is a Maclaurin series.

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Function Maple Mathematica

Error Function erf(x) Erf[x]

Complementary Error Function erfc(x) Erfc[x]

Inverse Error Function fslove(erf(x)=s) InverseErf[s]

Inverse Complementary fslove(erfc(x)=s) InverseErfc[s]Error Function

where s is a numerical value and we solve for x


1. Diffusion: Transient conduction in a semi-infinite solid is governed by the diffusionequation, given as

2T

x2

=1

T

t

where is thermal diffusivity. The solution to the diffusion equation is a function ofeither the erf x or erfc x depending on the boundary condition used. For instance,for constant surface temperature, where T(0, t) = Ts

T(x, t) TsTi Ts

= erfc

x

2

t

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Complimentary Error Function

The complementary error function is defined as

erfc x = 1

erf x

=2

x

et2

dt (2.8)

3 2 1 0 1 2 3

x

0.5

1

1.5

2

erfx

Figure 2.4: Plot of the Complimentary Error Function

and similar to the error function, the complimentary error function can be written in termsof the incomplete gamma functions as follows:

erfc x =1

1

2, x2

(2.9)

As shown in Figure 2.5, the superposition of the error function and the complimentary errorfunction when the argument is greater than zero produces a constant value of unity.


1. Diffusion: In a similar manner to the transient conduction problem described for theerror function, the complimentary error function is used in the solution of the diffusionequation when the boundary conditions are constant surface heat flux, where qs = q0

T(x, t) Ti =2q0(t/)

1/2

kexp

x24t

q0x

kerfc

x

2

t

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0 0.5 1 1.5 2 2.5 3

x

0.2

0.4

0.6

0.8

1

er

fx

Erf xErfc

Erf x

Erfc x

Figure 2.5: Superposition of the Error and Complimentary Error Functions

and surface convection, where k Tx

x=0

= h[T T(0, t)]

T(x, t) TiT Ti

= erfc

x

2

t

exp

hx

k+

h2t

k2

erfc

x

2

t+

h

t

k

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Relations and Selected Values of Error Functions

erf (x) = erf x erfc (x) = 2 erfc x

erf 0 = 0 erfc 0 = 1

erf = 1 erfc = 0

erf () = 10

erfc x dx = 1/

0

erfc2 x dx = (2 2)/

Ten decimal place values for selected values of the argument appear in Table 2.1.

Table 2.1 Ten decimal place values of erf x

x erf x x erf x

0.0 0.00000 00000 2.5 0.99959 304800.5 0.52049 98778 3.0 0.99997 790951.0 0.84270 07929 3.5 0.99999 925691.5 0.96610 51465 4.0 0.99999 998462.0 0.99532 22650 4.5 0.99999 99998

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Approximations

Power Series for Small x (x < 2)

Since

erf x =2

x0

et2

dt =2

x0

n=0

(1)nt2nn!

dt (2.10)

and the series is uniformly convergent, it may be integrated term by term. Therefore

erf x =2

n=0

(1)nx2n+1(2n + 1)n!

(2.11)

= 2

x

1 0! x3

3 1! + x5

5 2! x7

7 3! + x9

9 4!

(2.12)

Asymptotic Expansion for Large x (x > 2)

Since

erfc x =2

x

et2

dt =2

x

1

tet

2

t dt

we can integrate by parts by letting

u =1

tdv = et

2

d dt

du = t2 dt v = 12

et2

therefore

x

1

tet

2

t dt =

uv

x

x

v du =

1

2tet

2

x

x

1

2

et2

t2dt

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Thus

erfc x =2

1

2xex

2 12

x

et2

t2dt (2.13)

Repeating the process n times yields

2erfc x =

1

2ex

2

1

x 1

2x3+

1 322x5

+ (1)n11 3 (2n 3)2n1x2n1

+

+(1)n 1 3 (2n 1)2n

x

et2

t2ndt (2.14)

Finally we can write

xex

2

erfc x = 1 +

n=1

(1)n 1 3 5 (2n 1)(2x2)n

(2.15)

This series does not converge, since the ratio of the nth term to the (n1)th does not remainless than unity as n increases. However, if we take n terms of the series, the remainder,

1 3 (2n 1)2n

x

et2

t2ndt

is less than the nth term because

x

et2

t2ndt < ex

2

0 (2.41)

0

et2

x

t2 + y2dt =

2yexy

2

erfc (xy) x > 0, y > 0 (2.42)

0

etx

(t + y)

tdt =

yexy erfc (xy) x > 0, y = 0 (2.43)

0

et xerf (

yt) dt =

y

x(x + y)1/2 (x + y) > 0 (2.44)

0

et xerf (y/t dt =1

xe2

xy x > 0, y > 0 (2.45)

a

erfc (t) dt = ierfc (a) + 2a = ierfc (a) (2.46)aa

erf (t) dt = 0 (2.47)

aa

erfc(t) dt = 2a (2.48)

aierfc (t) dt = i2 erfc (

a) =

1

2+ a

i2 erfc (a) (2.49)

a

in erfc

t + c

b

dt = bin+1 erfc

a + c

b

(2.50)

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Numerical Computation of Error Functions

The power series form of the error function is not recommended for numerical computationswhen the argument approaches and exceeds the value x = 2 because the large alternat-ing terms may cause cancellation, and because the general term is awkward to compute

recursively. The function can, however, be expressed as a confluent hypergeometric series.

erf x =2

x ex2

M

1,

3

2, x2

(2.51)

in which all terms are positive, and no cancellation can occur. If we write

erf x = b

n=0

an 0 x 2 (2.52)

with

b =2x

ex

2

a0 = 1 an =x2

(2n + 1)/2an1 n 1

then erf x can be computed very accurately (e.g. with an absolute error less that 109).Numerical experiments show that this series can be used to compute erf x up to x = 5to the required accuracy; however, the time required for the computation of erf x is muchgreater due to the large number of terms which have to be summed. For x 2 an alternatemethod that is considerably faster is recommended which is based upon the asymptoticexpansion of the complementary error function.

erfc x =2

x

et2

dt

=ex

2

x

2 Fo

1

2, 1, 1

x2

x (2.53)

which cannot be used to obtain arbitrarily accurate values for any x. An expression thatconverges for all x > 0 is obtained by converting the asymptotic expansion into a continuedfraction

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ex

2

erfc x =1

x +1/2

x +1

x + 3/2

x +2

x +5/2

x + . . .

x > 0 (2.54)

which for convenience will be written as

erfc x =ex

2

1

x+

1/2

x+

1

x+

3/2

x+

2

x+ x > 0 (2.55)It can be demonstrated experimentally that for x 2 the 16th approximant gives erfc xwith an absolute error less that 109. Thus we can write

erfc x =ex

2

1

x+

1/2

x+

1

x+

3/2

x+ 8

x

x 2 (2.56)

Using a fixed number of approximants has the advantage that the continued fraction can beevaluated rapidly beginning with the last term and working upward to the first term.

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Rational Approximations of the Error Functions (0 x < )Numerous rational approximations of the error functions have been developed for digitalcomputers. The approximations are often based upon the use of economized Chebyshevpolynomials and they give values of erf x from 4 decimal place accuracy up to 24 decimal

place accuracy.

Two approximations by Hastings et al.11 are given below.

erf x = 1 [t(a1 + t(a2 + a3t))] ex2 + (x) 0 x (2.57)

where

t =1

1 + px

and the coefficients are

p = 0.47047

a1 = 0.3480242

a2 = 0.0958798

a3 = 0.7478556

This approximation has a maximum absolute error of |(x)| < 2.5 105.

Another more accurate rational approximation has been developed for example

erf x = 1 [t(a1 + t(a2 + t(a3 + t(a4 + a5t))))] ex2 + (x) (2.58)

where

t =1

1 + px

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and the coefficients are

p = 0.3275911

a1 = 0.254829592

a2 = 0.284496736

a3 = 1.421413741

a4 = 1.453152027

a5 = 1.061405429

This approximation has a maximum absolute error of|(x)

|< 1.5

107.

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Assigned Problems

Problem Set for Error and Due Date: February 12, 2004Complementary Error Function

1. Evaluate the following integrals to four decimal places using either power series, asymp-totic series or polynomial approximations:

a)

20

ex2

dx b)

0.0020.001

ex2

dx

c) 21.5

ex2

dx d) 2105

ex2

dx

e)

1.51

1

2ex

2

dx f)

2

1

1

2ex

2

dx

2. The value of erf 2 is 0.995 to three decimal places. Compare the number of termsrequired in calculating this value using:

a) the convergent power series, andb) the divergent asymptotic series.

Compare the approximate errors in each case after two terms; after ten terms.

3. For the function ierfc(x) compute to four decimal places when x = 0, 0.2, 0.4, 0.8,and 1.6.

4. Prove that

i)

erf(x) =

1

2, x2

ii)

erfc(x) =

1

2, x2

where

1

2, x2

and

1

2, x2

are the incomplete Gamma functions defined as:

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(a, y) =

y0

euua1 du

and

(a, y) =

y

euua1 du

5. Show that (x, t) = 0 erfc(x/2

t) is the solution of the following diffusionproblem:

2

x2=

1

tx 0, t > 0

and

(0, t) = 0, constant

(x, t)

0 as x

6. Given (x, t) = 0 erf x/2

t:

i) Obtain expressions for

tand

xat any x and all t > 0

ii) For the function

2

x

0

x

show that it has a maximum value when x/2

t = 1/

2 and the maximum value

is 1/

2e .

7. Given the transient point source solution valid within an isotropic half space

T =q

2krerfc(r/2

t), dA = r dr d

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derive the expression for the transient temperature rise at the centroid of a circulararea (a2) which is subjected to a uniform and constant heat flux q. Superposition ofpoint source solutions allows one to write

T0 =

a

0

2

0

T dA

8. For a dimensionless time F o < 0.2 the temperature distribution within an infiniteplate L x L is given approximately by

T( , F o) TsT0

Ts= 1 erfc

1 2

F o+ erfc

1 +

2

F ofor 0 1 where = x/L and F o = t/L2.Obtain the expression for the mean temperature (T(F o) Ts)/(T0 Ts) where

T =

10

T( , F o) d

The initial and surface plate temperature are denoted by T0 and Ts, respectively.

9. Compare the approximate short time (F o < 0.2) solution:

( , F o) = 1 3

n=1

(1)n+1

erfc(2n 1)

2

F o+ erfc

(2n 1) + 2

F o

and the approximate long time (F o > 0.2) solution

( , F o) =3

n=1

2(1)n+1n

e2n

F o cos(n)

with n = (2n 1)/2.For the centerline ( = 0) compute to four decimal places (0, F o)ST and (0, F o)LTfor F o = 0.02, 0.06, 0.1, 0.4, 1.0 and 2.0 and compare your values with the exactvalues given in Table 1.

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Table 1: Exact values of (0, F o) for the Infinite Plate

F o (0, F o)

0.02 1.0000

0.06 0.99220.10 0.94930.40 0.47451.0 0.10802.0 0.0092

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References

1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover,New York, 1965.

2. Fletcher, A., Miller, J.C.P., Rosehead, L. and Comrie, L.J., An Index of

Mathematical Tables, Vols. 1 and 2, 2 edition, Addison-Wesley, Reading, Mass., 1962.

3. Hochsadt, H., Special Functions of Mathematical Physics, Holt, Rinehart and Win-ston, New York, 1961.

4. Jahnke, E., Emdw, F. and Losch, F., Tables of Higher Functions, 6th Edition,McGraw-Hill, New York, 1960.

5. Lebedev, A.V. and Fedorova, R.M., A Guide to Mathematical Tables, PergamonPress, Oxford, 1960.

6. Lebedev, N.N., Special Functions and Their Applications, Prentice-Hall, EnglewoodCliffs, NJ, 1965.

7. Magnus, W., Oberhettinger, F. and Soni, R.P., Formulas and Theorems for theFunctions of Mathematical Physics, 3rd Edition, Springer-Verlag, New York, 1966.

8. Rainville, E.D., Special Functions, MacMillan, New York, 1960.

9. Sneddon, I.N., Special Functions of Mathematical Physics and Chemistry, 2nd Edi-tion, Oliver and Boyd, Edinburgh, 1961.

10. National Bureau of Standards, Applied Mathematics Series 41, Tables of Error

Function and Its Derivatives, 2nd Edition, Washington, DC, US Government PrintingOffice, 1954.

11. Hastings, C., Approximations for Digital Computers, Princeton University Press,Princeton, NJ, 1955.

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Elliptic Integrals, Elliptic Functions and

Theta Functions

Reading Problems


Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Elliptic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

Theta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

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Background

This chapter deals with the Legendre elliptic integrals, the Theta functions and the Jaco-bian elliptic functions. These elliptic integrals and functions find many applications in thetheory of numbers, algebra, geometry, linear and non-linear ordinary and partial differential

equations, dynamics, mechanics, electrostatics, conduction and field theory.

An elliptic integral is any integral of the general form

f(x) =

A(x) + B(x)

C(x) + D(x)

S(x)dx

where A(x), B(x), C(x) and D(x) are polynomials in x and S(x) is a polynomial ofdegree 3 or 4. Elliptic integrals can be viewed as generalizations of the inverse trigonometric

functions. Within the scope of this course we will examine elliptic integrals of the first andsecond kind which take the following forms:

First Kind

If we let the modulus k satisfy 0 k2 < 1 (this is sometimes written in terms of theparameter m k2 or modular angle sin1 k). The incomplete elliptic integral of thefirst kind is written as

F(, k) =sin0

dt(1 t2)(1 k2t2), 0 k

2 1 and 0 sin 1

if we let t = sin and dt = cos d =

1 t2 d, then

F(, k) =

0

d1 k2 sin2

, 0 k2 1 and 0 /2

This is referred to as the incomplete Legendre elliptic integral. The complete elliptic integralcan be obtained by setting the upper bound of the integral to its maximum range, i.e.sin = 1 or = /2 to give

K(k) =

1

0

dt(1 t2)(1 k2t2)

=

/20

d

1 k2 sin2

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Second Kind

E(, k) =

sin

0

1 k2t2

1 t2 dt

=

0

1 k2 sin2 d

Similarly, the complete elliptic integral can be obtained by setting the upper bound of inte-gration to the maximum value to get

E(k) =

1

0

1 k2t2

1 t2 dt

=

/20

1 k2 sin2 t dt

Another very useful class of functions can be obtained by inverting the elliptic integrals. Asan example of the Jacobian elliptic function sn we can write

u(x = sin , k) = F(, k) = sin

0

dt(1 t2)(1 k2t2)

If we wish to find the inverse of the elliptic integral

x = sin = sn(u, k)

or

u =

sn0

dt(1 t2)(1 k2t2)

While there are 12 different types of Jacobian elliptic functions based on the number of polesand the upper limit on the elliptic integral, the three most popular are the copolar trio of sineamplitude, sn(u, k), cosine amplitude, cn(u, k) and the delta amplitude elliptic function,dn(u, k) where

3


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sn2 + cn2 = 1 and k2sn2 + dn2 = 1

Historical Perspective

The first reported study of elliptical integrals was in 1655 when John Wallis began to studythe arc length of an ellipse. Both John Wallis (1616-1703) and Isaac Newton (1643-1727)published an infinite series expansion for the arc length of the ellipse. But it was not until thelate 1700s that Legendre began to use elliptic functions for problems such as the movementof a simple pendulum and the deflection of a thin elastic bar that these types of problemscould be defined in terms of simple functions.

Adrien-Marie Legendre (1752-1833), a French mathematician, is remembered mainly for the

Legendre symbol and Legendre functions which bear his name but he spent more than fortyyears of his life working on elliptic functions, including the classification of elliptic integrals.His first published writings on elliptic integrals consisted of two papers in the Memoires delAcadmie Francaise in 1786 based on elliptic arcs. In 1792 he presented to the Acadmie amemoir on elliptic transcendents.

Legendres major work on elliptic functions appeared in 3 volumes 5 in 1811-1816. In thefirst volume Legendre introduced basic properties of elliptic integrals as well as propertiesfor beta and gamma functions. More results on beta and gamma functions appeared in thesecond volume together with applications of his results to mechanics, the rotation of theEarth, the attraction of ellipsoids and other problems. The third volume contained the veryelaborate and now well-known tables of elliptic integrals which were calculated by Legendrehimself, with an account of the mode of their construction. He then repeated much of thiswork again in a three volume set 6 in 1825-1830.

Despite forty years of dedication to elliptic functions, Legendres work went essentially unno-ticed by his contemporaries until 1827 when two young and as yet unknown mathematiciansAbel and Jacobi placed the subject on a new basis, and revolutionized it completely.

In 1825, the Norwegian government funded Abel on a scholarly visit to France and Germany.Abel then traveled to Paris, where he gave an important paper revealing the double period-

icity of the elliptic functions. Among his other accomplishments, Abel wrote a monumentalwork on elliptic functions7 which unfortunately was not discovered until after his death.

Jacobi wrote the classic treatise8 on elliptic functions, of great importance in mathematicalphysics, because of the need to integrate second order kinetic energy equations. The motion

5Exercises du Calcul Intgral6Trait des Fonctions Elliptiques7Abel, N.H. Recherches sur les fonctions elliptiques. J. reine angew. Math. 3, 160-190, 1828.8Jacobi, C.G.J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum

Borntraeger, 1829.

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equations in rotational form are integrable only for the three cases of the pendulum, thesymmetric top in a gravitational field, and a freely spinning body, wherein solutions are interms of elliptic functions.

Jacobi was also the first mathematician to apply elliptic functions to number theory, for

example, proving the polygonal number theorem of Pierre de Fermat. The Jacobi thetafunctions, frequently applied in the study of hypergeometric series, were named in his honor.

In developments of the theory of elliptic functions, modern authors mostly follow Karl Weier-strass. The notations of Weierstrasss elliptic functions based on his p-function are conve-nient, and any elliptic function can be expressed in terms of these. The elliptic functionsintroduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are morecomplex but important both for the history and for general theory.

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Theory

1. Elliptic Integrals

There are three basic forms of Legendre elliptic integrals that will be examined here; first,

second and third kind. In their most general form, elliptic integrals are presented in a formreferred to as incomplete integrals where the bounds of the integral representation rangefrom0 sin 1 or 0 /2.

a) First Kind: The incomplete elliptic integral can be written as

F(sin , k) =

sin

0

dt

(1 t2)(1 k2t2), 0 k

2 1 (3.1)

0 sin 1

by letting t = sin , Eq. 3.1 becomes

F(, k) =

0

d(1 k2 sin2 )

, 0 k2 < 1 (3.2)

0 < 2

The parameter k is called the modulus of the elliptic integral and is the amplitudeangle.

The complete elliptic integral is obtained by setting the amplitude = /2 orsin = 1, the maximum range on the upper bound of integration for the ellipticintegral.

F =

2

, k = F (sin = 1, k) = K(k) = K (3.3)

A complementary form of the elliptical integral can be obtained by letting the modulusbe

(k)2 = 1 k2 (3.4)

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If we let v = tan and in turn dv = sec2 d = (1 + v2) d, then

F(, k) =

tan

0

dv

(1 + v2)1 k2v2

1 + v

2=

tan

0

dv1 + v2

(1 + v2 k2v2

=

tan

0

dv(1 + v2)(1 + kv2)

(3.5)

The complementary, complete elliptic integral can then be written as

F

= 2

, k

= F(sin = 1, k) = K(k) = K (3.6)

b) Second Kind:

E(, k) =

sin

0

1 k2t2

1 t2 dt, 0 k2 1 (3.7)

or its equivalent

E(, k) =

0

1 k2 sin2 d, 0 k2 1 (3.8)

0 2

And similarly, the complete elliptic integral of the second kind can be written as

E

=

2 , k

= E(sin = 1, k) = E(k) = E (3.9)

and the complementary complete integral of the second kind

E

=

2, k

= E(sin = 1, k) = E(k) = E (3.10)

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c) Third Kind:

(,n,k) =

sin

0

dt

(1 + nt)2

(1 t2)(1 k2t2) , 0 k

2 1(3.11)

or its equivalent

(,n,k) =

0

d

(1 + n sin2 )

(1 k2 sin2 ), 0 k2 1 (3.12)

0 2

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More than any other special function, you need to be very careful about the argumentsyou give to elliptic integrals and elliptic functions. There are several conventions incommon use in competing Computer Algebra Systems and it is important that you

check the documentation provided with the CAS to determine which convection isincorporated.

Function Mathematica

Elliptic Integral of the first kind, F[|m] EllipticF[,m]- amplitude and modulus m = k2

Complete Elliptic Integral EllipticK[m]of the first kind, K(m)

Elliptic Integral of the second kind, E[|m] EllipticE[,m]- amplitude and modulus m = k2

Complete Elliptic Integral EllipticE[m]of the second kind, E(m)

Elliptic Integral of the third kind, [n; |k] EllipticPi[n, ,m]- amplitude and modulus m = k2

Complete Elliptic Integral EllipticPi[n,m]

of the third kind, (n|m)


Determining the arc length of a circle is easily achieved using trigonometric functions,however elliptic integrals must be used to find the arc length of an ellipse.

Tracing the arc of a pendulum can be achieved for small angles using trigonometric

functions but to determine the full path of the pendulum elliptic integrals must beused.

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Relations and Selected Values of Elliptic Integrals

Complete Elliptic Integrals of the First and Second Kind, K, K, E , E

The four elliptic integrals K, K, E, and E, satisfy the following identity attributed

to Legendre

KE + KE KK = 2

(3.13)

The elliptic integrals K and E as functions of the modulus k are connected by meansof the following equations:

dE

dk=

1

k(E K) (3.14)

dK

dk=

1

k(k)2[E (k)2K] (3.15)

Incomplete Elliptic Integrals of the First and Second Kind, F(, k), E(, k)

It is convenient to introduce another frequently encountered elliptic integral which isrelated to E and F.

D(, k) =

0

sin2

d =

F Ek2

(3.16)

where

=

1 k2 sin2 (3.17)

Therefore

F = E + k2D (3.18)

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Other incomplete integrals expressed by D, E, and F are

0

cos2

d = F D (3.19)

0

tan2

d = tan E(k)2

(3.20)

0

d

cos2 =

tan + k2(D F)(k)2

(3.21)

0

sin2

2d =

F D(k)2

sin cos (k)2

(3.22)

0

cos2

2d = D +

sin cos

(3.23)

0

tan2 d = tan + F 2E (3.24)

Special Values of the Elliptic Integrals

E(0, k) = 0 (3.25)

F(0, k) = 0 (3.26)

(0, 2, k) = 0 (2 = n) (3.27)

E(, k) = (3.28)

F(, k) = (3.29)

(, 2, 0) = (if n = 0) (3.30)

= arctan

(1 2

) tan

1 2 , if

2 < 1 (3.31)

Especial Functions

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Transcript of Especial Functions