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11: 13.2 Definitions and Basic Properties
This equation has a regular singularity at the origin with indices 0 and 1 b , and an irregular singularity at infinity of rank one. … In general, U ( a , b , z ) has a branch point at z = 0 . The principal branch corresponds to the principal value of z a in (13.2.6), and has a cut in the z -plane along the interval ( , 0 ] ; compare §4.2(i). … Except when z = 0 each branch of U ( a , b , z ) is entire in a and b . Unless specified otherwise, however, U ( a , b , z ) is assumed to have its principal value. …
12: 16.8 Differential Equations
is a value z 0 of z at which all the coefficients f j ( z ) , j = 0 , 1 , , n 1 , are analytic. …
13: 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 … In general the series (16.2.1) diverges for all nonzero values of z . …
14: 10.72 Mathematical Applications
In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . …
15: 28.5 Second Solutions fe n , ge n
when a = a n ( q ) , n = 0 , 1 , 2 , , and by …For m = 0 , 1 , 2 , , we have … As q 0 with n 0 , C n ( q ) 0 , S n ( q ) 0 , C n ( q ) f n ( z , q ) sin n z , and S n ( q ) g n ( z , q ) cos n z . … For q = 0 , … …
16: 28.2 Definitions and Basic Properties
se n ( z , 0 ) = sin ( n z ) , n = 1 , 2 , 3 , .
17: 4.13 Lambert W -Function
The other branches W k ( z ) are single-valued analytic functions on ( , 0 ] , have a logarithmic branch point at z = 0 , and, in the case k = ± 1 , have a square root branch point at z = e 1 0 i respectively. …
18: 1.10 Functions of a Complex Variable
If the path circles a branch point at z = a , k times in the positive sense, and returns to z 0 without encircling any other branch point, then its value is denoted conventionally as F ( ( z 0 a ) e 2 k π i + a ) . …
19: 15.6 Integral Representations
In (15.6.2) the point 1 / z lies outside the integration contour, t b 1 and ( t 1 ) c b 1 assume their principal values where the contour cuts the interval ( 1 , ) , and ( 1 z t ) a = 1 at t = 0 . …
20: 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 } . …