Handbook of mathematical functions (without numerical tables)

Release notes......Page 0001.djvu
Errata Notice......Page 0002.djvu
Preface......Page 0003.djvu
Foreword......Page 0004.djvu
Table of contents (short)......Page 0005.djvu
Introduction......Page 0006.djvu
Acknowledgments......Page 0007.djvu
Table of Contents (hyperlinked)......Page 0009.djvu
A......Page 0022.djvu
B......Page 0023.djvu
C......Page 0025.djvu
D......Page 0028.djvu
E......Page 0029.djvu
F......Page 0032.djvu
G......Page 0033.djvu
H......Page 0034.djvu
I......Page 0035.djvu
J......Page 0037.djvu
K......Page 0038.djvu
L......Page 0039.djvu
M......Page 0040.djvu
O......Page 0043.djvu
P......Page 0044.djvu
Q......Page 0046.djvu
S......Page 0047.djvu
U......Page 0050.djvu
W......Page 0051.djvu
Index of Notations (hyperlinked)......Page 0052.djvu
Title page......Page I.djvu
Errata Notice......Page II.djvu
Preface......Page III.djvu
Preface to the Ninth Printing......Page IIIa.djvu
Foreword......Page V.djvu
Table of Contents (not hyperlinked)......Page VII.djvu
2. Accuracy of the Tables......Page IX.djvu
4. Interpolation......Page X.djvu
5. Inverse Interpolation......Page XII.djvu
7. Generation of Functions from Recurrence Relations......Page XIII.djvu
8. Acknowledgments......Page XIV.djvu
2. Physical Constants and Conversion Factors......Page 5.djvu
Table 2.2. Names and Conversion Factors for Electric and Magnetic Units......Page 6.djvu
Table 2.3. Adjusted Values of Constants......Page 7.djvu
Table 2.6. Geodetic Constants......Page 8.djvu
3. Elementary analytical methods......Page 9.djvu
3.2. Inequalities......Page 10.djvu
3.3. Rules for Differentiation and Integration......Page 11.djvu
3.4. Limits, Maxima and Minima......Page 13.djvu
3.6. Infinite Series......Page 14.djvu
3.7. Complex Numbers and Functions......Page 16.djvu
3.8. Algebraic Equations......Page 17.djvu
3.9. Successive Approximation Methods......Page 18.djvu
3.12. Computing Techniques......Page 19.djvu
References......Page 23.djvu
4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions......Page 65.djvu
4.1. Logarithmic Function......Page 67.djvu
4.2. Exponential Function......Page 69.djvu
4.3. Circular Functions......Page 71.djvu
4.4. Inverse Circular Functions......Page 79.djvu
4.5. Hyperbolic Functions......Page 83.djvu
4.6. Inverse Hyperbolic Functions......Page 86.djvu
Numerical Methods. 4.7. Use and Extension of the Tables......Page 89.djvu
References......Page 93.djvu
5. Exponential Integral and Related Functions......Page 227.djvu
5.1. Exponential Integral......Page 228.djvu
5.2. Sine and Cosine Integrals......Page 231.djvu
Numerical Methods. 5.3. Use and Extension of the Tables......Page 233.djvu
References......Page 235.djvu
6. Gamma Function and Related Functions......Page 253.djvu
6.1. Gamma Function......Page 255.djvu
6.3. Psi (Digamma) Function......Page 258.djvu
6.5. Incomplete Gamma Function......Page 260.djvu
Numerical Methods. 6.7. Use and Extension of the Tables......Page 263.djvu
6.8. Summation of Rational Series by Means of Polygamma Functions......Page 264.djvu
References......Page 265.djvu
7. Error Function and Fresnel Integrals......Page 295.djvu
7.1. Error Function......Page 297.djvu
7.2. Repeated Integrals of the Error Function......Page 299.djvu
7.3. Fresnel Integrals......Page 300.djvu
7.4. Definite and Indefinite Integrals......Page 302.djvu
Numerical Methods. 7.5. Use and Extension of the Tables......Page 304.djvu
References......Page 308.djvu
Complex zeros, maxima, minima: asymptotics......Page 329.djvu
8. Legendre function......Page 331.djvu
8.1. Differential Equation......Page 332.djvu
8.4. Explicit Expressions......Page 333.djvu
8.6. Special Values......Page 334.djvu
8.10. Asymptotic Expansions......Page 335.djvu
8.11. Toroidal Functions......Page 336.djvu
8.14. Integrals......Page 337.djvu
Numerical Methods. 8.15. Use and Extension of the Tables......Page 339.djvu
References......Page 340.djvu
9. Bessel Functions of Integer Order......Page 355.djvu
Bessel Functions J and Y. 9.1. Definitions and Elementary Properties......Page 358.djvu
9.2. Asymptotic Expansions for Large Arguments......Page 364.djvu
9.3. Asymptotic Expansions for Large Orders......Page 365.djvu
9.4. Polynomial Approximations......Page 369.djvu
9.5. Zeros......Page 370.djvu
Modified Bessel Functions I and K. 9.6. Definitions and Properties......Page 374.djvu
9.7. Asymptotic Expansions......Page 377.djvu
9.8. Polynomial Approximations......Page 378.djvu
Kelvin Functions. 9.9. Definitions and Properties......Page 379.djvu
9.10. Asymptotic Expansions......Page 381.djvu
9.11. Polynomial Approximations......Page 384.djvu
Numerical Methods. 9.12. Use and Extension of the Tables......Page 385.djvu
References......Page 388.djvu
10. Bessel Functions of Fractional Order......Page 435.djvu
10.1. Spherical Bessel Functions......Page 437.djvu
10.2. Modified Spherical Bessel Functions......Page 443.djvu
10.3. Riccati-Bessel Functions......Page 445.djvu
10.4. Airy Functions......Page 446.djvu
Numerical Methods. 10.5. Use and Extension of the Tables......Page 452.djvu
References......Page 455.djvu
11. Integrals of Bessel Functions......Page 479.djvu
11.1. Simple Integrals of Bessel Functions......Page 480.djvu
11.2. Repeated Integrals of J_n(z) and K_0(z)......Page 482.djvu
11.3. Reduction Formulas for Indefinite Integrals......Page 483.djvu
11.4. Definite Integrals......Page 485.djvu
Numerical Methods. 11.5. Use and Extension of the Tables......Page 488.djvu
References......Page 490.djvu
12. Struve Functions and Related Functions......Page 495.djvu
12.1. Struve Function H_n(s)......Page 496.djvu
12.3. Anger and Weber Functions......Page 498.djvu
Numerical Methods. 12.4. Use and Extension of the Tables......Page 499.djvu
References......Page 500.djvu
Explanations of numerical methods to compute Struve functions......Page 502.djvu
13. Confluent Hypergeometric Functions......Page 503.djvu
13.1. Definitions of Kummer and Whittaker Functions......Page 504.djvu
13.2. Integral Representations......Page 505.djvu
13.3. Connections With Bessel Functions......Page 506.djvu
13.5. Asymptotic Expansions and Limiting Forms......Page 508.djvu
13.6. Special Cases......Page 509.djvu
13.7. Zeros and Turning Values......Page 510.djvu
Numerical Methods. 13.8. Use and Extension of the Tables......Page 511.djvu
References......Page 514.djvu
14. Coulomb Wave Functions......Page 537.djvu
14.1. Differential Equation, Series Expansions......Page 538.djvu
14.4. Bessel Function Expansions......Page 539.djvu
14.5. Asymptotic Expansions......Page 540.djvu
14.6. Special Values and Asymptotic Behavior......Page 542.djvu
Numerical Methods. 14.7. Use and Extension of the Tables......Page 543.djvu
References......Page 544.djvu
15. Hypergeometric Functions......Page 555.djvu
15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument......Page 556.djvu
15.2. Differentiation Formulas and Gauss' Relations for Contiguous Functions......Page 557.djvu
Integral Representations and Transformation Formulas......Page 558.djvu
15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre Functions......Page 561.djvu
15.5. The Hypergeometric Differential Equation......Page 562.djvu
15.6. Riemann's Differential Equation......Page 564.djvu
15.7. Asymptotic Expansions. References......Page 565.djvu
16. Jacobian Elliptic Functions and Theta Functions......Page 567.djvu
16.1. Introduction......Page 569.djvu
16.3. Relation of the Jacobian Functions to the Copolar Trio......Page 570.djvu
16.6. Jacobian Functions when m=0 or 1......Page 571.djvu
16.8. Change of Argument......Page 572.djvu
16.14. Ascending Landen Transformation......Page 573.djvu
16.20. Jacobi's Imaginary Transformation......Page 574.djvu
16.24. Integrals of the Twelve Jacobian Elliptic Functions......Page 575.djvu
16.29. Logarithmic Derivatives of the Theta Functions......Page 576.djvu
16.33. Addition of Quarter-Periods to Jacobins Eta and Theta Functions......Page 577.djvu
16.36. Neville's Notation for Theta Functions......Page 578.djvu
Numerical Methods. 16.39. Use and Extension of the Tables......Page 579.djvu
References......Page 581.djvu
17. Elliptic Integrals......Page 587.djvu
17.1. Definition of Elliptic Integrals. 17.2. Canonical Forms......Page 589.djvu
17.3. Complete Elliptic Integrals of the First and Second Kinds......Page 590.djvu
17.4. Incomplete Elliptic Integrals of the First and Second Kinds......Page 592.djvu
17.5. Landen's Transformation......Page 597.djvu
17.6. The Process of the Arithmetic-Geometric Mean......Page 598.djvu
17.7. Elliptic Integrals of the Third Kind......Page 599.djvu
Numerical Methods. 17.8. Use and Extension of the Tables......Page 600.djvu
References......Page 606.djvu
18. Weierstrass Elliptic and Related Functions......Page 627.djvu
18.1. Definitions, Symbolism, Restrictions and Conventions......Page 629.djvu
18.2. Homogeneity Relations, Reduction Formulas and Processes......Page 631.djvu
18.3. Special Values and Relations......Page 633.djvu
18.5. Series Expansions......Page 635.djvu
18.6. Derivatives and Differential Equations......Page 640.djvu
18.7. Integrals......Page 641.djvu
18.8. Conformal Mapping......Page 642.djvu
18.9. Relations with Complete Elliptic Integrals K and K' and Their Parameter m and with Jacobi's Elliptic Functions......Page 649.djvu
18.10. Relations with Theta Functions......Page 650.djvu
18.11. Expressing any Elliptic Function in Terms of P and P'......Page 651.djvu
18.13. Equianharmonic Case (g_2=0, g_3=1)......Page 652.djvu
18.14. Lemniscatic Case (g_2=1, g_3=0)......Page 658.djvu
18.15. Pseudo-Lemniscatic Case (g_2=-1, g_3=0)......Page 662.djvu
Numerical Methods. 18.16. Use and Extension of the Tables......Page 663.djvu
References......Page 670.djvu
19. Parabolic Cylinder Functions......Page 685.djvu
19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations......Page 686.djvu
19.7 to 19.11. Asymptotic Expansions......Page 689.djvu
19.12 to 19.15. Connections With Other Functions......Page 691.djvu
The Equation d^2y/dx^2+(x^2/4-a)y=0. 19.16 to 19.19. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations......Page 692.djvu
19.20 to 19.24. Asymptotic Expansions......Page 693.djvu
19.25. Connections With Other Functions......Page 695.djvu
19.26. Zeros......Page 696.djvu
19.27. Bessel Functions of Order +-1/4, +-3/4 as Parabolic Cylinder Functions. Numerical Methods. 19.28. Use and Extension of the Tables......Page 697.djvu
References......Page 700.djvu
20. Mathieu Functions......Page 721.djvu
20.2. Determination of Characteristic Values......Page 722.djvu
20.3. Floquet's Theorem and Its Consequences......Page 727.djvu
20.4. Other Solutions of Mathieu's Equation......Page 730.djvu
20.6. Solutions of Mathieu's Modified Equation for Integral nu......Page 732.djvu
20.7. Representations by Integrals and Some Integral Equations......Page 735.djvu
20.8. Other Properties......Page 738.djvu
20.9. Asymptotic Representations......Page 740.djvu
20.10. Comparative Notations......Page 744.djvu
References......Page 745.djvu
21. Spheroidal Wave Functions......Page 751.djvu
21.5. Wave Equation in Prolate and Oblate Spheroidal Coordinates......Page 752.djvu
21.7. Prolate Angular Functions......Page 753.djvu
21.9. Radial Spheroidal Wave Functions......Page 756.djvu
21.10. Joining Factors for Prolate Spheroidal Wave Functions......Page 757.djvu
21.11. Notation......Page 758.djvu
References......Page 759.djvu
22. Orthogonal Polynomials......Page 771.djvu
22.1. Definition of Orthogonal Polynomials......Page 773.djvu
22.2. Orthogonality Relations......Page 774.djvu
22.3. Explicit Expressions......Page 775.djvu
22.5. Interrelations......Page 777.djvu
22.6. Differential Equations......Page 781.djvu
22.7. Recurrence Relations......Page 782.djvu
22.9. Generating Functions......Page 783.djvu
22.10. Integral Representations......Page 784.djvu
22.13. Integrals Involving Orthogonal Polynomials......Page 785.djvu
22.14. Inequalities......Page 786.djvu
22.16. Zeros......Page 787.djvu
Numerical Methods. 22.18. Use and Extension of the Tables......Page 788.djvu
22.19. Least Square Approximations......Page 790.djvu
References......Page 792.djvu
23. Bernoulli and Euler Polynomials, Riemann Zeta Function......Page 803.djvu
23.1. Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula......Page 804.djvu
23.2. Riemann Zeta Function and Other Sums of Reciprocal Powers......Page 807.djvu
References......Page 808.djvu
24. Combinatorial Analysis......Page 821.djvu
24.1.1. Binomial Coefficients......Page 822.djvu
24.1.2. Multinomial Coefficients......Page 823.djvu
24.1.4. Stirling Numbers of the Second Kind......Page 824.djvu
24.2.2. Partitions Into Distinct Parts......Page 825.djvu
24.3.2. The Euler Function......Page 826.djvu
24.3.4. Primitive Roots. References......Page 827.djvu
25. Numerical Interpolation, Differentiation, and Integration......Page 875.djvu
25.1. Differences......Page 877.djvu
25.2. Interpolation......Page 878.djvu
25.3. Differentiation......Page 882.djvu
25.4. Integration......Page 885.djvu
25.5. Ordinary Differential Equations......Page 896.djvu
References......Page 898.djvu
26. Probability Functions......Page 925.djvu
26.1. Definitions and Properties......Page 927.djvu
26.2. Normal or Gaussian Probability Function......Page 931.djvu
26.3. Bivariate Normal Probability Function......Page 936.djvu
26.4. Chi-Square Probability Function......Page 940.djvu
26.5. Incomplete Beta Function......Page 944.djvu
26.6. F-(Variance-Ratio) Distribution Function......Page 946.djvu
26.7. Student's t-Distribution......Page 948.djvu
Numerical Methods. 26.8. Methods of Generating Random Numbers and Their Applications......Page 949.djvu
26.9. Use and Extension of the Tables......Page 953.djvu
References......Page 961.djvu
27. Miscellaneous Functions......Page 997.djvu
27.1. Debye functions......Page 998.djvu
27.3. Einstein Functions......Page 999.djvu
27.4. Sievert Integral......Page 1000.djvu
27.5. $f_m(x)=int_0^infty t^m e^{-t^2-x/t} dt$ and Related Integrals......Page 1001.djvu
27.6. $f(x)=int_0^infty e^{-t^2}/(t+x) dt$......Page 1003.djvu
27.7 Dilogarithm (Spence's Integral)......Page 1004.djvu
27.8. Clausen's Integral and Related Summations......Page 1005.djvu
27.9. Vector-Addition Coefficients......Page 1006.djvu
29. Laplace Transforms......Page 1019.djvu
29.2. Operations for the Laplace Transform......Page 1020.djvu
29.3. Table of Laplace Transforms......Page 1021.djvu
29.4. Table of Laplace-Stieltjes Transforms......Page 1029.djvu
References......Page 1030.djvu
A-B-......Page 1031.djvu
-B-C-......Page 1032.djvu
-C-D-......Page 1033.djvu
-D-E-......Page 1034.djvu
-E-F-G-H-......Page 1035.djvu
-H-I-......Page 1036.djvu
-I-J-K-L-......Page 1037.djvu
-L-M-......Page 1038.djvu
-M-N-O-......Page 1039.djvu
-O-P-......Page 1040.djvu
-P-Q-R-S-......Page 1041.djvu
-S-T-U-V-W-......Page 1042.djvu
-W-Z......Page 1043.djvu
Index of Notations......Page 1044.djvu
Notation -- Greek Letters. Miscellaneous Notations......Page 1046.djvu