a Zed?d@Z edAdBZ!edCdDZ"edEdFZ#edkdHdIZ$edJdKZ%edLdMZ&edNdOZ'edPdQZ(dRdSZ)d3dGl*Z*d3dGl+Z+dTdUZ,dVdWZ-edldXdYZ.edmdZd[Z/dnd\d]Z0ed^d_Z1ed`daZ2edbdcZ3edodddeZ4edpdfdgZ5dGS)q)xrangec@seZdZdZiZdZddZeddZddZ d d Z d d Z d dZ ddZ ddZddZddZddZddZddZddZdS) SpecialFunctionsa This class implements special functions using high-level code. Elementary and some other functions (e.g. gamma function, basecase hypergeometric series) are assumed to be predefined by the context as "builtins" or "low-level" functions. gP?c Cs.|j}|jD] }|j|\}}||||q |d|_|d|_|d|_|d|_|d|_|d|_ |d|_ |d|_ |d |_ |d |_ |d |_|d |_|d |_|d|_|d|_|d|_|d|_i|_|jdddddddd||j|_dS)N)rr)rr)r)r)r)rr )r)rr)r)r r)rr)rr)rr)r)r r )r rargconjrootpsizetaZfibZfac)phase conjugateZnthrootZ polygammaZhurwitzZ fibonacci factorial) __class__defined_functions _wrap_specfunZ_mpqZmpq_1Zmpq_0Zmpq_1_2Zmpq_3_2Zmpq_1_4Zmpq_1_16Zmpq_3_16Zmpq_5_2Zmpq_3_4Zmpq_7_4Zmpq_5_4Zmpq_1_3Zmpq_2_3Zmpq_4_3Zmpq_1_6Zmpq_5_6Zmpq_5_3Z_misc_const_cache_aliasesupdatememoizeZzetazeroZzetazero_memoized)selfclsnamefwrapr j/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/mpmath/functions/functions.py__init__s@                  zSpecialFunctions.__init__cCst|||dSN)setattr)rrrrr r r!r=szSpecialFunctions._wrap_specfuncCstdSr#NotImplementedErrorctxnzr r r!_besseljDzSpecialFunctions._besseljcCstdSr#r%r(r*r r r!_erfEr,zSpecialFunctions._erfcCstdSr#r%r-r r r!_erfcFr,zSpecialFunctions._erfccCstdSr#r%)r(r*ar r r!_gamma_upper_intGr,z!SpecialFunctions._gamma_upper_intcCstdSr#r%r'r r r! _expint_intHr,zSpecialFunctions._expint_intcCstdSr#r%r(sr r r!_zetaIr,zSpecialFunctions._zetacCstdSr#r%)r(r4r0r)Z derivativesZreflectr r r! _zetasum_fastJr,zSpecialFunctions._zetasum_fastcCstdSr#r%r-r r r!_eiKr,zSpecialFunctions._eicCstdSr#r%r-r r r!_e1Lr,zSpecialFunctions._e1cCstdSr#r%r-r r r!_ciMr,zSpecialFunctions._cicCstdSr#r%r-r r r!_siNr,zSpecialFunctions._sicCstdSr#r%r3r r r!_altzetaOr,zSpecialFunctions._altzetaN)__name__ __module__ __qualname____doc__rZ THETA_Q_LIMr" classmethodrr+r.r/r1r2r5r6r7r8r9r:r;r r r r!rs$+ rcCs|dftj|j<|S)NTrrr<rr r r! defun_wrappedQsrCcCs|dftj|j<|S)NFrArBr r r!defunUsrDcCstt|j||Sr#)r$rr<rBr r r! defun_staticYsrEcCs|j||Sr#)onetanr-r r r!cot]srHcCs|j||Sr#)rFcosr-r r r!sec`srJcCs|j||Sr#)rFsinr-r r r!csccsrLcCs|j||Sr#)rFtanhr-r r r!cothfsrNcCs|j||Sr#)rFcoshr-r r r!sechisrPcCs|j||Sr#)rFsinhr-r r r!cschlsrRcCs"|s|jdS||j|SdS)N?)piatanrFr-r r r!acotos rVcCs||j|Sr#)acosrFr-r r r!asecvsrXcCs||j|Sr#)asinrFr-r r r!acscysrZcCs"|s|jdS||j|SdS)Ny?)rTatanhrFr-r r r!acoth|s r\cCs||j|Sr#)acoshrFr-r r r!asechsr^cCs||j|Sr#)asinhrFr-r r r!acschsr`cCsH||}|r||r|S||r<|dkr4|jS|j S|t|S)Nr)convertisnanZ _is_real_typerFabsr(xr r r!signs  rfrcCs2|dkr||S||}||}|||SNr)Zagm1raZ_agm)r(r0br r r!agms    ricCs,||rd|S|s|dS|||Srg)isinfrKrdr r r!sincs  rkcCs2||rd|S|s|dS|||j|Srg)rjZsinpirTrdr r r!sincpis  rlcsBs jSj kr,ddSfdddS)NrSrcstdgSN)iterexpr rdr r!r,zexpm1..r)zeromagprecsum_accuratelyrdr rdr!expm1s rvcCsH|s |jS|||j kr,|d|dS||jd|d|jdS)NrSrrrt)rrrsrtlogZfaddrdr r r!log1ps ryc s|j}|j}|}||}|dkr,|S|sJrFdvrJ|rJ|S|}|}|} ||| |j kr| | ddS|fdddS)Ni)rrny?yrcstdgSrm)ror reyr r!rqr,zpowm1..r)rsrFZisintlnrtru) r(rer{rsrFwMx1ZmagyZlnxr rzr!powm1s  rcCsxt|}t|}||;}|s"|jSd||kr6|j Sd||krH|jSd|d|kr`|j S|d|||S)Nrrr)intrFjZexpjpimpf)r(kr)r r r!_rootof1s  rrcCst|}||}|r|d@rVd||dkrV||sV||dkrV|| | S|j}z2|jd7_|||d|||}W||_n||_0| S|||S)Nrrr )rraimrerrtrZ_nthroot)r(rer)rrtvr r r!rs 0rFcstjj}zPjd7_|r<fddtD}nfddtD}W|_n|_0dd|DS)Nrcs&g|]}|dkr|qS)rr.0rr(gcdr)r r! r,zunitroots..csg|]}|qSr rr)r(r)r r!rr,cSsg|] }| qSr r )rrer r r!rr,)Z_gcdrtrange)r(r)Z primitivertrr rr! unitrootssrcCs*||}||}||}|||Sr#)ra_re_imatan2)r(rerrr r r!r s   r cCst||Sr#)rcrardr r r!fabssrcCs||}t|dr|jS|S)Nreal)rahasattrrrdr r r!rs  rcCs ||}t|dr|jS|jS)Nimag)rarrrrrdr r r!rs  rcCs0||}z |WSty*|YS0dSr#)rarAttributeErrorrdr r r!r s    rcCs||||fSr#)rr r-r r r!polar(srcCs||j||Sr#)ZmpcZcos_sin)r(rphir r r!rect,srNcCs8|dur||S|jd}|j||d|j||dS)Nrw)r|rt)r(rerhwpr r r!rx0s  rxcCs ||dS)Nr)rxrdr r r!log107srcCs||||Sr#)ra)r(rer{r r r!fmod;srcCs ||jSr#Zdegreerdr r r!degrees?srcCs ||jSr#rrdr r r!radiansCsrcCsv|s|s |S|j|S||jkrD|dkr,|S|d||j|jS||jkrl| d|d|j|jS||S)Nrrr)ZninfinfrTrr|)r(r*rr r r!_lambertw_specialGs   rcCsd}t|dr8t|j}|j}|r.d|dk}t|}nt|}d}d}|sPd}t||}|dkrd|krzdkrnnd|krdkrnn|r|d krd d |d S|d krdd|dS|dkrdd|dS|dkrdd|dS|dkr2|dkr"dd|dSdd|dSd}|sJ||krJ|}|dkrtdd ||d!d"||S|d!kr|Sd#d$|S|s|dkrt|}t|}nt|}t|}n|dkrd}|s||krdkrnn|}|dkrP|d%krPd&|kr,dkrPnn dd ||d!d"||S|sd|krndkrnnt| }|t| S|dkr|s|dkrt|d'}nt|d(}t|}||||||d)d)|d)S)*Nrrrnggg@gg@?yx&1?p= ף?yh|?5?ʡEƿy?@g?y)\(?&1?y?L7A`y??yx&1?p= ףyh|?5?ʡE?y?gпy)\(?&1ʿy?L7A`?y?gy'1ZԿq= ףp?yM`"ry'1ZԿq= ףpyM`"?g2,6V׿gɿg4@rSg}tp?g?g333333?g?g333333y-DT! @y-DT!@r)rfloatrrcomplexmathrxcmath)r*rZ imag_signrer{rL1L2r r r!_lambertw_approx_hybridZs`      6      "0 "  rc s}d|kr dkr>nnd|krnn|dkrtddkr|dks|d kr|dks|dkrdkrfd d }| }j|7_d jd}j|8_d dd d d d }|dkr&| }j} t t d |D]Ɖvr fddt d D|<dd d |d dd|d dd<|} | | 7} | | kr| dfSd7q:j|d 7_| dfS|dks0|d kr>t |dfS|dkrx|d krbddfS } | } nz|d krЈsdkrdkrnn  } | | dfS dj|} | } | | | | | | d d | d dfS)z Return rough approximation for W_k(z) from an asymptotic series, sufficiently accurate for the Halley iteration to converge to the correct value. iiiirg,6V?g?rrncsdgSrm)rpr r-r r!rqr,z"_lambertw_series..rrc3s&|]}|d|VqdS)rNr )rr)lur r! r,z#_lambertw_series..rTFg,6V׿y@)rsrcrrurtsqrterrrrmaxfsumrr|rrT) r(r*rtolZmagzdeltaZ cancellationpr0r4Ztermrrr )r(rrr*r!_lambertw_seriessT 8    $T      8  rcCs||}t|}||s(t|||S|j}|jd||p@d7_|j}|d}t||||\}}|s |d}tdD]p} | |} || } | |} || | | ||| |||} || ||| |kr| }qq| }q| dkr | d|||_| S)Nrrr rdz1Lambert W iteration failed to converge for z = %s) rarZisnormalrrtrsrrrrpwarn)r(r*rrtrrr}doneZtwoiewZwewZwewzZwnr r r!lambertws0      ( rcCs||}|s(||r|St|dS||sP||sP||sP||rX||S|dkrd|S|dkrx||dS|dkr||St|||d||S)NrrrT)rarbtyperjrl_polyexprp)r(r)rer r r!bells   ( rcs fdd}j|ddS)Nc3s@rV}d}||V|d7}||}qdSrg)rl)trr(extrar)rer r!_termss z_polyexp.._termsr)Z check_step)ru)r(r)rerrr rr!rs rcCs||s(||s(||s(||r0||S|dkr@||S|dkrR||S|dkrh|||S|dkr||||dSt|||S)Nrrr)rjrbrvrpr)r(r4r*r r r!polyexps(rc Cst|}|dkrtd|j}|dkr*|S|dkr:||S|dkrJ||Sd}d}d}d}td|dD]l}||sh|||} ||| } | r|| | 9}qh| dkr||9}|d7}qh| dkrh||9}|d7}qh|r||kr|d9}n||9}||}|S)Nrzn cannot be negativerrrn)r ValueErrorrFrZmoebiusr) r(r)r*rZa_prodZb_prodZ num_zerosZ num_polesdr}rhr r r! cyclotomics@   rcCst|}|dkr|jS|ddkr@||d@dkr:|j S|jSdD]N}||sD||d}}|dkrt||\}}|r^|jSq^||SqD||r||S|dkrtd}t|d|d}|dkr|jS|||kr||r||S|d7}qdS) a Evaluates the von Mangoldt function `\Lambda(n) = \log p` if `n = p^k` a power of a prime, and `\Lambda(n) = 0` otherwise. **Examples** >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> [mangoldt(n) for n in range(-2,3)] [0.0, 0.0, 0.0, 0.0, 0.6931471805599453094172321] >>> mangoldt(6) 0.0 >>> mangoldt(7) 1.945910149055313305105353 >>> mangoldt(8) 0.6931471805599453094172321 >>> fsum(mangoldt(n) for n in range(101)) 94.04531122935739224600493 >>> fsum(mangoldt(n) for n in range(10001)) 10013.39669326311478372032 rrr) rr r l73Me'rrSN)rrrZln2divmodr|Zisprimer&)r(r)rqrrr r r!mangoldt;s6       rcCs.|t|t|}|r t|S||SdSr#)Z _stirling1rrr(r)rexactrr r r! stirling1wsrcCs.|t|t|}|r t|S||SdSr#)Z _stirling2rrrr r r! stirling2sr)r)r)F)N)r)r)F)F)F)6Z libmp.backendrobjectrrCrDrErHrJrLrNrPrRrVrXrZr\r^r`rfrirkrlrvryrrrrr rrrrrrrxrrrrrrrrrrrrrrrrrr r r r!s N                                  ?6    * ;