[LONG] If you like or *need* Mathematics ...


... then you'll probably find this a very interesting resource.

Hello, all:

    The resource in question is this:
          "Handbook of Mathematical Functions with Formulas, Graphs,
    and Mathematical Tables
    (Milton Abramowitz and Irene A. Stegun, Editors)"
    and it's a monumental, 1059-page book choke-full of everything there is to know about most mathematical functions (including all important ones for the applications), such as definition, formulae, tables, graphs, algorithms for their numerical computation, the works !
    Also included are number theory functions, statistical and probability functions, conversion functions, etc.

    I've used it very extensively in the past and still do today when I need some information on a particular function, such as formulae and/or suitable numerical methods to implement it, so it might be the case that you find it useful too, if you didn't know about it in advance, most specially if you want to program your HP33S, say, to compute some advanced functions related to your engineering or statistical field (say you need elliptic functions for some electrical engineering problem) and you'd like to get comprehensive coverage from a single source. Even the tables will be extremely useful to check your results against them. You might also find useful the physical constants and conversion factors.

    The full table of contents and index I've prepared follows, for you to have a
    look at what's available and whether you're interested in getting it. Should that be the case, you can download it from here, as a 67 Mb PDF document.

    Best regards from V.

Table of contents

1.Mathematical Constants
2.Physical Constants and Conversion Factors
3.Elementary Analytical Methods
4.Elementary Transcendental Functions
5.Exponential Integral and Related Functions
6.Gamma Function and Related Functions
7.Error Function and Fresnel Integrals
8.Legendre Functions
9.Bessel Functions of Integer Order
10.Integrals of Bessel Function
11.Struve Functions of Fractal Order
12.Confluent Hypergeometric Functions
13.Coulomb Wave Functions
14.Hypergeometric Functions
15.Jacobian Elliptic Functions and Theta Functions
16.Elliptic Integrals
17.Weierstrass Elliptic and Related Functions
18.Parabolic Cylinder Functions
19.Mathieu Functions
20.Spheroidal Wave Functions
21.Orthogonal Polynomials
22.Bernoulli and Euler Polynomials, Riemann Zeta Function
23.Combinatorial Analysis
24.Numerical Interpolation, Differentiation and Integration
25.Probability Functions
26.Miscellaneous Functions
27.Scales of Notation
28.Laplace Transformations
29.Subject Index
30.Index of Notations

Table of Contents - Expanded

1. Introduction.
1.1. Introduction.
1.2. Accuracy of the Tables.
1.3. Auxiliary Functions and Arguments.
1.4. Interpolation
1.5. Inverse Interpolation
1.6. Bivariate Interpolation.
1.7. Generation of Functions from Recurrence Relations

2. Physical Constants and Conversion Factors
2.1. Common Units and Conversion Factors.
2.2. Names and Conversion Factors for Electric and Magnetic Units
2.3. Adjusted Values of Constants
2.4. Miscellaneous Conversion Factors.
2.5. Conversion Factors for Customary U.S. Units to Metric Units.
2.6. Geodetic Constants

3. Elementary analytical methods
3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means.
3.2. Inequalities
3.3. Rules for Differentiation and Integration
3.4. Limits, Maxima and Minima
3.5. Absolute and Relative Errors.
3.6. Infinite Series
3.7. Complex Numbers and Functions
3.8. Algebraic Equations
3.9. Successive Approximation Methods
3.10. Theorems on Continued Fractions. Numerical Methods.
3.11. Use and Extension of the Tables.
3.12. Computing Techniques

4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions
Mathematical Properties.
4.1. Logarithmic Function
4.2. Exponential Function
4.3. Circular Functions
4.4. Inverse Circular Functions
4.5. Hyperbolic Functions
4.6. Inverse Hyperbolic Functions
Numerical Methods.
4.7. Use and Extension of the Tables

5. Exponential Integral and Related Functions
Mathematical Properties.
5.1. Exponential Integral
5.2. Sine and Cosine Integrals
Numerical Methods.
5.3. Use and Extension of the Tables

6. Gamma Function and Related Functions
Mathematical Properties.
6.1. Gamma Function
6.2. Beta Function.
6.3. Psi (Digamma) Function
6.4. Polygamma Functions.
6.5. Incomplete Gamma Function
6.6. Incomplete Beta Function. Numerical Methods.
6.7. Use and Extension of the Tables
6.8. Summation of Rational Series by Means of Polygamma Functions

7. Error Function and Fresnel Integrals
Mathematical Properties.
7.1. Error Function
7.2. Repeated Integrals of the Error Function
7.3. Fresnel Integrals
7.4. Definite and Indefinite Integrals
Numerical Methods.
7.5. Use and Extension of the Tables
Complex zeros, maxima, minima of the error function and Fresnel integrals: asymptotics

8. Legendre function
Mathematical Properties. Notation.
8.1. Differential Equation
8.2. Relations Between Legendre Functions.
8.3. Values on the Cut.
8.4. Explicit Expressions
8.6. Special Values
8.7. Trigonometric Expansions.
8.8. Integral Representations.
8.9. Summation Formulas.
8.10. Asymptotic Expansions
8.11. Toroidal Functions
8.12. Conical Functions.
8.13. Relation to Elliptic Integrals.
8.14. Integrals
Numerical Methods.
8.15. Use and Extension of the Tables

9. Bessel Functions of Integer Order
Mathematical Properties. Notation. Bessel Functions J and Y.
9.1. Definitions and Elementary Properties
9.2. Asymptotic Expansions for Large Arguments
9.3. Asymptotic Expansions for Large Orders
9.4. Polynomial Approximations
9.5. Zeros
Modified Bessel Functions I and K.
9.6. Definitions and Properties
9.7. Asymptotic Expansions
9.8. Polynomial Approximations
Kelvin Functions.
9.9. Definitions and Properties
9.10. Asymptotic Expansions
9.11. Polynomial Approximations
Numerical Methods.
9.12. Use and Extension of the Tables

10. Bessel Functions of Fractional Order
Mathematical Properties.
10.1. Spherical Bessel Functions
10.2. Modified Spherical Bessel Functions
10.3. Riccati-Bessel Functions
10.4. Airy Functions
Numerical Methods.
10.5. Use and Extension of the Tables

11. Integrals of Bessel Functions
Mathematical Properties.
11.1. Simple Integrals of Bessel Functions
11.2. Repeated Integrals of Jn(z) and K0(z)
11.3. Reduction Formulas for Indefinite Integrals
11.4. Definite Integrals
Numerical Methods.
11.5. Use and Extension of the Tables

12. Struve Functions and Related Functions
Mathematical Properties.
12.1. Struve Function Hn(s)
12.2. Modified Struve Function Lnu(z).
12.3. Anger and Weber Functions
Numerical Methods.
12.4. Use and Extension of the Tables
Explanations of numerical methods to compute Struve functions

13. Confluent Hypergeometric Functions
Mathematical Properties.
13.1. Definitions of Kummer and Whittaker Functions
13.2. Integral Representations
13.3. Connections With Bessel Functions
13.5. Asymptotic Expansions and Limiting Forms
13.6. Special Cases
13.7. Zeros and Turning Values
Numerical Methods.
13.8. Use and Extension of the Tables

14. Coulomb Wave Functions
Mathematical Properties.
14.1. Differential Equation, Series Expansions
14.2. Recurrence and Wronskian Relations.
14.3. Integral Representations. 14.4. Bessel Function Expansions
14.5. Asymptotic Expansions
14.6. Special Values and Asymptotic Behavior
Numerical Methods.
14.7. Use and Extension of the Tables

15. Hypergeometric Functions
Mathematical Properties.
15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument
15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions
Integral Representations and Transformation Formulas
15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre Functions
15.5. The Hypergeometric Differential Equation
15.6. Riemann's Differential Equation
15.7. Asymptotic Expansions. References

16. Jacobian Elliptic Functions and Theta Functions
Mathematical Properties.
16.1. Introduction
16.2. Classification of the Twelve Jacobian Elliptic Functions.
16.3. Relation of the Jacobian Functions to the Copolar Trio
16.4. Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M.).
16.5. Special Arguments.
16.6. Jacobian Functions when m=0 or 1
16.7. Principal Terms.
16.8. Change of Argument
16.9. Relations Between the Squares of the Functions.
16.10. Change of Parameter.
16.11. Reciprocal Parameter (Jacobi's Real Transformation).
16.12. Descending Landen Transformation (Gauss' Transformation).
16.13. Approximation in Terms of Circular Functions.
16.14. Ascending Landen Transformation
16.15. Approximation in Terms of Hyperbolic Functions.
16.16. Derivatives.
16.17. Addition Theorems.
16.18. Double Arguments.
16.19. Half Arguments.
16.20. Jacobi's Imaginary Transformation
16.21. Complex Arguments.
16.22. Leading Terms of the Series in Ascending Powers of u.
16.23. Series Expansion in Terms of the Nome q and the Argument v.
16.24. Integrals of the Twelve Jacobian Elliptic Functions
16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions.
16.26. Integrals in Terms of the Elliptic Integral of the Second Kind.
16.27. Theta Functions; Expansions in Terms of the Nome q.
16.28. Relations Between the Squares of the Theta Functions.
16.29. Logarithmic Derivatives of the Theta Functions
16.30. Logarithms of Theta Functions of Sum and Difference.
16.31. Jacobi's Notation for Theta Functions.
16.32. Calculation of Jacobi's Theta Function Theta(u|m) by Use of the Arithmetic-Geometric Mean.
16.33. Addition of Quarter-Periods to Jacobins Eta and Theta Functions
16.34. Relation of Jacobi's Zeta Function to the Theta Functions.
16.35. Calculation of Jacobi's Zeta Function Z(u|m) by Use of the Arithmetic-Geometric Mean.
16.36. Neville's Notation for Theta Functions
16.37. Expression as Infinite Products.
16.38. Expression as Infinite Series. Numerical Methods.
16.39. Use and Extension of the Tables

17. Elliptic Integrals
Mathematical Properties.
17.1. Definition of Elliptic Integrals.
17.2. Canonical Forms
17.3. Complete Elliptic Integrals of the First and Second Kinds
17.4. Incomplete Elliptic Integrals of the First and Second Kinds
17.5. Landen's Transformation
17.6. The Process of the Arithmetic-Geometric Mean
17.7. Elliptic Integrals of the Third Kind
Numerical Methods.
17.8. Use and Extension of the Tables

18. Weierstrass Elliptic and Related Functions
Mathematical Properties.
18.1. Definitions, Symbolism, Restrictions and Conventions
18.2. Homogeneity Relations, Reduction Formulas and Processes
18.3. Special Values and Relations
18.4. Addition and Multiplication Formulas. 18.5. Series Expansions
18.6. Derivatives and Differential Equations
18.7. Integrals
18.8. Conformal Mapping
18.9. Relations with Complete Elliptic Integrals K and K' and Their Parameter m and with Jacobins Elliptic Functions
18.10. Relations with Theta Functions
18.11. Expressing any Elliptic Function in Terms of P and P'
18.13. Equianharmonic Case (g2=0, g3=1)
18.14. Lemniscatic Case (g2=1, g3=0)
18.15. Pseudo-Lemniscatic Case (g2=-1, g3=0)
Numerical Methods.
18.16. Use and Extension of the Tables

19. Parabolic Cylinder Functions
Mathematical Properties.
19.1. The Parabolic Cylinder Functions, Introductory. The Equation d2y/dx2-(x2/4+a)y=0.
19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations
19.7 to 19.11. Asymptotic Expansions
19.12 to 19.15. Connections With Other Functions The Equation d2y/dx2+(x2/4-a)y=0.
19.16 to 19.19. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations
19.20 to 19.24. Asymptotic Expansions
19.25. Connections With Other Functions
19.26. Zeros
19.27. Bessel Functions of Order ±1/4, ±3/4 as Parabolic Cylinder Functions. Numerical Methods.
19.28. Use and Extension of the

20. Mathieu Functions
Mathematical Properties.
20.1. Mathieu's Equation.
20.2. Determination of Characteristic Values
20.3. Floquet's Theorem and Its Consequences
20.4. Other Solutions of Mathieu's Equation
20.5. Properties of Orthogonality and Normalization.
20.6. Solutions of Mathieu's Modified Equation for Integral nu
20.7. Representations by Integrals and Some Integral Equations
20.8. Other Properties
20.9. Asymptotic Representations
20.10. Comparative Notations

21. Spheroidal Wave Functions
Mathematical Properties.
21.1. Definition of Elliptical Coordinates.
21.2. Definition of Prolate Spheroidal Coordinates.
21.3. Definition of Oblate Spheroidal Coordinates.
21.4. Laplacian in Spheroidal Coordinates.
21.5. Wave Equation in Prolate and Oblate
Spheroidal Coordinates
21.6. Differential Equations for Radial and Angular Spheroidal Wave Functions.
21.7. Prolate Angular Functions
21.8. Oblate Angular Functions. 21.9. Radial Spheroidal Wave Functions
21.10. Joining Factors for Prolate Spheroidal Wave Functions
21.11. Notation

22. Orthogonal Polynomials
Mathematical Properties.
22.1. Definition of Orthogonal Polynomials
22.2. Orthogonality Relations
22.3. Explicit Expressions
22.4. Special Values.
22.5. Interrelations
22.6. Differential Equations
22.7. Recurrence Relations
22.8. Differential Relations.
22.9. Generating Functions
22.10. Integral Representations
22.11. Rodrigues' Formula.
22.12. Sum Formulas.
22.13. Integrals Involving Orthogonal Polynomials
22.14. Inequalities
22.15. Limit Relations.
22.16. Zeros
22.17. Orthogonal Polynomials of a Discrete Variable. Numerical Methods.
22.18. Use and Extension of the Tables
22.19. Least Square Approximations

23. Bernoulli and Euler Polynomials, Riemann Zeta Function
Mathematical Properties.
23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula
23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers

24. Combinatorial Analysis
Mathematical Properties.
24.1. Basic Numbers.
24.1.1. Binomial Coefficients
24.1.2. Multinomial Coefficients
24.1.3. Stirling Numbers of the First Kind.
24.1.4. Stirling Numbers of the Second Kind
24.2. Partitions.
24.2.1. Unrestricted Partitions.
24.2.2. Partitions Into Distinct Parts
24.3. Number Theoretic Functions.
24.3.1. The Mobius Function.
24.3.2. The Euler Function
24.3.3. Divisor Functions.
24.3.4. Primitive Roots. References

25. Numerical Interpolation, Differentiation, and Integration
25.1. Differences
25.2. Interpolation
25.3. Differentiation
25.4. Integration
25.5. Ordinary Differential Equations

26. Probability Functions
Mathematical Properties.
26.1. Probability Functions: Definitions and Properties
26.2. Normal or Gaussian Probability Function
26.3. Bivariate Normal Probability Function
26.4. Chi-Square Probability Function
26.5. Incomplete Beta Function
26.6. F-(Variance-Ratio) Distribution Function
26.7. Student's t-Distribution
Numerical Methods.
26.8. Methods of Generating Random Numbers and Their Applications
26.9. Use and Extension of the Tables

27. Miscellaneous Functions
27.1. Debye functions
27.2. Planck's Radiation Function.
27.3. Einstein Functions
27.4. Sievert Integral
27.5. $f_m(x)=\int_0^\infinity t^m e^{-t^2-x/t} dt$ and Related Integrals
27.6. $f(x)=\int_0^\infinity e^{-t^2}/(t+x) dt$
27.7 Dilogarithm (Spence's Integral)
27.8. Clausen's Integral and Related Summations
27.9. Vector-Addition Coefficients

29. Laplace Transforms
29.1. Definition of the Laplace Transform.
29.2. Operations for the Laplace Transform
29.3. Table of Laplace Transforms
29.4. Table of Laplace-Stieltjes Transforms
Index of Notations
Notation -- Greek Letters. Miscellaneous Notations


Thanks, Valentin!
I downloaded 'the Bible' right away!



I was lucky enough to stumble across a copy in a used book store some years ago. Don't remember much of the reasons it was useful, but along with Numerical Recipes, I'm comforted knowing it is on my shelf if needed.


I bought a copy of that book from the Government Printing Office at roughly the same time as Hewlett-Packard introduced the HP-35. It's still listed in the GPO catalog.


Unfortunately, it's listed as "out of print", so it may be a while before they print another batch.

The book is big, about the size of a large phone book or dictionary.


A (large) paperback reprint of this book is still available from Dover Publications. Their identifier is 0-486-61272-4, and their list price is $34.95. However, Amazon is currently offering it new for $22.02, and used starting at $8.00.

I have both the GPO and the Dover versions, since I wanted one at work and one at home, and it is definitely too big to pack conveniently back and forth in a brief case. The book can be extremely useful, and of course the download would be both portable and economic even compared to a used hard copy. However the prospect of working with a (unbookmarked?) 67 MB PDF file might still turn some toward one of the available hard copies.


Thanks for maning the PDf available and free of charge. I found another PDF on the net last year, just one day after I had paid for another PDF version of the same book!!

So free is nice!



Thanks Valentin, I have fond memories of this book from my studies in Austria! Your pdf is very generous, I'm delighted to have a searchable abramowitz now (my hardcopy was lost at home a long time ago). Now I might have a slightly better chance to participate in the SSMC (sorry that i missed the last two, I was too much under water at work to play)

Another "handbook" (not sure why they call those monster books hand-book, one needs at least two arms to log them around) is the Brohnstein & Semendyayev . Unfortunately I think the german edition is a bit better translation, but it also served me very well during the night-owl hours of quantum physics...




Hi Valentin!

I also keep a nicely preserved copy of the Dover paperback edition in my bookshelf. I have it open in my lap as I write this.

The book brings very fond memories of the HP calc programming days, back in the late 70's. It was an invaluable source of ideas and information to create programs, in many cases simply for the fun of programming.

Thanks for the PDF version!


Hi, Fer !

Nice to see you posting here, so unusual ... !

I'm glad you like the PDF version, I also own a printed copy, a very useful reference indeed though calling it a "handbook" is quite an stretch of imagination.

By the way, I'm still waiting for your attempt at my latest challenge, I've seen you easily solve much harder ones. Also, I'm sure you'll like my article featured in this month's issue of Datafile, due in a few days.

Back to my Easter vacations !

Best regards from V.


Thanks, Valentin!

I used to have a printed copy of this "handbook" ("desktopbook"?),
but it seems that it must've grown feet and walked away a few
years ago. I guess that I should've looked at it more, for more
reasons than one. Even though I didn't often use it, there have
been occasions when I really did need some information available
in it.

That PDF file looks a bit impractical to download over my dial-up

I found an online version as well as an HTML/JPEG (tar.gz package)
copy at 43 MiB, and of course the PDF file, at

I figure that I'll download the tar.gz file to a flash drive the
next time I get a chance to borrow my sister-in-law's PC with the
cable connection.

I suspect that those who participate in the Short and Sweet Math
Challenges may find the handbook to be very helpful.


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