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asymptotic approximations to zeros


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1: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the q -Laguerre and continuous q - 1 -Hermite polynomials see Chen and Ismail (1998).
2: 13.22 Zeros
Asymptotic approximations to the zeros when the parameters κ and/or μ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …
3: 10.74 Methods of Computation
Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of 2 D graphs in (e. …
4: 5.4 Special Values and Extrema
As n , …
5: 18.26 Wilson Class: Continued
For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). …
6: 3.8 Nonlinear Equations
Initial approximations to the zeros can often be found from asymptotic or other approximations to f ( z ) , or by application of the phase principle or Rouché’s theorem; see §1.10(iv). …
7: 36.15 Methods of Computation
§36.15(ii) Asymptotics
Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from x = 0 , the uniform asymptotic approximations of §36.12 can be used. … Direct numerical evaluation can be carried out along a contour that runs along the segment of the real t -axis containing all real critical points of Φ and is deformed outside this range so as to reach infinity along the asymptotic valleys of exp ( i Φ ) . … This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of Φ , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
8: 13.9 Zeros
§13.9 Zeros
For fixed a and z in , U ( a , b , z ) has two infinite strings of b -zeros that are asymptotic to the imaginary axis as | b | .
9: 7.13 Zeros
§7.13 Zeros
As n As n the x n and y n corresponding to the zeros of C ( z ) satisfy … In consequence of (7.5.5) and (7.5.10), zeros of ( z ) are related to zeros of erfc z . …
10: 28.34 Methods of Computation
  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).