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1: 16.18 Special Cases
§16.18 Special Cases
The F 1 1 and F 1 2 functions introduced in Chapters 13 and 15, as well as the more general F q p functions introduced in the present chapter, are all special cases of the Meijer G -function. …
2: 19.14 Reduction of General Elliptic Integrals
§19.14(ii) General Case
3: 8.18 Asymptotic Expansions of I x ( a , b )
General Case
4: Guide to Searching the DLMF
DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. …
5: 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 . … In the general case, P VI  has rational solutions if …
6: 16.2 Definition and Analytic Properties
§16.2(ii) Case p q
§16.2(iii) Case p = q + 1
§16.2(iv) Case p > q + 1
Polynomials
7: 20.2 Definitions and Periodic Properties
Figure 20.2.1: z -plane. …Left-hand diagram is the rectangular case ( τ purely imaginary); right-hand diagram is the general case. …
8: 23.22 Methods of Computation
  • (a)

    In the general case, given by c d 0 , we compute the roots α , β , γ , say, of the cubic equation 4 t 3 - c t - d = 0 ; see §1.11(iii). These roots are necessarily distinct and represent e 1 , e 2 , e 3 in some order.

    If c and d are real, and the discriminant is positive, that is c 3 - 27 d 2 > 0 , then e 1 , e 2 , e 3 can be identified via (23.5.1), and k 2 , k 2 obtained from (23.6.16).

    If c 3 - 27 d 2 < 0 , or c and d are not both real, then we label α , β , γ so that the triangle with vertices α , β , γ is positively oriented and [ α , γ ] is its longest side (chosen arbitrarily if there is more than one). In particular, if α , β , γ are collinear, then we label them so that β is on the line segment ( α , γ ) . In consequence, k 2 = ( β - γ ) / ( α - γ ) , k 2 = ( α - β ) / ( α - γ ) satisfy k 2 0 k 2 (with strict inequality unless α , β , γ are collinear); also | k 2 | , | k 2 | 1 .

    Finally, on taking the principal square roots of k 2 and k 2 we obtain values for k and k that lie in the 1st and 4th quadrants, respectively, and 2 ω 1 , 2 ω 3 are given by

    23.22.1 2 ω 1 M ( 1 , k ) = - 2 i ω 3 M ( 1 , k ) = π 3 c ( 2 + k 2 k 2 ) ( k 2 - k 2 ) d ( 1 - k 2 k 2 ) ,

    where M denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 2 possible choices of the square root.

  • (b)

    If d = 0 , then

    23.22.2 2 ω 1 = - 2 i ω 3 = ( Γ ( 1 4 ) ) 2 2 π c 1 / 4 .

    There are 4 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when c > 0 and ω 1 > 0 .

  • (c)

    If c = 0 , then

    23.22.3 2 ω 1 = 2 e - π i / 3 ω 3 = ( Γ ( 1 3 ) ) 3 2 π d 1 / 6 .

    There are 6 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when d > 0 and ω 1 > 0 .

  • 9: 25.16 Mathematical Applications
    H ( s ) is the special case H ( s , 1 ) of the function …
    10: 16.8 Differential Equations
    Analytical continuation formulas for F q q + 1 ( a ; b ; z ) near z = 1 are given in Bühring (1987b) for the case q = 2 , and in Bühring (1992) for the general case. …