Neumann%20polynomial
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11: 20 Theta Functions
Chapter 20 Theta Functions
…12: 18.5 Explicit Representations
13: Errata
Originally the contour of integration written incorrectly as , has been corrected to be .
Reported by Mark Dunster on 2021-03-22
14: 24.2 Definitions and Generating Functions
§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
… ►15: 1.11 Zeros of Polynomials
§1.11 Zeros of Polynomials
… ►Horner’s Scheme
… ►§1.11(ii) Elementary Properties
… ►The discriminant of is defined by … ►§1.11(v) Stable Polynomials
…16: 3.8 Nonlinear Equations
§3.8(iv) Zeros of Polynomials
►The polynomial … ►Example. Wilkinson’s Polynomial
… ►Consider and . We have and . …17: Bibliography I
18: 6.20 Approximations
Cody and Thacher (1968) provides minimax rational approximations for , with accuracies up to 20S.
Cody and Thacher (1969) provides minimax rational approximations for , with accuracies up to 20S.
MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
19: 24.18 Physical Applications
§24.18 Physical Applications
►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).20: 7.24 Approximations
Hastings (1955) gives several minimax polynomial and rational approximations for , and the auxiliary functions and .
Cody (1969) provides minimax rational approximations for and . The maximum relative precision is about 20S.
Cody et al. (1970) gives minimax rational approximations to Dawson’s integral (maximum relative precision 20S–22S).