第二十三讲柱函数( 第3页 Zeros of the functions J,(a)& Nv(z) 1. Real zeros When v is real, the functions Jv(a)& Nv(z)each have an infinite number of zeros, all of which are simple with the possible exception of z=0. For non-negative v the sth positive zeros of these functions are denoted by ju,,s and nu, s respectively 3s2 3.831710.89358 7.015593.95768 312 38.6537310.173477.068058.59601 411.7915313.3236910.2223511.74915 514.9309216.4706313.3611014.89744 618.0710619.6158616.5009218.04340 721.2116422.7600819.6413121.18807 824.3524725.9036722.7820324.33194 927.4934829.0468325.9229627.47529 030.6346132.1896829.0640330.61829 2. McMahons expansions for large zeros -14(-1)(71-31)32(1-1)(8312-982+3779) 3(86) 64(-1)(89-1535415436272372-…,s>n,H=4D2 105(86 6= 3. Complex zeros of J,(a) When 12-l the zeros of J,(z) are all real. If v< -l and v is not an integer the number of complex zeros of J,(z) is twice the integer part of (-v; if the integer part of (-v)is odd two of these zeros lie on the imaginary axis 4. Complex zeros of Nv(a When v is real the pattern of the complex zeros of N,(z)depends on the non-integer part of Attention is confined here to the case v=n.a tive integer or zeroWu Chong-shi ￾✁✂✄☎ ✆ ✝ ✞ (✁ ) ➼ 3 ➽ Zeros of the functions Jν(z) & Nν (z) 1. Real zeros When ν is real, the functions Jν(z) & Nν (z) each have an infinite number of zeros, all of which are simple with the possible exception of z = 0. For non-negative ν the sth positive zeros of these functions are denoted by jν,s and nν,s respectively. s j0,s j1,s n0,s n1,s 1 2.40483 3.83171 0.89358 2.19714 2 5.52008 7.01559 3.95768 5.42968 3 8.65373 10.17347 7.06805 8.59601 4 11.79153 13.32369 10.22235 11.74915 5 14.93092 16.47063 13.36110 14.89744 6 18.07106 19.61586 16.50092 18.04340 7 21.21164 22.76008 19.64131 21.18807 8 24.35247 25.90367 22.78203 24.33194 9 27.49348 29.04683 25.92296 27.47529 10 30.63461 32.18968 29.06403 30.61829 2. McMahon’s expansions for large zeros jν,s, nν,s ∼ β − µ − 1 8β − 4(µ − 1)(7µ − 31) 3(8β) 3 − 32(µ − 1)(83µ 2 − 982µ + 3779) 15(8β) 5 − 64(µ − 1)(6949µ 3 − 153855µ 2 + 1585743µ − 6277237) 105(8β) 7 − · · · · · · , s  ν, µ = 4ν 2 , β =     s + ν 2 − 1 4  π, for jν,s  s + ν 2 − 3 4  π, for nν,s 3. Complex zeros of Jν(z) When ν ≥ −1 the zeros of Jν(z) are all real. If ν < −1 and ν is not an integer the number of complex zeros of Jν(z) is twice the integer part of (−ν); if the integer part of (−ν) is odd two of these zeros lie on the imaginary axis. 4. Complex zeros of Nν (z) When ν is real the pattern of the complex zeros of Nν(z) depends on the non-integer part of ν. Attention is confined here to the case ν = n, a positive integer or zero