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1: 23.13 Zeros
§23.13 Zeros
β–ΊFor information on the zeros of ⁑ ( z ) see Eichler and Zagier (1982).
2: 34.10 Zeros
§34.10 Zeros
β–ΊIn a 3 ⁒ j symbol, if the three angular momenta j 1 , j 2 , j 3 do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the 3 ⁒ j symbol is zero. …Such zeros are called trivial zeros. …Such zeros are called nontrivial zeros. β–ΊFor further information, including examples of nontrivial zeros and extensions to 9 ⁒ j symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).
3: 10.58 Zeros
§10.58 Zeros
β–ΊFor n 0 the m th positive zeros of 𝗃 n ⁑ ( x ) , 𝗃 n ⁑ ( x ) , 𝗒 n ⁑ ( x ) , and 𝗒 n ⁑ ( x ) are denoted by a n , m , a n , m , b n , m , and b n , m , respectively, except that for n = 0 we count x = 0 as the first zero of 𝗃 0 ⁑ ( x ) . … β–Ί
a n , m = j n + 1 2 , m ,
β–Ί
b n , m = y n + 1 2 , m ,
β–ΊHowever, there are no simple relations that connect the zeros of the derivatives. …
4: 9.9 Zeros
§9.9 Zeros
β–ΊThey are denoted by a k , a k , b k , b k , respectively, arranged in ascending order of absolute value for k = 1 , 2 , . β–Ί
§9.9(ii) Relation to Modulus and Phase
β–Ί
§9.9(iv) Asymptotic Expansions
β–Ί
§9.9(v) Tables
5: 13.22 Zeros
§13.22 Zeros
β–ΊAsymptotic approximations to the zeros when the parameters ΞΊ and/or ΞΌ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if ΞΌ ( 0 ) is fixed and ΞΊ ( > 0 ) is large, then the r th positive zero Ο• r of M ΞΊ , ΞΌ ⁑ ( z ) is given by …where j 2 ⁒ ΞΌ , r is the r th positive zero of the Bessel function J 2 ⁒ ΞΌ ⁑ ( x ) 10.21(i)). …
6: 6.13 Zeros
§6.13 Zeros
β–ΊThe function Ei ⁑ ( x ) has one real zero x 0 , given by β–Ί
6.13.1 x 0 = 0.37250 74107 81366 63446 19918 66580 ⁒ .
β–Ί Ci ⁑ ( x ) and si ⁑ ( x ) each have an infinite number of positive real zeros, which are denoted by c k , s k , respectively, arranged in ascending order of absolute value for k = 0 , 1 , 2 , . Values of c 1 and c 2 to 30D are given by MacLeod (1996b). …
7: 16.9 Zeros
§16.9 Zeros
β–ΊThen F p p ⁑ ( 𝐚 ; 𝐛 ; z ) has at most finitely many zeros if and only if the a j can be re-indexed for j = 1 , , p in such a way that a j b j is a nonnegative integer. … β–ΊThen F p p ⁑ ( 𝐚 ; 𝐛 ; z ) has at most finitely many real zeros. … β–ΊFor further information on zeros see Hille (1929).
8: 24.12 Zeros
§24.12 Zeros
β–Ί
§24.12(i) Bernoulli Polynomials: Real Zeros
β–Ί
§24.12(ii) Euler Polynomials: Real Zeros
β–Ί
§24.12(iii) Complex Zeros
β–Ί
§24.12(iv) Multiple Zeros
9: 10.42 Zeros
§10.42 Zeros
β–ΊFor example, if Ξ½ is real, then the zeros of I Ξ½ ⁑ ( z ) are all complex unless 2 ⁒ β„“ < Ξ½ < ( 2 ⁒ β„“ 1 ) for some positive integer β„“ , in which event I Ξ½ ⁑ ( z ) has two real zeros. … β–ΊThe zeros in the sector 1 2 ⁒ Ο€ ph ⁑ z 3 2 ⁒ Ο€ are their conjugates. … β–ΊFor z -zeros of K Ξ½ ⁑ ( z ) , with complex Ξ½ , see Ferreira and Sesma (2008). … β–Ί
10: 8.13 Zeros
§8.13 Zeros
β–Ί
  • (a)

    one negative zero x ⁑ ( a ) and no positive zeros when 1 2 ⁒ n < a < 2 2 ⁒ n ;

  • β–Ί
  • (b)

    one negative zero x ⁑ ( a ) and one positive zero x + ⁑ ( a ) when 2 ⁒ n < a < 1 2 ⁒ n .

  • β–ΊAs x increases the positive zeros coalesce to form a double zero at ( a n * , x n * ). The values of the first six double zeros are given to 5D in Table 8.13.1. …