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11: 3.2 Linear Algebra
β–ΊIf 𝐀 is nondefective and Ξ» is a simple zero of p n ⁑ ( Ξ» ) , then the sensitivity of Ξ» to small perturbations in the matrix 𝐀 is measured by the condition number
12: 29.12 Definitions
β–ΊThe polynomial P ⁑ ( ΞΎ ) is of degree n and has m zeros (all simple) in ( 0 , 1 ) and n m zeros (all simple) in ( 1 , k 2 ) . …
13: 19.2 Definitions
β–ΊLet s 2 ⁒ ( t ) be a cubic or quartic polynomial in t with simple zeros, and let r ⁑ ( s , t ) be a rational function of s and t containing at least one odd power of s . …
14: 2.8 Differential Equations with a Parameter
β–ΊIn Case II f ⁑ ( z ) has a simple zero at z 0 and g ⁑ ( z ) is analytic at z 0 . …
15: 28.31 Equations of Whittaker–Hill and Ince
β–ΊThey are real and distinct, and can be ordered so that C p m ⁑ ( z , ΞΎ ) and S p m ⁑ ( z , ΞΎ ) have precisely m zeros, all simple, in 0 z < Ο€ . …
16: 10.58 Zeros
β–ΊHowever, there are no simple relations that connect the zeros of the derivatives. …
17: 10.21 Zeros
β–ΊThe zeros of any cylinder function or its derivative are simple, with the possible exceptions of z = 0 in the case of the functions, and z = 0 , ± Ξ½ in the case of the derivatives. … β–ΊAll of these zeros are simple, provided that Ξ½ 1 in the case of J Ξ½ ⁑ ( z ) , and Ξ½ 1 2 in the case of Y Ξ½ ⁑ ( z ) . When all of their zeros are simple, the m th positive zeros of these functions are denoted by j Ξ½ , m , j Ξ½ , m , y Ξ½ , m , and y Ξ½ , m respectively, except that z = 0 is counted as the first zero of J 0 ⁑ ( z ) . … β–Ίare simple and the asymptotic expansion of the m th positive zero as m is given by …
18: 9.12 Scorer Functions
β–ΊAll zeros, real or complex, of Gi ⁑ ( z ) and Hi ⁑ ( z ) are simple. …
19: 18.2 General Orthogonal Polynomials
β–ΊAll n zeros of an OP p n ⁑ ( x ) are simple, and they are located in the interval of orthogonality ( a , b ) . …
20: Bibliography R
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  • W. P. Reinhardt (2021a) Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.
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  • W. P. Reinhardt (2021b) Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.
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  • Hans-J. Runckel (1971) On the zeros of the hypergeometric function. Math. Ann. 191 (1), pp. 53–58.
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  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.