About the Project

representations%20by%20Euler%E2%80%93Maclaurin%20formula

AdvancedHelp

(0.004 seconds)

1—10 of 596 matching pages

1: 24.1 Special Notation
Unless otherwise noted, the formulas in this chapter hold for all values of the variables x and t , and for all nonnegative integers n . …
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 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. …
3: 20 Theta Functions
Chapter 20 Theta Functions
4: 25.12 Polylogarithms
The notation Li 2 ( z ) was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): …
See accompanying text
Figure 25.12.1: Dilogarithm function Li 2 ( x ) , 20 x < 1 . Magnify
See accompanying text
Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
Integral Representation
Sometimes the factor 1 / Γ ( s + 1 ) is omitted. …
5: 32.8 Rational Solutions
In the general case assume γ δ 0 , so that as in §32.2(ii) we may set γ = 1 and δ = 1 . … For determinantal representations see Kajiwara and Masuda (1999). … For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999). …
  • (c)

    α = 1 2 a 2 , β = 1 2 ( a + n ) 2 , and γ = m , with m + n even.

  • For determinantal representations see Masuda et al. (2002). …
    6: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.
  • 7: 25.5 Integral Representations
    §25.5 Integral Representations
    For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). In (25.5.15)–(25.5.19), 0 < s < 1 , ψ ( x ) is the digamma function, and γ is Euler’s constant (§5.2). …
    25.5.19 ζ ( m + s ) = ( 1 ) m 1 Γ ( s ) sin ( π s ) π Γ ( m + s ) 0 ψ ( m ) ( 1 + x ) x s d x , m = 1 , 2 , 3 , .
    §25.5(iii) Contour Integrals
    8: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • Zhang and Jin (1996, Table 3.8) tabulates γ ( a , x ) for a = 0.5 , 1 , 3 , 5 , 10 , 25 , 50 , 100 , x = 0 ( .1 ) 1 ( 1 ) 3 , 5 ( 5 ) 30 , 50 , 100 to 8D or 8S.

  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 9: 8 Incomplete Gamma and Related
    Functions
    10: 28 Mathieu Functions and Hill’s Equation