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1: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 1 2 2 z 2 + 5 3 4 2 z 2 + 9 , z > 0 ,
7.9.3 w ( z ) = i π 1 z 1 2 z 1 z 3 2 z 2 z , z > 0 .
2: 1.12 Continued Fractions
§1.12(iv) Contraction and Extension
If C n = C 2 n , n = 0 , 1 , 2 , , then C is called the even part of C . The even part of C exists iff b 2 k 0 , k = 1 , 2 , , and up to equivalence is given by … and the even and odd parts of the continued fraction converge to finite values. …
3: 18.17 Integrals
18.17.41 0 e a x 𝐻𝑒 n ( x ) x z 1 d x = Γ ( z + n ) a n 2 F 2 2 ( 1 2 n , 1 2 n + 1 2 1 2 z 1 2 n , 1 2 z 1 2 n + 1 2 ; 1 2 a 2 ) , a > 0 . Also, z > 0 , n even; z > 1 , n odd.
4: 28.23 Expansions in Series of Bessel Functions
When j = 2 , 3 , 4 the series in the even-numbered equations converge for z > 0 and | cosh z | > 1 , and the series in the odd-numbered equations converge for z > 0 and | sinh z | > 1 . …
5: 20.1 Special Notation
m , n integers.
q ( ) the nome, q = e i π τ , 0 < | q | < 1 . Since τ is not a single-valued function of q , it is assumed that τ is known, even when q is specified. Most applications concern the rectangular case τ = 0 , τ > 0 , so that 0 < q < 1 and τ and q are uniquely related.
6: 19.38 Approximations
They are valid over parts of the complex k and ϕ planes. The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for ϕ near π / 2 with the improvements made in the 1970 reference. …
7: 4.37 Inverse Hyperbolic Functions
4.37.11 arccosh ( z ) = ± π i + arccosh z , z 0 .
8: 14.17 Integrals
(When l + m + n is even the condition | m n | < l < m + n is not needed.) …
14.17.18 1 P ν ( x ) Q λ ( x ) d x = 1 ( λ ν ) ( ν + λ + 1 ) , λ > ν > 0 .
14.17.19 1 Q ν ( x ) Q λ ( x ) d x = ψ ( λ + 1 ) ψ ( ν + 1 ) ( λ ν ) ( λ + ν + 1 ) , ( λ + ν ) > 1 , λ ν , λ and ν 1 , 2 , 3 , .
14.17.20 1 ( Q ν ( x ) ) 2 d x = ψ ( ν + 1 ) 2 ν + 1 , ν > 1 2 .
9: 18.32 OP’s with Respect to Freud Weights
where Q ( x ) is real, even, nonnegative, and continuously differentiable, where x Q ( x ) increases for x > 0 , and Q ( x ) as x , see Freud (1969). …See the early survey by Nevai (1986, Part 2). …
10: 20.2 Definitions and Periodic Properties
For fixed τ , each θ j ( z | τ ) is an entire function of z with period 2 π ; θ 1 ( z | τ ) is odd in z and the others are even. For fixed z , each of θ 1 ( z | τ ) / sin z , θ 2 ( z | τ ) / cos z , θ 3 ( z | τ ) , and θ 4 ( z | τ ) is an analytic function of τ for τ > 0 , with a natural boundary τ = 0 , and correspondingly, an analytic function of q for | q | < 1 with a natural boundary | q | = 1 . …