[LONG] If you like or *need* Mathematics ... « Next Oldest | Next Newest »

 ▼ Valentin Albillo Posting Freak Posts: 1,755 Threads: 112 Joined: Jan 2005 04-06-2006, 08:51 AM ... 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 References 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 Tables References 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 References 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 References 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 References 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 References 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 References 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 References 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 References Index of Notations Notation -- Greek Letters. Miscellaneous Notations  ▼ Werner Member Posts: 163 Threads: 7 Joined: Jul 2007 04-06-2006, 09:57 AM Thanks, Valentin! I downloaded 'the Bible' right away! Werner Bill Smith Member Posts: 87 Threads: 10 Joined: Apr 2006 04-06-2006, 01:10 PM 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. John Limpert Member Posts: 172 Threads: 13 Joined: Jul 2005 04-06-2006, 01:46 PM 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. http://bookstore.gpo.gov/actions/GetPublication?stocknumber=003-003-00279-8 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. ▼ Gordon Strickland Junior Member Posts: 18 Threads: 0 Joined: Sep 2007 04-06-2006, 03:09 PM 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. Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 04-06-2006, 03:03 PM 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! Namir PeterP Senior Member Posts: 564 Threads: 72 Joined: Sep 2005 04-06-2006, 09:04 PM 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... Cheers Peter Fernando del Rey Junior Member Posts: 30 Threads: 0 Joined: Sep 2005 04-07-2006, 04:41 PM 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! ▼ Valentin Albillo Posting Freak Posts: 1,755 Threads: 112 Joined: Jan 2005 04-10-2006, 09:48 PM 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. James M. Prange (Michigan) Posting Freak Posts: 1,041 Threads: 15 Joined: Jan 2005 04-14-2006, 01:37 AM 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 connection. 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 http://www.math.sfu.ca/~cbm/aands/. 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. Regards,James

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