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1: 4.46 Tables
For 40D values of the first 500 roots of tan x = x , see Robinson (1972). (These roots are zeros of the Bessel function J 3 / 2 ( x ) ; see §10.21.) For 10S values of the first five complex roots of sin z = a z , cos z = a z , and cosh z = a z , for selected positive values of a , see Fettis (1976). …
2: 2.2 Transcendental Equations
Then for y > f ( a ) the equation f ( x ) = y has a unique root x = x ( y ) in ( a , ) , and
2.2.2 x ( y ) y , y .
2.2.3 t 2 - ln t = y .
2.2.4 t = y 1 2 ( 1 + o ( 1 ) ) , y .
2.2.5 t 2 = y + ln t = y + 1 2 ln y + o ( 1 ) ,
3: 1.11 Zeros of Polynomials
Roots of f ( z ) = 0 are 2 + 4 3 + 2 3 , 2 + 4 3 ρ + 2 3 ρ 2 , 2 + 4 3 ρ 2 + 2 3 ρ . … The square roots are chosen so that …
§1.11(iv) Roots of Unity and of Other Constants
The roots of … The roots of …
4: Tom H. Koornwinder
Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …
5: 23.3 Differential Equations
§23.3(i) Invariants, Roots, and Discriminant
The lattice roots satisfy the cubic equation …
23.3.4 Δ = g 2 3 - 27 g 3 2 = 16 ( e 2 - e 3 ) 2 ( e 3 - e 1 ) 2 ( e 1 - e 2 ) 2 .
23.3.5 e 1 + e 2 + e 3 = 0 ,
23.3.7 g 3 = 4 e 1 e 2 e 3 = 4 3 ( e 1 3 + e 2 3 + e 3 3 ) .
6: 19.38 Approximations
Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …
7: 23.7 Quarter Periods
23.7.2 ( 1 2 ω 2 ) = e 2 - i ( e 1 - e 2 ) ( e 2 - e 3 ) = e 2 - i ω 1 - 2 ( K ( k ) ) 2 k k ,
23.7.3 ( 1 2 ω 3 ) = e 3 - ( e 1 - e 3 ) ( e 2 - e 3 ) = e 3 - ω 1 - 2 ( K ( k ) ) 2 k ,
where k , k and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.
8: 2.9 Difference Equations
2.9.4 ρ j n n α j s = 0 a s , j n s , j = 1 , 2 ,
where ρ 1 , ρ 2 are the roots of the characteristic equationWhen the roots of (2.9.5) are equal we denote them both by ρ . Assume first 2 g 1 f 0 f 1 . … Then the indices α 1 , α 2 are the roots of …
9: 23.21 Physical Applications
Ellipsoidal coordinates ( ξ , η , ζ ) may be defined as the three roots ρ of the equation
23.21.1 x 2 ρ - e 1 + y 2 ρ - e 2 + z 2 ρ - e 3 = 1 ,
23.21.3 f ( ρ ) = 2 ( ( ρ - e 1 ) ( ρ - e 2 ) ( ρ - e 3 ) ) 1 / 2 .
Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots23.3(i)), and setting …
10: 4.43 Cubic Equations
The roots of … Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …