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1: 25.11 Hurwitz Zeta Function
ζ ( s , a ) has a meromorphic continuation in the s -plane, its only singularity in being a simple pole at s = 1 with residue 1 . …
See accompanying text
Figure 25.11.1: Hurwitz zeta function ζ ( x , a ) , a = 0. …8, 1, 20 x 10 . … Magnify
25.11.13 ζ ( 0 , a ) = 1 2 a .
§25.11(xii) a -Asymptotic Behavior
Similarly, as a in the sector | ph a | 1 2 π δ ( < 1 2 π ) , …
2: 15.10 Hypergeometric Differential Equation
15.10.3 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( 1 c ) z c ( 1 z ) c a b 1 .
15.10.5 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( a + b c ) z c ( 1 z ) c a b 1 .
15.10.7 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( a b ) z c ( z 1 ) c a b 1 .
(c) If the parameter c in the differential equation equals 2 n = 0 , 1 , 2 , , then fundamental solutions in the neighborhood of z = 0 are given by z n 1 times those in (a) and (b), with a and b replaced throughout by a + n 1 and b + n 1 , respectively. … The ( 6 3 ) = 20 connection formulas for the principal branches of Kummer’s solutions are: …
3: 36.2 Catastrophes and Canonical Integrals
36.2.28 Ψ ( E ) ( 0 , 0 , z ) = Ψ ( E ) ( 0 , 0 , z ) ¯ = 2 π π z 27 exp ( 2 27 i z 3 ) ( J 1 / 6 ( 2 27 z 3 ) + i J 1 / 6 ( 2 27 z 3 ) ) , z 0 ,
36.2.29 Ψ ( H ) ( 0 , 0 , z ) = Ψ ( H ) ( 0 , 0 , z ) ¯ = 2 1 / 3 3 exp ( 1 27 i z 3 ) Ψ ( E ) ( 0 , 0 , z 2 2 / 3 ) , < z < .
4: 28.16 Asymptotic Expansions for Large q
28.16.1 λ ν ( h 2 ) 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
5: 20.7 Identities
See Lawden (1989, pp. 19–20). …
§20.7(viii) Transformations of Lattice Parameter
20.7.28 θ 3 ( z | τ + 1 ) = θ 4 ( z | τ ) ,
20.7.29 θ 4 ( z | τ + 1 ) = θ 3 ( z | τ ) .
20.7.34 θ 1 ( z , q 2 ) θ 3 ( z , q 2 ) θ 1 ( z , i q ) = θ 2 ( z , q 2 ) θ 4 ( z , q 2 ) θ 2 ( z , i q ) = i 1 / 4 θ 2 ( 0 , q 2 ) θ 4 ( 0 , q 2 ) 2 .
6: 12.11 Zeros
12.11.9 u a , 1 2 1 2 μ ( 1 1.85575 708 μ 4 / 3 0.34438 34 μ 8 / 3 0.16871 5 μ 4 0.11414 μ 16 / 3 0.0808 μ 20 / 3 ) ,
7: 30.9 Asymptotic Approximations and Expansions
30.9.1 λ n m ( γ 2 ) γ 2 + γ q + β 0 + β 1 γ 1 + β 2 γ 2 + ,
2 20 β 5 = 527 q 7 61529 q 5 10 43961 q 3 22 41599 q + 32 m 2 ( 5739 q 5 + 1 27550 q 3 + 2 98951 q ) 2048 m 4 ( 355 q 3 + 1505 q ) + 65536 m 6 q .
For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982). …
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.
  • 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.
  • N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • 9: 11.6 Asymptotic Expansions
    For fixed λ ( > 1 )
    11.6.6 𝐊 ν ( λ ν ) ( 1 2 λ ν ) ν 1 π Γ ( ν + 1 2 ) k = 0 k ! c k ( λ ) ν k , | ph ν | 1 2 π δ ,
    11.6.7 𝐌 ν ( λ ν ) ( 1 2 λ ν ) ν 1 π Γ ( ν + 1 2 ) k = 0 k ! c k ( i λ ) ν k , | ph ν | 1 2 π δ .
    c 3 ( λ ) = 20 λ 6 4 λ 4 ,
    10: Bibliography F
  • P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.
  • FDLIBM (free C library)
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.