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doubly-periodic forms

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11: 30.2 Differential Equations
§30.2(ii) Other Forms
The Liouville normal form of equation (30.2.1) is …
12: 29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
29.2.4 ( 1 k 2 cos 2 ϕ ) d 2 w d ϕ 2 + k 2 cos ϕ sin ϕ d w d ϕ + ( h ν ( ν + 1 ) k 2 cos 2 ϕ ) w = 0 ,
we have …
13: 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 ,
7.9.3 w ( z ) = i π 1 z 1 2 z 1 z 3 2 z 2 z , z > 0 .
14: 15.14 Integrals
Integrals of the form x α ( x + t ) β F ( a , b ; c ; x ) d x and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), Gradshteyn and Ryzhik (2000, §7.5) and Koornwinder (2015). …
15: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5 Limiting Forms for Small ρ , Small | η | , or Large
§33.5(i) Small ρ
§33.5(iii) Small | η |
§33.5(iv) Large
16: 8.23 Statistical Applications
Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). …
17: 10.29 Recurrence Relations and Derivatives
For results on modified quotients of the form z 𝒵 ν ± 1 ( z ) / 𝒵 ν ( z ) see Onoe (1955) and Onoe (1956).
§10.29(ii) Derivatives
18: 10.30 Limiting Forms
§10.30 Limiting Forms
§10.30(i) z 0
19: 13.27 Mathematical Applications
The elements of this group are of the form
20: 26.5 Lattice Paths: Catalan Numbers
§26.5(iv) Limiting Forms