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case λ=0

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11: 20.11 Generalizations and Analogs
In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). …
12: 28.35 Tables
  • Blanch and Clemm (1965) includes values of Mc n ( 2 ) ( x , q ) , Mc n ( 2 ) ( x , q ) for n = 0 ( 1 ) 7 , x = 0 ( .02 ) 1 ; n = 8 ( 1 ) 15 , x = 0 ( .01 ) 1 . Also Ms n ( 2 ) ( x , q ) , Ms n ( 2 ) ( x , q ) for n = 1 ( 1 ) 7 , x = 0 ( .02 ) 1 ; n = 8 ( 1 ) 15 , x = 0 ( .01 ) 1 . In all cases q = 0 ( .05 ) 1 . Precision is generally 7D. Approximate formulas and graphs are also included.

  • 13: 10.67 Asymptotic Expansions for Large Argument
    §10.67(ii) Cross-Products and Sums of Squares in the Case ν = 0
    14: 28.4 Fourier Series
    §28.4(iv) Case q = 0
    15: 30.2 Differential Equations
    This equation has regular singularities at z = ± 1 with exponents ± 1 2 μ and an irregular singularity of rank 1 at z = (if γ 0 ). … With ζ = γ z Equation (30.2.1) changes to …
    §30.2(iii) Special Cases
    If γ = 0 , Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …If γ = 0 , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).
    16: 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 . … In the general case assume δ 0 , so that as in §32.2(ii) we may set δ = 1 2 . … For the case δ = 0 see Airault (1979) and Lukaševič (1968). …
    17: 18.14 Inequalities
    18.14.3_5 ( 1 2 ( 1 + x ) ) β / 2 | P n ( α , β ) ( x ) | P n ( α , β ) ( 1 ) = ( α + 1 ) n n ! , 1 x 1 , α , β 0 .
    18.14.8 e 1 2 x | L n ( α ) ( x ) | L n ( α ) ( 0 ) = ( α + 1 ) n n ! , 0 x < , α 0 .
    The case β = 0 of (18.14.26) is the Askey–Gasper inequality (18.38.3). …
    18: 19.26 Addition Theorems
    §19.26(ii) Case x = 0
    19: 19.17 Graphics
    The cases x = 0 or y = 0 correspond to the complete integrals. …
    20: 10.43 Integrals
    10.43.2 z ν 𝒵 ν ( z ) d z = π 1 2 2 ν 1 Γ ( ν + 1 2 ) z ( 𝒵 ν ( z ) 𝐋 ν 1 ( z ) 𝒵 ν 1 ( z ) 𝐋 ν ( z ) ) , ν 1 2 .
    10.43.25 0 K ν ( b t ) exp ( p 2 t 2 ) d t = π 4 p sec ( 1 2 π ν ) exp ( b 2 8 p 2 ) K 1 2 ν ( b 2 8 p 2 ) , | ν | < 1 , ( p 2 ) > 0 .