Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE

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1042 INDEX Spheroidal wave functions- Continued Page characteristic values for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 753,756 differential equations _____ _ _ _ _ _ _ _ _ _ _ _ __ _ - _ __ _ _ 753 evaluation of coefficients for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 755 expansions for _______________________________ 755 joining factors for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ 757 normalization of _ _ _ _ _ _ _ - - _ - _ _ _ _ _ - _ __ _ __ __ _ _ _ _ 755 tables of ___________________________________ ‘_ 766 table of eigenvalues of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 760 table of prolate joining factors _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 769 Stirling numbers- - _ _ _ _ _ _ _ . . . . . . . . . . . . . . . . . . . . . 824 table of the first kind _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 833 table of the second kind . . . . . . . . . . . . . . . . . . . . . . 835 Stirling’s formula-_- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 257 Struve’s functions- ____________________________ 495 asymptotic expansions for large orders- _ _ _ - __ _ _ 498 asymptotic expansions for large 121 ___________ 497,498 computation of ______________________________ 499 differential equation- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 496 graphs o f _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ - _ _ _ _ - - 496 integral representations of - _ - - _ - - _ - _ _ - _ - _ _ _ _ 496,498 integrals__________________-_-_____-_______ 497,498 modified___________________________________- 498 power series expansion for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 496,498 recurrence relations_-- . . . . . . . . . . . . . . . . . . . . . 496,498 relation to Weber’s function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 498 special properties of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 497 tables of ____________________________________ 501 Student’s &distribution ____ - _ - _ _ _ _ - - _ __ _ _ ___ _ _ - _ 948 approximations to- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 949 asymptotic expansion of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 949 limiting distribution- - - - _ _ - _ - _ __ _ - _ _ - _ _ _ _ - _ _ - 949 non-central______________________________-_- 949 series expansions for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 948 statistical properties of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 948 Subtabulation _________________________________ 881 Summable series _______________________________ 1005 Summation of rational series . . . . . . . . . . . . . . . . . . . . 264 Sums of positive powers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 813 Sums of powers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 804 Sums of reciprocal powers- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 807,811 Systems of differential equations of first order__--_ 897 T Taylor expansion____________-__-_______-__-___ 880 Taylor’s formula ______________________________ 14 Tesseral harmonics- _ - - _ - _ _ - - - - _ - - - - - _ - _ _ _ - _ _ - _ 332 Tetrachoric functions- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 934 Tetragamma function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 260 (see polygamma functions) Theta functions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - - _ _ 576 addition of quarter-periods - - _ - - _ _ - _ _ - - - - - - - _ _ 577 calculation by use of the arithmetic-geometric mean--____-___________________________ 577,580 expansions in terms of the nome p _ _ _ _ _ _ _ _ _ _ _ _ _ 576 Jacobi’s notation for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 577 logarithmic derivatives of _ _ _ _ _ _ - _ - _ _ _ _ _ _--__ - _ 576 logarithms of sum and difference _ _ _ _ _ _ _ _ _ _ _ _ _ _ 577 Neville’s notation for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 578,582 relations between squares of the functions- - - - _ - 576 relation to Jacobi’s zeta function _ _ _ _ _ _ _ _ _ _ _ _ _ _ 578 relation with Weierstrass elliptic functions_- - - _ - 650 Page Thiele’s interpolation formula-- _ __ - _ - __ __ __ - - - _ - 881 Toroidal functions _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 336 Toronto function-- - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 509 Trapezoidal rule _______________________________ 885 Triangle inequality- - - - _ _ _ - _ __ _ - - - _ _ _ _ _ - _ - _ - _ - _ 11 Trigamma function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 260 Trigonometric functions--- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 71 Truncated exponential function- - _ - _ - - - __ - - _ - _ - 70,262 U (see orthogonal polynomials) (see polygampa functions) (see circular functions) Ultraspherical polynomials _ _ _ _ - _ - _ _ _ - _ - _ - _ - _ - _ - _ 774 coefficients for and P in terms of _ _ _ _ _ _ _ _ _ _ _ _ 794 graphs of_____--____----____----_-_-----_-- 776 Unit step function _____________________________ 1020 V Variance________-_-_-__-_-_-_--__-_-_-_-_---_ 928 Variance-ratio distribution function-- - _ - _ - _ - _ - _ _ - 946 Vector-addition coefficients- _ - - _ _ - _ - _ _ - _ - _ - _ - _ - _ 1006 (see F-distribution function) W Wallis’ formula ________________________________ 258 Wave equation in prolate and oblate spheroidal coordinates..--..- 752 Weber’s function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 498 relation to Anger’s function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 498 relation to Struve’s function _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 498 Weierstrass elliptic functions-- - _ - _ - - _ - _ - _ _ - _ - _ - _ 627 addition formulas for _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 635 case A = O _ _ _ - _ - - _ - _ - - _ _ - _ _ - - _ - _ _ _ _ - - _ - 651 computation of ______________________________ 663 conformal mappiw of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 642,654,659 definitions_______-_-_-_---__-_-____----___-_-_ 629 derivatives of _______________________________ 640 determination of periods from given invariants- - 665 determination of values a t half-periods, etc., from given periods _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 664 differential equation- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 629,640 discriminant___--__--_-____-___-_-___--_-_-_ 629 equianharmonic case__- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 652 expressing any elliptic function in terms of 8 a n d 8‘______________________________________ 651 fundamental period parallelogram- _ _ - _ _ - - - - _ - - 629 fundamental rectangle- _ - - _ _ _ - _ _ _ _ _ _ _ _ _ - _ _ _ _ _ 630 homogeneity relations _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ 631 integrals___--_--_--__--_-_-_-_---------_----_- 641 invariants- - - _ _ _ _ - _ _ - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ 629 Lek-ndre’s relation- - - _ _ _ _ _ _ _ _ - - - _ _ _ _ _ _ _ _ _ - _ _ 634 lemniscatic case- _ _ _ _ _ - - - _ _ - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 658 maps o f _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - - _ - - - - - - - - 642,654,659 multiplication formulas _ _ _ _ _ - - - - - - - - - - - - _ - - - - - 635 other series involving 8, 8’, 5 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 639 pseudo-lemnisctltic case - _ - _ - _ _ _ - - _ _ - _ - _ - _ - _ - - 662 reduction formulas _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 631 relation with complete elliptic integrals-- - - _ - _ - - 649 relations with Jacobi’s elliptic functions- - - - - - - _ 649 relations with theta functions _ _ _ c _ _ _ _ _ _ _ _ _ _ _ _ _ 650 . .