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1: 27.2 Functions
Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … Such a set is a reduced residue system modulo n . …
Table 27.2.2: Functions related to division.
n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84
7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72
2: 27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod m k ) , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod m ), where m is the product of the moduli. … Their product m has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod m 1 ), (mod m 2 ), (mod m 3 ), and (mod m 4 ), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers a 1 ( mod m 1 ) , a 2 ( mod m 2 ) , a 3 ( mod m 3 ) , and a 4 ( mod m 4 ) , where each a j has no more than five digits. These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result ( mod m ) , which is correct to 20 digits. …
3: 20 Theta Functions
Chapter 20 Theta Functions
4: 12.19 Tables
  • Miller (1955) includes W ( a , x ) , W ( a , x ) , and reduced derivatives for a = 10 ( 1 ) 10 , x = 0 ( .1 ) 10 , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

  • Murzewski and Sowa (1972) includes D n ( x ) ( = U ( n 1 2 , x ) ) for n = 1 ( 1 ) 20 , x = 0 ( .05 ) 3 , 7S.

  • 5: Wolter Groenevelt
    Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …
    6: 13.28 Physical Applications
    §13.28(i) Exact Solutions of the Wave Equation
    The reduced wave equation 2 w = k 2 w in paraboloidal coordinates, x = 2 ξ η cos ϕ , y = 2 ξ η sin ϕ , z = ξ η , can be solved via separation of variables w = f 1 ( ξ ) f 2 ( η ) e i p ϕ , where …
    §13.28(iii) Other Applications
    For dynamics of many-body systems see Meden and Schönhammer (1992); for tomography see D’Ariano et al. (1994); for generalized coherent states see Barut and Girardello (1971); for relativistic cosmology see Crisóstomo et al. (2004).
    7: Bibliography R
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • REDUCE (free interactive system)
  • S. O. Rice (1954) Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.
  • H. Rosengren (2004) Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.
  • 8: Bibliography M
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • 9: 36.5 Stokes Sets
    36.5.4 80 x 5 40 x 4 55 x 3 + 5 x 2 + 20 x 1 = 0 ,
    36.5.7 X = 9 20 + 20 u 4 Y 2 20 u 2 + 6 u 2 sign ( z ) ,
    10: Software Index
    Open Source With Book Commercial
    4.48(iii) General Precision a REDUCE
    20 Theta Functions
  • Open Source Collections and Systems.

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