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1: 1.17 Integral and Series Representations of the Dirac Delta
§1.17 Integral and Series Representations of the Dirac Delta
§1.17(iii) Series Representations
By analogy with §1.17(ii) we have the formal series representation
Legendre Polynomials (§§14.7(i) and 18.3)
Laguerre Polynomials (§18.3)
2: 25.16 Mathematical Applications
25.16.1 ψ ( x ) = m = 1 p m x ln p ,
25.16.5 H ( s ) = n = 1 h ( n ) n s ,
25.16.6 H ( s ) = - ζ ( s ) + γ ζ ( s ) + 1 2 ζ ( s + 1 ) + r = 1 k ζ ( 1 - 2 r ) ζ ( s + 2 r ) + n = 1 1 n s n B ~ 2 k + 1 ( x ) x 2 k + 2 d x ,
25.16.7 H ( s ) = 1 2 ζ ( s + 1 ) + ζ ( s ) s - 1 - r = 1 k ( s + 2 r - 2 2 r - 1 ) ζ ( 1 - 2 r ) ζ ( s + 2 r ) - ( s + 2 k 2 k + 1 ) n = 1 1 n n B ~ 2 k + 1 ( x ) x s + 2 k + 1 d x .
25.16.11 H ( s , z ) = n = 1 1 n s m = 1 n 1 m z , ( s + z ) > 1 ,
3: 25.13 Periodic Zeta Function
25.13.1 F ( x , s ) n = 1 e 2 π i n x n s ,
4: 25.11 Hurwitz Zeta Function
§25.11(iv) Series Representations
25.11.8 ζ ( s , 1 2 a ) = ζ ( s , 1 2 a + 1 2 ) + 2 s n = 0 ( - 1 ) n ( n + a ) s , s > 0 , s 1 , 0 < a 1 .
25.11.9 ζ ( 1 - s , a ) = 2 Γ ( s ) ( 2 π ) s n = 1 1 n s cos ( 1 2 π s - 2 n π a ) , s > 1 , 0 < a 1 .
§25.11(x) Further Series Representations
25.11.36 L ( s , χ ) = n = 1 χ ( n ) n s = k - s r = 1 k - 1 χ ( r ) ζ ( s , r k ) , s > 1 ,
5: 25.12 Polylogarithms
25.12.7 Li 2 ( e i θ ) = n = 1 cos ( n θ ) n 2 + i n = 1 sin ( n θ ) n 2 .
25.12.10 Li s ( z ) = n = 1 z n n s .
25.12.12 Li s ( z ) = Γ ( 1 - s ) ( ln 1 z ) s - 1 + n = 0 ζ ( s - n ) ( ln z ) n n ! , s 1 , 2 , 3 , , | ln z | < 2 π ,
6: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
7: 25.15 Dirichlet L -functions
25.15.1 L ( s , χ ) = n = 1 χ ( n ) n s , s > 1 ,
8: 25.14 Lerch’s Transcendent
25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .
9: Bibliography E
  • J. A. Ewell (1990) A new series representation for ζ ( 3 ) . Amer. Math. Monthly 97 (3), pp. 219–220.
  • 10: 13.12 Products
    For integral representations, integrals, and series containing products of M ( a , b , z ) and U ( a , b , z ) see Erdélyi et al. (1953a, §6.15.3).