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1: 33.13 Complex Variable and Parameters
These functions may also be continued analytically to complex values of ρ , η , and . …
2: 28.7 Analytic Continuation of Eigenvalues
As functions of q , a n ( q ) and b n ( q ) can be continued analytically in the complex q -plane. …
3: 1.10 Functions of a Complex Variable
The function f 1 ( z ) on D 1 is said to be analytically continued along the path z ( t ) , a t b , if there is a chain ( f 1 , D 1 ) , ( f 2 , D 2 ) , , ( f n , D n ) . … Then f ( z ) can be continued analytically across AB by reflection, that is, …
4: 2.5 Mellin Transform Methods
Similarly, if f ( 1 - z ) and h ( z ) can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for I ( x ) for large values of x . … Furthermore, f 1 ( z ) can be continued analytically to a meromorphic function on the entire z -plane, whose singularities are simple poles at - α s , s = 0 , 1 , 2 , , with principal part … Similarly, if κ = 0 in (2.5.18), then h 2 ( z ) can be continued analytically to a meromorphic function on the entire z -plane with simple poles at β s , s = 0 , 1 , 2 , , with principal part …Alternatively, if κ 0 in (2.5.18), then h 2 ( z ) can be continued analytically to an entire function. Since h 1 ( z ) is analytic for z > - c by Table 2.5.1, the analytically-continued h 2 ( z ) allows us to extend the Mellin transform of h via …
5: 18.2 General Orthogonal Polynomials
It is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …
6: 19.7 Connection Formulas
Provided the functions in these identities are correctly analytically continued in the complex β -plane, then the identities will also hold in the complex β -plane. …
7: 33.22 Particle Scattering and Atomic and Molecular Spectra
The Coulomb functions given in this chapter are most commonly evaluated for real values of ρ , r , η , ϵ and nonnegative integer values of , but they may be continued analytically to complex arguments and order as indicated in §33.13. …
8: Bibliography J
  • L. Jacobsen, W. B. Jones, and H. Waadeland (1986) Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.
  • W. B. Jones and W. J. Thron (1980) Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.
  • 9: 10.20 Uniform Asymptotic Expansions for Large Order
    The function ζ = ζ ( z ) given by (10.20.2) and (10.20.3) can be continued analytically to the z -plane cut along the negative real axis. …
    10: 2.7 Differential Equations
    Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution w j ( z ) can be continued analytically into any other sector. … In a finite or infinite interval ( a 1 , a 2 ) let f ( x ) be real, positive, and twice-continuously differentiable, and g ( x ) be continuous. …