§172 Neumann 第10页 Zeros of the functions Jv(a)& nv(z) 1. Real zeros When v is real, the functions Jv(a)& N(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 these functions are denoted by jv s and nv. s respectively s J0. s J1,s n0,s n1,8 12.404833.831710.893582.19714 25.520087.015593.957685.42968 38.6537310.173477.068058.59601 411.7915313.3236910.2223511.74915 514.9309216.4706313.3611014.8974 618.0710619.6158616.5009218.0 721.2116422.7600819.6413121 824.3524725.90367227820324.33194 927.4934829.0468325.9229627.47529 1030.6346132.1896829.0640330.61829 2. McMahons expansions for large zeros 4-14(-1)(7-31) 32(-1)(83y2-9821+3779) 64(-1)(694 55x2+15857431-627723 105(8)7 D 1 B fo Jv, s v 3 24 3. Complex zeros of Jv(z) When v2-1 the zeros of Jv(a) 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(z) When v is real the pattern of the complex zeros of Nv(z) depends on the non-inte art of v. Attention is confined here to the case v= n, a positive integer or zero
§17.2 Neumann 函数 第 10 页 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 À ν, µ = 4ν 2 , β = µ s + ν 2 − 1 4 ¶ π, for jν,s β = µ 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
§172 Neumann 第11页 The figure above shows the approximate distribution of the complex zeros of Nn(a)in the region arg a<T. The figure is symmetrical about the real axis. The two curves on the left extend to infinity, having the asymptotes Imz=±=ln3=±0.5493 There are an infinite number of zeros near each of these curves The two curves extending from z = -n to z= n and bounding an eye-shaped domain intersect the imaginary axis at the points ti(na +b), where t-1=0.66274 f2ln2=0.191 and to=1.19968 is the positive root of cotht= t. There are n zeros near each of these curves. Complex zeros of No(2 Complex zeros of Ni(2 Real part Imaginary par Real part Imaginary part 2.40302 0.50274 0.78624 5.519880.54718 -3.833530.56236 -8.653670.54841 7.015900.55339
§17.2 Neumann 函数 第 11 页 Zeros of Nn(z) The figure above shows the approximate distribution of the complex zeros of Nn(z) in the region | arg z| ≤ π. The figure is symmetrical about the real axis. The two curves on the left extend to infinity, having the asymptotes Im z = ± 1 2 ln 3 = ±0.54931 . . . . . . There are an infinite number of zeros near each of these curves. The two curves extending from z = −n to z = n and bounding an eye-shaped domain intersect the imaginary axis at the points ±i(na + b), where a = q t 2 0 − 1 = 0.66274 . . . . . . b = 1 2 q 1 − t −2 0 ln 2 = 0.19146 . . . . . . and t0 = 1.19968 . . . . . . is the positive root of coth t = t. There are n zeros near each of these curves. Complex zeros of N0(z) Complex zeros of N1(z) Real part Imaginary part Real part Imaginary part −2.40302 0.53988 −0.50274 0.78624 −5.51988 0.54718 −3.83353 0.56236 −8.65367 0.54841 −7.01590 0.55339
第12页 §17.3柱函数 凡是满足递推关系 d x"C(x)=x"C-1(x) d 的函数{Cu(x)},统称为柱函数.前面介绍的 Bessel函数和 Neumann函数都是柱函数 柱函数一定是 Bessel方程的解 首先,把递推关系改写成 C(x)+C(x)=C-1(x) C(x)-=C(a)=-C+1(x) 将(型)式微商,得 C"(x)+=C"(x)-C(x)=C"-1(x) (##) 再将(#)式中的改写为v-1,并将()式代入, u-1(x)-C(x) (x)+C,(x)-C(x) 再代入(##)式,即得 ca)+c()-2xc(x)+2c(x)-C(x) 稍加整理,就得到 C"(x)+C(x)+ C 这就证明了柱函数{Cu(x)}一定是 Bessel方程的解.口
§17.3 柱 函 数 第 12 页 §17.3 柱 函 数 凡是满足递推关系 d dx [x νCν(x)] = x νCν−1(x), d dx h x −νCν(x) i = − x −νCν+1(x) 的函数{Cν(x)}，统称为柱函数．前面介绍的Bessel函数和Neumann 函数都是柱函数． 柱函数一定是Bessel方程的解． 首先，把递推关系改写成 C 0 ν(x) + ν x Cν(x) = Cν−1(x) (z) C 0 ν(x) − ν x Cν(x) = −Cν+1(x). (#) 将(z)式微商，得 C 00 ν (x) + ν x C 0 ν(x) − ν x2 Cν(x) = C 0 ν−1(x). (##) 再将(#)式中的ν改写为ν − 1，并将(z)式代入， C 0 ν−1(x) = ν − 1 x Cν−1(x) − Cν(x) = ν − 1 x h C 0 ν(x) + ν x Cν(x) i − Cν(x). 再代入(##)式，即得 C 00 ν (x) + ν x C 0 ν(x) − ν x2 Cν(x) = ν − 1 x C 0 ν(x) + ν(ν − 1) x2 Cν(x) − Cν(x), 稍加整理，就得到 C 00 ν (x) + 1 x C 0 ν(x) + µ 1 − ν 2 x2 ¶ Cν(x) = 0. 这就证明了柱函数{Cν(x)}一定是Bessel方程的解．