... then you'll probably find this a very interesting resource.
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.
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Table of contents
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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
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Table of Contents - Expanded
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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
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