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1: 20.4 Values at z = 0
§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 ) .
2: 12.2 Differential Equations
§12.2(ii) Values at z = 0
3: 18.6 Symmetry, Special Values, and Limits to Monomials
Laguerre
4: 12.14 The Function W ( a , x )
§12.14(ii) Values at z = 0 and Wronskian
5: 6.4 Analytic Continuation
Analytic continuation of the principal value of E 1 ( z ) yields a multi-valued function with branch points at z = 0 and z = . …
6: 4.2 Definitions
This is a multivalued function of z with branch point at z = 0 . …
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: 10.25 Definitions
In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . … It has a branch point at z = 0 for all ν . The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in ( - , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . Both I ν ( z ) and K ν ( z ) are real when ν is real and ph z = 0 . For fixed z ( 0 ) each branch of I ν ( z ) and K ν ( z ) is entire in ν . …
9: 10.2 Definitions
This differential equation has a regular singularity at z = 0 with indices ± ν , and an irregular singularity at z = of rank 1 ; compare §§2.7(i) and 2.7(ii). … This solution of (10.2.1) is an analytic function of z , except for a branch point at z = 0 when ν is not an integer. … Whether or not ν is an integer Y ν ( z ) has a branch point at z = 0 . … Each solution has a branch point at z = 0 for all ν . The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the z -plane along the interval ( - , 0 ] . …
10: 13.14 Definitions and Basic Properties
In general M κ , μ ( z ) and W κ , μ ( z ) are many-valued functions of z with branch points at z = 0 and z = . …