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1: 5.5 Functional Relations
§5.5(iii) Multiplication
Gauss’s Multiplication Formula
5.5.7 k = 1 n - 1 Γ ( k n ) = ( 2 π ) ( n - 1 ) / 2 n - 1 / 2 .
2: 24.4 Basic Properties
§24.4(v) Multiplication Formulas
Next, …
3: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
§3.8(iv) Zeros of Polynomials
has n zeros in , counting each zero according to its multiplicity. …No explicit general formulas exist when n 5 . … The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of q ( z ) . …
4: Bibliography O
  • F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
  • F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
  • 5: 27.20 Methods of Computation: Other Number-Theoretic Functions
    To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
    6: 2.8 Differential Equations with a Parameter
    §2.8(v) Multiple and Fractional Turning Points
    The approach used in preceding subsections for equation (2.8.1) also succeeds when z 0 is a multiple or fractional turning point. For the former f ( z ) has a zero of multiplicity λ = 2 , 3 , 4 , and g ( z ) is analytic. …For results, including error bounds, see Olver (1977c). For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978). …
    7: Bibliography S
  • D. M. Smith (1998) Algorithm 786: Multiple-precision complex arithmetic and functions. ACM Trans. Math. Software 24 (4), pp. 359–367.
  • D. M. Smith (1989) Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.
  • D. M. Smith (1991) Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.
  • D. Sornette (1998) Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.
  • A. H. Stroud (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 8: 18.18 Sums
    §18.18(iii) Multiplication Theorems
    Laguerre
    Hermite
    §18.18(iv) Connection Formulas
    §18.18(v) Linearization Formulas
    9: 27.5 Inversion Formulas
    §27.5 Inversion Formulas
    The multiplicative functions are a subgroup of this group. …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … Special cases of Möbius inversion pairs are: … Other types of Möbius inversion formulas include: …
    10: 17.6 ϕ 1 2 Function
    Related formulas are (17.7.3), (17.8.8) and … For similar formulas see Verma and Jain (1983). …
    17.6.13 ϕ 1 2 ( a , b ; c ; q , q ) + ( q / c , a , b ; q ) ( c / q , a q / c , b q / c ; q ) ϕ 1 2 ( a q / c , b q / c ; q 2 / c ; q , q ) = ( q / c , a b q / c ; q ) ( a q / c , b q / c ; q ) ,
    where | z | < 1 , | ph ( - z ) | < π , and the contour of integration separates the poles of ( q 1 + ζ , c q ζ ; q ) / sin ( π ζ ) from those of 1 / ( a q ζ , b q ζ ; q ) , and the infimum of the distances of the poles from the contour is positive. …