a d?d@dAdBdCdDdEfZ5e6e5dFZ7dGZ8dHZ9dIdJZ:dKdLZ;ee:e;ZdQdRZ?dSdTZ@gdUZAdVdWZBdXdYZCeeBeCZDgdZddd[ZEgd\ddd[ZFd]d^ZGd_d`ZHdadbZIdcddZJdedfZKdgdhZLgdiZMdjZNd{dldmZOd|dndoZPd}dpdqZQdrdsZRgdtZSgduddd[ZTgdvddd[ZUgdwZVgdxZWdydzZXdS)~z This module complements the math and cmath builtin modules by providing fast machine precision versions of some additional functions (gamma, ...) and wrapping math/cmath functions so that they can be called with either real or complex arguments. Ng-DT! @giW @g;f?gw@gw?g9B.?gUk@gox?g]O?g m.f__name__rr rrr r _mathfun_reals rcsfdd}j|_|S)Nc sJt|tur|Szt|WSttfyDt|YS0dSr)rrrrrr r rrr+s  z_mathfun..frrrr r_mathfun*srcsfdd}j|_|S)Nc sBzdd|DWSttfy<dd|DYS0dS)Ncss|]}t|VqdSr)r.0r rrr 8z(_mathfun_n..f..css|]}t|VqdSrrrrrrr:r)rr)argsr r rrr6sz_mathfun_n..frrrr r _mathfun_n5srgcCs|dkrtdt|SNzmath domain error)rmathlogr rrrmath_logBsr#cCs|dkrtdt|Sr)rr sqrtr"rrr math_sqrtFsr%cCs t||Srr)r yrrrNrr'cCstt|jt|jSr)rr floorrealimagzrrrr'`rcCstt|jt|jSr)rr ceilr)r*r+rrrr'brcCst|t|fSr)r cossinr"rrrr'ercCst|t|fSr)cmathr.r/r+rrrr'frcCs|dSNgUUUUUU?rr"rrrr'hrcCs|dSr1rr+rrrr'hrc Cs>d|}zt||WSttfy8t||YS0dSN?)rrrrr nrrrrnthrootjs r7cCs|dkrt|  St|d\}}|t9}|d;}|dkrDt|S|dkrVt|S|dkrjt| S|dkr~t| SdSNr?) _sinpi_realdivmodpir r/r.r4rrrr>qs r>cCs||dkr| }t|d\}}|t9}|d;}|dkr>t|S|dkrRt| S|dkrft| S|dkrxt|SdSr8)r?r@r r.r/r4rrr _cospi_real|srAcCs|jdkrt|  St|jd\}}tt||j}|d;}|dkrPt|S|dkrbt|S|dkrvt| S|dkrt| SdSr8) r)_sinpi_complexr?r@rr*r0r/r.r,r5r6rrrrBs  rBcCs|jdkr| }t|jd\}}tt||j}|d;}|dkrJt|S|dkr^t| S|dkrrt| S|dkrt|SdSr8)r)r?r@rr*r0r.r/rCrrr_cospi_complexs rDcCsTzt|t|WStyNt|jdkr4YdSt|jdkrHYdSYn0dS)N ?)sinpicospi OverflowErrorrr*r"rrrtanpis rKcCsTzt|t|WStyNt|jdkr4YdSt|jdkrHYdSYn0dS)NrErGrF)rIrHrJrr*r"rrrcotpis rLggg?P5>cCst|}||kr0|dkr td|tkr0t|S|dkrPtt|td|S|d8}td}tdt dD]}|t|||7}qn|t d}d||dt | |SdS)Nrzgamma function poler9r;r3r<' @) intZeroDivisionError_max_exact_gamma _exact_gammar@r> _gamma_real _lanczos_prange _lanczos_gr exp)r _intxr6itrrrrUs rUcCs|jstt|jS|jdkr6tt|td|S|d8}td}tdt dD]}|t|||7}qT|t d}d||dt | |SdS)Nr9r;r3rr<rP) r*rrUr)r@rB_gamma_complexrVrWrXr0rY)r r6r[r\rrrr]s  r]cCs.zdt|WSty(|dYS0dS)Nr3r)gammarRr"rrrrgammas r_cCs t|dSr2)r^r"rrr factorialsr`cCs(t|turtd|St|j|jS)Nr)rrr atan2r*r)r"rrrargs  rbc CsRt|ttfvrd|}tt|}nd}|dkr`|d|8}|d7}qB|d}|}tD]$}|||8}|dkrq||9}qp|t|d|S) Nrzpolygamma poler9r3r$@#B ;)rQrRr@rL _psi_coeffr#)r rZrnx2r\crrr _digamma_real;s&    rucCs|jstt|jS|jdkr4d|}tt|}nd}t|dkrZ|d|8}|d7}q8|d}|}tD](}|||8}t|dkrq||9}qj|t |d|S)Nr9r3rrorprq) r*rrur)r@rLrgrrr0r!)r rnrsr\rtrrr_digamma_complexQs"       rv) gS?g}^.@gCGb@g߸G?g1R5?gߵy?gLy~F?goO$y?gŕX3=?rc) r3ga o @ghr@g-@gF@gv#o?g,?gE㦰u?g󟿑?gan$hI?cCs*|d}|ddD]}|||}q|S)Nrr;r)Zcoeffsr rmrtrrr_polyvalsrwcCsz||}|}}d}t|dkrr|||9}||||d8}|d7}|||9}||||d7}|d7}qd|S)Nr;FFg ?KX?)g^> ǿr)g\S;S?QSβ?)g\S;Sr)g|?M _6?)g|ꢿrgD@Fc Csd|}d}}d}|||9}||7}t|dkr6q@|d7}q|t||}|rt|turn|j}|j}n|}d}|dkr|dkr|td7}n8t|tur|jdkr|td7}|jdkr|td8}|S)Nr3r;gؗҜ             rc Csvt|}t|ttfvrLzt|}t}Wn"ttfyJt|}t}Yn0|turf|jsft|jd}t| dd S)NrT)r)rrrrrr*r)r)r,rrrre1Ls  r)grgSbQ?rgH*2Q?g|ʎA?gE G?g92"?gcj?g9?g[:?gH1?g,[ ?g?g+9@?g ?gL?g9?gNe?ga;?g?g֕?g1@?g ?gZ>?g?g?) gA gpgp6g|Upg%gE3+Ӿgf{g'( gѲG]) r3g/" =g{%gh{srg9D"glXxkվg%ygծ*geuy-) g]u:=gh#%.Jg`5'>>g^ҹgU3 gPF; ?g2Mg=Ƨ_ΚcgviT{?g\?gH$?) ggI ;g.k^>gPNg/wogS' ?g7z2Mg)cg?g|ɐ?r9c Cst|ttfstz t|}WnXttfyrz&t|}|jsPtt|jWYSWnttfyhYn0t Yn0|dkrtd|dkrdd| d| St|}||kr|dkrt |S|dsd S|d krd|t |dt d |t d|td|S|dkrJ|dkr8tt||dStt||dStt|tt|}dd| d| d | |S) z. Riemann zeta function, real argument r;z zeta(1) poler3rMg@rr<rr9g@) isinstancerrQrrrr*zetar)NotImplementedError _zeta_intr@r>rUrw_zeta_0_zeta_1_zeta_P_zeta_Q)rnr5r,rrrrs:   8  r)F)F)F)Y__doc__operatorr r0r@eZsqrt2Zsqrt5phiZln2Zln10rcatalanZkhinchinZaperyrirrrr!r#r%rrr$powrYr.r/tanacosasinatancoshsinhtanhr(r-Zcos_sinZcbrtr7r>rArBrDrIrHrKrLrZNINFZNANZEPSrTlenrSrXrVrUr]r^r_r`rbrjrrrurvZdigammarzr{rwryr|r~rrrrrrrrrrrrrrrrrrs                 7      ,   %