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relation to infinite partial fractions

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1: 1.9 Calculus of a Complex Variable
If f ( z ( t 0 ) ) = , a t 0 b , then the integral is defined analogously to the infinite integrals in §1.4(v). …
§1.9(v) Infinite Sequences and Series
This sequence converges pointwise to a function f ( z ) if …
§1.9(vii) Inversion of Limits
2: 1.10 Functions of a Complex Variable
§1.10(ix) Infinite Products
The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. …
§1.10(x) Infinite Partial Fractions
Mittag-Leffler’s Expansion
3: 16.7 Relations to Other Functions
§16.7 Relations to Other Functions
4: 3.10 Continued Fractions
§3.10(ii) Relations to Power Series
can be converted into a continued fraction C of type (3.10.1), and with the property that the n th convergent C n = A n / B n to C is equal to the n th partial sum of the series in (3.10.3), that is, …
Stieltjes Fractions
The A n and B n of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … This forward algorithm achieves efficiency and stability in the computation of the convergents C n = A n / B n , and is related to the forward series recurrence algorithm. …
5: 6.11 Relations to Other Functions
§6.11 Relations to Other Functions
Incomplete Gamma Function
Confluent Hypergeometric Function
6.11.2 E 1 ( z ) = e - z U ( 1 , 1 , z ) ,
6: 4.22 Infinite Products and Partial Fractions
§4.22 Infinite Products and Partial Fractions
4.22.1 sin z = z n = 1 ( 1 - z 2 n 2 π 2 ) ,
4.22.2 cos z = n = 1 ( 1 - 4 z 2 ( 2 n - 1 ) 2 π 2 ) .
4.22.3 cot z = 1 z + 2 z n = 1 1 z 2 - n 2 π 2 ,
4.22.4 csc 2 z = n = - 1 ( z - n π ) 2 ,
7: 4.36 Infinite Products and Partial Fractions
§4.36 Infinite Products and Partial Fractions
4.36.1 sinh z = z n = 1 ( 1 + z 2 n 2 π 2 ) ,
4.36.2 cosh z = n = 1 ( 1 + 4 z 2 ( 2 n - 1 ) 2 π 2 ) .
4.36.3 coth z = 1 z + 2 z n = 1 1 z 2 + n 2 π 2 ,
4.36.4 csch 2 z = n = - 1 ( z - n π i ) 2 ,
8: Bibliography R
  • E. M. Rains (1998) Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.
  • Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.
  • M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
  • R. R. Rosales (1978) The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
  • 9: 15.19 Methods of Computation
    However, by appropriate choice of the constant z 0 in (15.15.1) we can obtain an infinite series that converges on a disk containing z = e ± π i / 3 . Moreover, it is also possible to accelerate convergence by appropriate choice of z 0 . …
    §15.19(iv) Recurrence Relations
    The relations in §15.5(ii) can be used to compute F ( a , b ; c ; z ) , provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …
    §15.19(v) Continued Fractions
    10: 25.13 Periodic Zeta Function
    §25.13 Periodic Zeta Function
    The notation F ( x , s ) is used for the polylogarithm Li s ( e 2 π i x ) with x real:
    25.13.1 F ( x , s ) n = 1 e 2 π i n x n s ,
    Also,
    25.13.2 F ( x , s ) = Γ ( 1 - s ) ( 2 π ) 1 - s ( e π i ( 1 - s ) / 2 ζ ( 1 - s , x ) + e π i ( s - 1 ) / 2 ζ ( 1 - s , 1 - x ) ) , 0 < x < 1 , s > 1 ,