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1: 28.5 Second Solutions fe n , ge n
2: 28.4 Fourier Series
§28.4(iv) Case q = 0
3: 28.2 Definitions and Basic Properties
se n ( z , 0 ) = sin ( n z ) , n = 1 , 2 , 3 , .
4: 20.4 Values at z = 0
20.4.12 θ 1 ′′′ ( 0 , q ) θ 1 ( 0 , q ) = θ 2 ′′ ( 0 , q ) θ 2 ( 0 , q ) + θ 3 ′′ ( 0 , q ) θ 3 ( 0 , q ) + θ 4 ′′ ( 0 , q ) θ 4 ( 0 , q ) .
5: 28.7 Analytic Continuation of Eigenvalues
The only singularities are algebraic branch points, with a n ( q ) and b n ( q ) finite at these points. …All real values of q are normal values. To 4D the first branch points between a 0 ( q ) and a 2 ( q ) are at q 0 = ± i 1.4688 with a 0 ( q 0 ) = a 2 ( q 0 ) = 2.0886 , and between b 2 ( q ) and b 4 ( q ) they are at q 1 = ± i 6.9289 with b 2 ( q 1 ) = b 4 ( q 1 ) = 11.1904 . For real q with | q | < | q 0 | , a 0 ( i q ) and a 2 ( i q ) are real-valued, whereas for real q with | q | > | q 0 | , a 0 ( i q ) and a 2 ( i q ) are complex conjugates. … Analogous statements hold for a 2 n + 1 ( q ) , b 2 n + 1 ( q ) , and b 2 n + 2 ( q ) , also for n = 0 , 1 , 2 , . …
6: 31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values q m , m = 0 , 1 , 2 , , of the accessory parameter q , the function H ( a , q ; α , β , γ , δ ; z ) is analytic at z = 1 , and hence also throughout the disk | z | < a . …
7: 31.3 Basic Solutions
H ( a , q ; α , β , γ , δ ; z ) denotes the solution of (31.2.1) that corresponds to the exponent 0 at z = 0 and assumes the value 1 there. …
8: 16.2 Definition and Analytic Properties
When p q the series (16.2.1) converges for all finite values of z and defines an entire function. … The branch obtained by introducing a cut from 1 to + on the real axis, that is, the branch in the sector | ph ( 1 z ) | π , is the principal branch (or principal value) of F q q + 1 ( 𝐚 ; 𝐛 ; z ) ; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z = 0 , 1 , and . … On the circle | z | = 1 the series (16.2.1) is absolutely convergent if γ q > 0 , convergent except at z = 1 if 1 < γ q 0 , and divergent if γ q 1 , where … (However, except where indicated otherwise in the DLMF we assume that when p > q + 1 at least one of the a k is a nonpositive integer.) …
9: 31.9 Orthogonality
Here ζ is an arbitrary point in the interval ( 0 , 1 ) . The integration path begins at z = ζ , encircles z = 1 once in the positive sense, followed by z = 0 once in the positive sense, and so on, returning finally to z = ζ . …The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … The right-hand side may be evaluated at any convenient value, or limiting value, of ζ in ( 0 , 1 ) since it is independent of ζ . … and the integration paths 1 , 2 are Pochhammer double-loop contours encircling distinct pairs of singularities { 0 , 1 } , { 0 , a } , { 1 , a } . …
10: 2.4 Contour Integrals
Except that λ is now permitted to be complex, with λ > 0 , we assume the same conditions on q ( t ) and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of z . …
  • (a)

    In a neighborhood of a

    2.4.11
    p ( t ) = p ( a ) + s = 0 p s ( t a ) s + μ ,
    q ( t ) = s = 0 q s ( t a ) s + λ 1 ,

    with λ > 0 , μ > 0 , p 0 0 , and the branches of ( t a ) λ and ( t a ) μ continuous and constructed with ph ( t a ) ω as t a along 𝒫 .

  • and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of p ( t ) and q ( t ) at t = t 0 . … Cases in which p ( t 0 ) 0 are usually handled by deforming the integration path in such a way that the minimum of ( z p ( t ) ) is attained at a saddle point or at an endpoint. … with p , q and their derivatives evaluated at t 0 . …