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Returns (r,err) where r is the estimated location of the root and err is a positive number estimating the error of the asymptotic expansion. rrrr)rriSiirrii%iXiO2iu_iRrii i i,i il"qiQYgr)rr3lenr6)rkindprimer>rHrprbs1s2Zs3Zs4Zs5rrverrir r r mcmahons6 ****(4@,8D rcs||dkrtd|d}gg|||fddDfddt|dD}t||krn|S|d}q dS)z Given f known to have exactly n simple roots within [a,b], return a list of n intervals isolating the roots and having opposite signs at the endpoints. 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Alternatively, with *derivative=1*, gives the first nonnegative simple zero `j'_{\nu,m}` of `J'_{\nu}(z)`. The indexing convention is that used by Abramowitz & Stegun and the DLMF. Note the special case `j'_{0,1} = 0`, while all other zeros are positive. In effect, only simple zeros are counted (all zeros of Bessel functions are simple except possibly `z = 0`) and `j_{\nu,m}` becomes a monotonic function of both `\nu` and `m`. The zeros are interlaced according to the inequalities .. math :: j'_{\nu,k} < j_{\nu,k} < j'_{\nu,k+1} j_{\nu,1} < j_{\nu+1,2} < j_{\nu,2} < j_{\nu+1,2} < j_{\nu,3} < \cdots **Examples** Initial zeros of the Bessel functions `J_0(z), J_1(z), J_2(z)`:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> besseljzero(0,1); besseljzero(0,2); besseljzero(0,3) 2.404825557695772768621632 5.520078110286310649596604 8.653727912911012216954199 >>> besseljzero(1,1); besseljzero(1,2); besseljzero(1,3) 3.831705970207512315614436 7.01558666981561875353705 10.17346813506272207718571 >>> besseljzero(2,1); besseljzero(2,2); besseljzero(2,3) 5.135622301840682556301402 8.417244140399864857783614 11.61984117214905942709415 Initial zeros of `J'_0(z), J'_1(z), J'_2(z)`:: 0.0 3.831705970207512315614436 7.01558666981561875353705 >>> besseljzero(1,1,1); besseljzero(1,2,1); besseljzero(1,3,1) 1.84118378134065930264363 5.331442773525032636884016 8.536316366346285834358961 >>> besseljzero(2,1,1); besseljzero(2,2,1); besseljzero(2,3,1) 3.054236928227140322755932 6.706133194158459146634394 9.969467823087595793179143 Zeros with large index:: >>> besseljzero(0,100); besseljzero(0,1000); besseljzero(0,10000) 313.3742660775278447196902 3140.807295225078628895545 31415.14114171350798533666 >>> besseljzero(5,100); besseljzero(5,1000); besseljzero(5,10000) 321.1893195676003157339222 3148.657306813047523500494 31422.9947255486291798943 >>> besseljzero(0,100,1); besseljzero(0,1000,1); besseljzero(0,10000,1) 311.8018681873704508125112 3139.236339643802482833973 31413.57032947022399485808 Zeros of functions with large order:: >>> besseljzero(50,1) 57.11689916011917411936228 >>> besseljzero(50,2) 62.80769876483536093435393 >>> besseljzero(50,100) 388.6936600656058834640981 >>> besseljzero(50,1,1) 52.99764038731665010944037 >>> besseljzero(50,2,1) 60.02631933279942589882363 >>> besseljzero(50,100,1) 387.1083151608726181086283 Zeros of functions with fractional order:: >>> besseljzero(0.5,1); besseljzero(1.5,1); besseljzero(2.25,4) 3.141592653589793238462643 4.493409457909064175307881 15.15657692957458622921634 Both `J_{\nu}(z)` and `J'_{\nu}(z)` can be expressed as infinite products over their zeros:: >>> v,z = 2, mpf(1) >>> (z/2)**v/gamma(v+1) * \ ... nprod(lambda k: 1-(z/besseljzero(v,k))**2, [1,inf]) ... 0.1149034849319004804696469 >>> besselj(v,z) 0.1149034849319004804696469 >>> (z/2)**(v-1)/2/gamma(v) * \ ... nprod(lambda k: 1-(z/besseljzero(v,k,1))**2, [1,inf]) ... 0.2102436158811325550203884 >>> besselj(v,z,1) 0.2102436158811325550203884 rrrr>rHr;r r r besseljzerospr cCst|d||| S)a For a real order `\nu \ge 0` and a positive integer `m`, returns `y_{\nu,m}`, the `m`-th positive zero of the Bessel function of the second kind `Y_{\nu}(z)` (see :func:`~mpmath.bessely`). Alternatively, with *derivative=1*, gives the first positive zero `y'_{\nu,m}` of `Y'_{\nu}(z)`. The zeros are interlaced according to the inequalities .. math :: y_{\nu,k} < y'_{\nu,k} < y_{\nu,k+1} y_{\nu,1} < y_{\nu+1,2} < y_{\nu,2} < y_{\nu+1,2} < y_{\nu,3} < \cdots **Examples** Initial zeros of the Bessel functions `Y_0(z), Y_1(z), Y_2(z)`:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> besselyzero(0,1); besselyzero(0,2); besselyzero(0,3) 0.8935769662791675215848871 3.957678419314857868375677 7.086051060301772697623625 >>> besselyzero(1,1); besselyzero(1,2); besselyzero(1,3) 2.197141326031017035149034 5.429681040794135132772005 8.596005868331168926429606 >>> besselyzero(2,1); besselyzero(2,2); besselyzero(2,3) 3.384241767149593472701426 6.793807513268267538291167 10.02347797936003797850539 Initial zeros of `Y'_0(z), Y'_1(z), Y'_2(z)`:: >>> besselyzero(0,1,1); besselyzero(0,2,1); besselyzero(0,3,1) 2.197141326031017035149034 5.429681040794135132772005 8.596005868331168926429606 >>> besselyzero(1,1,1); besselyzero(1,2,1); besselyzero(1,3,1) 3.683022856585177699898967 6.941499953654175655751944 10.12340465543661307978775 >>> besselyzero(2,1,1); besselyzero(2,2,1); besselyzero(2,3,1) 5.002582931446063945200176 8.350724701413079526349714 11.57419546521764654624265 Zeros with large index:: >>> besselyzero(0,100); besselyzero(0,1000); besselyzero(0,10000) 311.8034717601871549333419 3139.236498918198006794026 31413.57034538691205229188 >>> besselyzero(5,100); besselyzero(5,1000); besselyzero(5,10000) 319.6183338562782156235062 3147.086508524556404473186 31421.42392920214673402828 >>> besselyzero(0,100,1); besselyzero(0,1000,1); besselyzero(0,10000,1) 313.3726705426359345050449 3140.807136030340213610065 31415.14112579761578220175 Zeros of functions with large order:: >>> besselyzero(50,1) 53.50285882040036394680237 >>> besselyzero(50,2) 60.11244442774058114686022 >>> besselyzero(50,100) 387.1096509824943957706835 >>> besselyzero(50,1,1) 56.96290427516751320063605 >>> besselyzero(50,2,1) 62.74888166945933944036623 >>> besselyzero(50,100,1) 388.6923300548309258355475 Zeros of functions with fractional order:: >>> besselyzero(0.5,1); besselyzero(1.5,1); besselyzero(2.25,4) 1.570796326794896619231322 2.798386045783887136720249 13.56721208770735123376018 rr r r r r besselyzerosYr N)r)r)r)r)r)F)r)rF)rT)rT)r)r)2Z functionsrrr r rrBrKrSrXrYr[r^r\rkrlr{rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr r r r r r s   B # #                       [ K   +     "'5 q