~9\c@ sddlmZmZddlmZddlmZmZddlm Z ddl m Z m Z ddl mZmZddlmZmZmZddlmZdd lmZdd lmZdd lmZmZdd lmZmZdd l m!Z!ddl"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*ddl+m,Z,m-Z-m.Z.ddl/m0Z0ddl1m2Z2ddl3m4Z4de fdYZ5dZ6ddZ7de5fdYZ8de5fdYZ9de5fdYZ:de5fdYZ;d e5fd!YZ<d"e<fd#YZ=d$e<fd%YZ>d&e fd'YZ?d(e fd)YZ@d*e@fd+YZAd,e@fd-YZBd.e@fd/YZCd0e@fd1YZDd2e@fd3YZEd4e@fd5YZFd6e@fd7YZGd8S(9i(tprint_functiontdivision(tAdd(tsympifytcacheit(trange(tFunctiontArgumentIndexError(t fuzzy_nottfuzzy_or(tigcdextRationaltpi(tNe(tS(tSymbol(t factorialtRisingFactorial(tlogtexp(tfloor(tacothtasinhtatanhtcoshtcothtHyperbolicFunctiontsinhttanh(tsqrttMintMax(t Piecewise(t FiniteSet(tnumbered_symbolstTrigonometricFunctioncB sJeZdZeZdZdZedZedZddZ RS(s(Base class for trigonometric functions. cC s\|j|j}|j|jkrQ|jdjrXt|jdjrXtSn|jSdS(Ni(tfunctargst is_rationalRtis_zerotFalse(tselfts((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_is_rational s &cC s|j|j}|j|jkr}t|jdjrN|jdjrNtSt|jd}|dk r|jrt Sn|jSdS(Ni( R$R%RR't is_algebraicR(t _pi_coefftNoneR&tTrue(R)R*tpi_coeff((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_is_algebraic(s&cK s*|jd||\}}||tjS(Ntdeep(t as_real_imagRt ImaginaryUnit(R)R2thintstre_parttim_part((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_expand_complex3scK s|jdjrW|r@t|d<|jdj||tjfS|jdtjfSn|r|jdj||j\}}n|jdj\}}||fS(Nitcomplex(R%tis_realR(texpandRtZeroR3(R)R2R5tretim((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt _as_real_imag7s  (cC s|jd}|dkr/t|jd}n|j|sEtjS||krU|S||jkr|jr|j|\}}||kr|t |Sn|j r|j|\}}|j|dt \}}||kr|t |Sqnt ddS(Nitas_Adds%Use the periodicity function instead.( R%R.ttuplet free_symbolsthasRR<tis_Multas_independenttabstis_AddR(tNotImplementedError(R)tgeneral_periodtsymboltftgthta((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_periodDs$       N( t__name__t __module__t__doc__R/t unbranchedR+R1R8R?R.RO(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR#s   cC sxztj|D]\}|tjkr2tj}Pq|jr|j\}}|tjkrl|jrlPqlqqW|tjfS|tj tj}|tj|}|||fS(s Split ARG into two parts, a "rest" and a multiple of pi/2. This assumes ARG to be an Add. The multiple of pi returned in the second position is always a Rational. Examples ======== >>> from sympy.functions.elementary.trigonometric import _peeloff_pi as peel >>> from sympy import pi >>> from sympy.abc import x, y >>> peel(x + pi/2) (x, pi/2) >>> peel(x + 2*pi/3 + pi*y) (x + pi*y + pi/6, pi/2) ( Rt make_argsRtPitOneRDt as_two_termst is_RationalR<tHalf(targRNtKtptm1tm2((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt _peeloff_pi^s    ic C srt|}|tjkr"tjS|s/tjS|jrn|jtj}|rn|j\}}|jrt |d}|dkrt t t |dj  }d|}||}t |} | |krt| |}||}qqtt |}||}n|jrg|d} | dkr6|S| s\|jdk rRtjStdS| |Sn|SndS(s  When arg is a Number times pi (e.g. 3*pi/2) then return the Number normalized to be in the range [0, 2], else None. When an even multiple of pi is encountered, if it is multiplying something with known parity then the multiple is returned as 0 otherwise as 2. Examples ======== >>> from sympy.functions.elementary.trigonometric import _pi_coeff as coeff >>> from sympy import pi, Dummy >>> from sympy.abc import x, y >>> coeff(3*x*pi) 3*x >>> coeff(11*pi/7) 11/7 >>> coeff(-11*pi/7) 3/7 >>> coeff(4*pi) 0 >>> coeff(5*pi) 1 >>> coeff(5.0*pi) 1 >>> coeff(5.5*pi) 3/2 >>> coeff(2 + pi) >>> coeff(2*Dummy(integer=True)*pi) 2 >>> coeff(2*Dummy(even=True)*pi) 0 iiiN(RRRURVR<RDtcoefft as_coeff_Multis_FloatRFtinttroundRtevalfR t is_integertis_evenR.( RZtcyclestcxtctxRKR\tmtcmtitc2((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR-s>$    "          tsincB seZdZddZddZedZee dZ dZ dZ dZ d Zd Zd Zd Zd ZdZdZdZdZedZdZdZdZdZRS(s The sine function. Returns the sine of x (measured in radians). Notes ===== This function will evaluate automatically in the case x/pi is some rational number [4]_. For example, if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6. Examples ======== >>> from sympy import sin, pi >>> from sympy.abc import x >>> sin(x**2).diff(x) 2*x*cos(x**2) >>> sin(1).diff(x) 0 >>> sin(pi) 0 >>> sin(pi/2) 1 >>> sin(pi/6) 1/2 >>> sin(pi/12) -sqrt(2)/4 + sqrt(6)/4 See Also ======== csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Sin .. [4] http://mathworld.wolfram.com/TrigonometryAngles.html cC s|jdt|S(Ni(ROR (R)RJ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pytperiodsicC s0|dkrt|jdSt||dS(Nii(tcosR%R(R)targindex((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pytfdiffs cC sddlm}ddlm}|jr|tjkr?tjS|tjkrUtjS|tjkss|tj kr|ddSn|tj krtjSt ||r|j |j }}t|dtj}|tj k r||dtj}n|tjk r ||dtj}n|||jttjddtjdtjk r|||jtdtjddtjdtjk r|ddS|||jttjddtjdtjk r|tt|t|dS|||jtdtjdd tjdtjk ri|dtt|t|S|tt|t|tt|t|Snt ||r|j|S|jr||  S|jtj}|dk rtjt|St|}|dk ra|jr.tjSd|jro|jrKtjS|jtkrotj |tj!Sn|j"s|tj} | |kr|| SdS|j"ra|d} | dkr|| dtj Sd| dkr|d| tjS|t#dddtj} t$| } t | t$s6| S|tj|krZ||tjSdSn|j%rt&|\} } | rt| t$| t$| t| Snt |t'r|j(d St |t)r|j(d } | t*d| dSt |t+r1|j(\} } | t*| d| dSt |t,r_|j(d } t*d| dSt |t-r|j(d } dt*dd| d| St |t.r|j(d } d| St |t/r|j(d } t*dd| dSdS( Ni(t AccumBounds(tSetExpriiiiiii(0tsympy.calculusRutsympy.sets.setexprRvt is_NumberRtNaNR<tInfinitytNegativeInfinitytComplexInfinityt isinstancetmintmaxRRUt intersectionR!tEmptySetRRpRt _eval_functcould_extract_minus_signtas_coefficientR4R.RR-RfRgR(t NegativeOneRYRXR RrRGR_tasinR%tatanRtatan2tacostacottacsctasec(tclsRZRuRvRRtdti_coeffR0tnargRktresultRlty((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pytevals 0    0 "4 "                  +      cG s|dks|ddkr#tjSt|}t|dkrd|d}| |d||dSd|d||t|SdS(Niiiii(RR<RtlenR(tnRktprevious_termsR\((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt taylor_termns  cK sntj}t|ts't|trI|j|jdjt}nt||t| |d|S(Nii( RR4R~R#RR$R%trewriteR(R)RZtkwargstI((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_exp|s "cK sJt|trFtj}|jd}||| d|||dSdS(Nii(R~RRR4R%(R)RZRRRk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_Pows  cK st|tjddtS(Nitevaluate(RrRRUR((R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_cosscK s'ttj|}d|d|dS(Nii(ttanRRY(R)RZRttan_half((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_tanscK st|t|t|S(N(RpRr(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_sincosscK s'ttj|}d|d|dS(Nii(tcotRRY(R)RZRtcot_half((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_cotscK s|jtjtS(N(RRrtpow(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_powscK s|jtjtS(N(RRrR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_sqrtscK sdt|S(Ni(tcsc(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_cscscK sdt|tjddtS(NiiR(tsecRRUR((R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_secscK s|t|S(N(tsinc(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_sincscC s|j|jdjS(Ni(R$R%t conjugate(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_conjugatescK sE|jd||\}}t|t|t|t|fS(NR2(R?RpRRrR(R)R2R5R=R>((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR3scK sddlm}ddlm}m}|jd}d}|jr|j\}}t |dt j }t |dt j } t |dt j } t |dt j } || | | S|j dt\} }| jrQ| jrd| dd|| t |S|d| ddt ||| dt |d t Snt|} | dk r| jr|jtSnt |S( Ni(t expand_mul(t chebyshevtt chebyshevuiRtrationaliiR2(tsympyRt#sympy.functions.special.polynomialsRRR%R.RGRWRpR(t_eval_expand_trigRrRaR/t is_Integertis_oddR-RXRR(R)R5RRRRZRkRtsxtsyRitcyRR0((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs,    #"   cC sbddlm}|jdj|}||jkrQ|d|j|rQ|S|j|SdS(Ni(tOrderii(RRR%tas_leading_termRBtcontainsR$(R)RkRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_as_leading_terms 'cC s|jdjrtSdS(Ni(R%R:R/(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt _eval_is_realscC s|jd}|jrtSdS(Ni(R%R:R/(R)RZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_is_finites  N(RPRQRRR.RqRtt classmethodRt staticmethodRRRRRRRRRRRRRRR/R3RRRR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRps.-  o               RrcB seZdZddZddZedZee dZ dZ dZ dZ d Zd Zd Zd Zd ZdZdZdZedZdZdZdZdZRS(s The cosine function. 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dZ dZ dZ d Zed Zd Zd Zd ZdZdZdZdZdZdZdZdZdZdZRS(s The tangent function. Returns the tangent of x (measured in radians). Notes ===== See :func:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import tan, pi >>> from sympy.abc import x >>> tan(x**2).diff(x) 2*x*(tan(x**2)**2 + 1) >>> tan(1).diff(x) 0 >>> tan(pi/8).expand() -1 + sqrt(2) See Also ======== sin, csc, cos, sec, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Tan cC s|jt|S(N(ROR (R)RJ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRqsicC s.|dkrtj|dSt||dS(Nii(RRVR(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRts cC stS(s7 Returns the inverse of this function. 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Returns the cotangent of x (measured in radians). Notes ===== See :func:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import cot, pi >>> from sympy.abc import x >>> cot(x**2).diff(x) 2*x*(-cot(x**2)**2 - 1) >>> cot(1).diff(x) 0 >>> cot(pi/12) sqrt(3) + 2 See Also ======== sin, csc, cos, sec, tan asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Cot cC s|jt|S(N(ROR (R)RJ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRqsicC s.|dkrtj|dSt||dS(Nii(RRR(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRts cC stS(s7 Returns the inverse of this function. (R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscC sddlm}|jrH|tjkr/tjS|tjkrHtjSn|tjkr^tjSt||rt|tj d S|j r||  S|j tj }|dk rtj t|St|d}|dk r|jrtjS|js&|tj }||kr"||SdS|jr|jdkr|jd r|tj d}t|t|tj d}}t|t rt|t rd||Snidd6dd 6dd 6dd 6dd6dd6dd6dd6}|j} |j| } | |kr|| tj || d|| tj || d} } d| ksgd| krkdSd| | | | S|tjdtjtj }t|t|tj d}}t|t rt|t r|dkrtjS||S||kr||Sqn|jrpt|\} }|rpt|}|tjkr_t| St|  Sqpnt|tr|jdSt|tr|jd} d| St|tr|j\}} | |St|tr|jd} td| d| St|tr8|jd} | td| dSt|t rn|jd} tdd| d| St|t!r|jd} dtdd| d| SdS(Ni(Ruiiiii iiiii iiii(i<ixi(ii(ii(ii(ii (ii(ii (ii(i(i<("RRuRyRRzR<R}R~RRURRR4R.RR-RfRXRRrR\RYRGR_RRR%RRRRRRR(RRZRuRR0RRRRRR\RRRkRltcotmR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs            $    ?$           cG sddlm}|dkr*dt|S|dksF|ddkrMtjSt|}||d}t|d}d|ddd|d||||SdS(Ni(Riii(RRRRR<R(RRkRRRR ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyROs  cC ss|jdj|dtj}|rQ|jrQ|jtj|d|d|S|jtj|d|d|S(NiRR!( R%R"RRURRRrR#R(R)RkRR!Rn((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR#_s "cC s|j|jdjS(Ni(R$R%R(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRescK s|jd||\}}|rgtd|td|}td| |td| |fS|j|tjfSdS(NR2i(R?RrRRpRR$RR<(R)R2R5R=R>R$((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR3hs (cK stj}t|ts't|trI|j|jdjt}nt| |t||}}|||||S(Ni( RR4R~R#RR$R%RR(R)RZRRR/R0((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRps  ""cK sPt|trLtj}|jd}| || |||| ||SdS(Ni(R~RRR4R%(R)RZRRRk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRws  cK s td|dt|dS(Ni(Rp(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR}scK s%t|t|tjddtS(NiR(RrRRUR((R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK st|t|S(N(RrRp(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK sdt|S(Ni(R(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s2t|j|}t|j|}||S(N(RrRRp(R)RZRR2R1((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s2t|j|}t|j|}||S(N(RrRRp(R)RZRR4R3((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s/|jtjt}|jtr+dS|S(N(RRrRRCR.(R)RZRR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s/|jtjt}|jtr+dS|S(N(RRrRRCR.(R)RZRR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscC sfddlm}|jdj|}||jkrU|d|j|rUd|S|j|SdS(Ni(Rii(RRR%RRBRR$(R)RkRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs 'cC s|jdjS(Ni(R%R:(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK sddlm}m}|jd}d}|jrGddlm}t|j}g}x6|jD]+}t|dt j } |j | qaWt d} gt |D]} t| ^q} ddg} xOt |ddD];} | || dc|| | d|| dd7R=i(R%RR&iiiRR'R((RR>R=R%R.RGR%RRR(RR R"RR)RR*RRaR/RRR4RR;(R)R5R>R=RZRkR%RtCXRiR-RnR&R\R`RRRR.((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs.   % 9( ,cC s|jd}|jrtSdS(Ni(R%R5R/(R)RZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs  cC s\||kr|S|jd}|j||}||krR|tjjrRtjSt|S(Ni(R%RRRURfR}R(R)toldtnewRZtargnew((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt _eval_subss  N(RPRQRRR.RqRtRRRRRRR#RR/R3RRRRRRRRRRRRRRR;(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs2#   i                tReciprocalTrigonometricFunctioncB seZdZdZdZdZedZdZ dZ dZ dZ ddZ dZd Zd Zd Zd Zd ZdZdZedZdZdZdZdZdZRS(s@Base class for reciprocal functions of trigonometric functions. cC s|jr8|jr || S|jr8||  Snt|}|dk rd|j r|jr|j}|jd|}||kr|dt j }|| Sd||krd|t j }|jr||S|jr|| Sqnt |dr#|j |kr#|j dS|jj|}|dkrE|Std|| fDrsd|jtStd|| fDrd|jtSd|SdS(NiiRics s|]}t|tVqdS(N(R~Rr(t.0Rn((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pys scs s|]}t|tVqdS(N(R~Rp(R=Rn((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pys s(Rt_is_event_is_oddR-R.RfRXRR\RRUthasattrRR%t_reciprocal_ofRtanyRRR(RRZR0RR\Rtt((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs<             !  cO s,|j|jd}t||||S(Ni(RAR%tgetattr(R)t method_nameR%Rto((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_call_reciprocalscO s-|j|||}|dk r)d|S|S(Ni(RGR.(R)RER%RRC((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_calculate_reciprocal scC s?|j||}|dk r;||j|kr;d|SdS(Ni(RGR.RA(R)RERZRC((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_rewrite_reciprocals!cC s#|jd}|j|j|S(Ni(R%RARq(R)RJRK((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyROs icC s|jd| |dS(NRti(RH(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRtscK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR"scK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR%scK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR(scK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR+scK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR.scK s|jd|S(NR(RI(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR1scC s|j|jdjS(Ni(R$R%R(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR4scK s$d|j|jdj||S(Nii(RAR%R3(R)R2R5((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR37scK s|jd|S(NR(RH(R)R5((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR;scC s|j|jdjS(Ni(RAR%R(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR>scC s!d|j|jdj|S(Nii(RAR%R(R)Rk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRAscC sd|j|jdjS(Nii(RAR%t is_finite(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRDscC s'd|j|jdj|||S(Nii(RAR%R#(R)RkRR!((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR#GsN(RPRQRRR.RAR>R?RRRGRHRIRORtRRRRRRRRR/R3RRRRR#(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR<s0%                  RcB s}eZdZeZeZd dZdZ dZ dZ dZ dZ dZdd Zeed ZRS( s{ The secant function. Returns the secant of x (measured in radians). Notes ===== See :func:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import sec >>> from sympy.abc import x >>> sec(x**2).diff(x) 2*x*tan(x**2)*sec(x**2) >>> sec(1).diff(x) 0 See Also ======== sin, csc, cos, tan, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Sec cC s |j|S(N(RO(R)RJ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRqqscK s$t|dd}|d|dS(Nii(R(R)RZRt cot_half_sq((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRtscK sdt|S(Ni(Rr(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRxscK st|t|t|S(N(RpRr(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR{scK sdt|j|S(Ni(RrR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR~scK sdt|j|S(Ni(RrR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK sttd|dtS(NiR(RR R((R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRsicC sA|dkr.t|jdt|jdSt||dS(Nii(RR%RR(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRts "cG s}ddlm}|dks,|ddkr3tjSt|}|d}d||d|td||d|SdS(Ni(teuleriii(t%sympy.functions.combinatorial.numbersRLRR<RR(RRkRRLtk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs   N(RPRQRRRrRAR/R>R.RqRRRRRRRtRRR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRKs!        RcB s}eZdZeZeZd dZdZ dZ dZ dZ dZ dZdd Zeed ZRS( s The cosecant function. Returns the cosecant of x (measured in radians). Notes ===== See :func:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import csc >>> from sympy.abc import x >>> csc(x**2).diff(x) -2*x*cot(x**2)*csc(x**2) >>> csc(1).diff(x) 0 See Also ======== sin, cos, sec, tan, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Csc cC s |j|S(N(RO(R)RJ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRqscK sdt|S(Ni(Rp(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK st|t|t|S(N(RrRp(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s$t|d}d|dd|S(Nii(R(R)RZRR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK sdt|j|S(Ni(RpR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK sttd|dtS(NiR(RR R((R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK sdt|j|S(Ni(RpR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRsicC sB|dkr/t|jd t|jdSt||dS(Nii(RR%RR(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRts #cG sddlm}|dkr*dt|S|dksF|ddkrMtjSt|}|dd}d|dddd|dd|d||d|dtd|SdS(Ni(Riii(RRRRR<R(RRkRRRN((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs  N(RPRQRRRpRAR/R?R.RqRRRRRRRtRRR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs!        RcB sDeZdZddZedZdZdZdZRS(sZRepresents unnormalized sinc function Examples ======== >>> from sympy import sinc, oo, jn, Product, Symbol >>> from sympy.abc import x >>> sinc(x) sinc(x) * Automated Evaluation >>> sinc(0) 1 >>> sinc(oo) 0 * Differentiation >>> sinc(x).diff() (x*cos(x) - sin(x))/x**2 * Series Expansion >>> sinc(x).series() 1 - x**2/6 + x**4/120 + O(x**6) * As zero'th order spherical Bessel Function >>> sinc(x).rewrite(jn) jn(0, x) References ========== .. [1] https://en.wikipedia.org/wiki/Sinc_function icC sL|jd}|dkr9|t|t||dSt||dS(Niii(R%RrRpR(R)RsRk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRts   cC s|jrtjS|jrR|tjtj gkr9tjS|tjkrRtjSn|tjkrhtjS|jr|| St |}|dk r|j rt |jrtjSqd|j rtj |tj|SndS(Ni(R'RRVRyR{R<RzR}RR-R.RfRRRY(RRZR0((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs$          cC s*|jd}t||j|||S(Ni(R%RpR#(R)RkRR!((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR#3s cK sddlm}|d|S(Ni(tjni(tsympy.functions.special.besselRO(R)RZRRO((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_jn7scK s,tt||t|dfdtfS(Nii(R RpR R/(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR;s( RPRQRRRtRRR#RQR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs &   tInverseTrigonometricFunctioncB seZdZRS(s/Base class for inverse trigonometric functions.(RPRQRR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRRDsRcB seZdZddZdZdZdZedZe e dZ dZ d Z d Zd Zd Zd ZdZdZddZRS(s The inverse sine function. Returns the arcsine of x in radians. Notes ===== asin(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1 and for some instances when the result is a rational multiple of pi (see the eval class method). Examples ======== >>> from sympy import asin, oo, pi >>> asin(1) pi/2 >>> asin(-1) -pi/2 See Also ======== sin, csc, cos, sec, tan, cot acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin icC s<|dkr)dtd|jddSt||dS(Niii(RR%R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRtns cC sF|j|j}|j|jkr;|jdjrBtSn|jSdS(Ni(R$R%R&R((R)R*((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR+ts cC s|jo|jdjS(Ni(RR%t is_positive(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_is_positive|scC s|jo|jdjS(Ni(RR%t is_negative(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_is_negativescC s|jr|tjkrtjS|tjkr<tjtjS|tjkrYtjtjS|tjkrotjS|tjkrtjdS|tj krtj dSn|tj krtj S|j r||  S|j ridt dd6dt d d6dt dd6dt d d6ddt d6ddt d6dt dt dd 6d t dt dd  6d tj6d tj 6d t dt dd6d t dt d d6dt ddd6ddt dd6dt ddt d6ddt dt d6tddt ddd6td dt dd d6}||krtj||Sn|jtj}|dk rtjt|St|trK|jd}|jrK|dt;}|tkrt|}n|tdkr%t|}n|t dkrDt |}n|Snt|tr|jd}|jrtdt|SndS(Niiiiiiiiiiiiii ii iiii(RyRRzR{R|R4R<RVRURR}Rt is_numberRRYRR.RR~RpR%t is_comparableR RrR(RRZt cst_tableRtang((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRsr       $         cG s|dks|ddkr#tjSt|}t|dkr{|dkr{|d}||dd||d|dS|dd}ttj|}t|}|||||SdS(Niiii(RR<RRRRYR(RRkRR\RNtRR ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs  $ cC sbddlm}|jdj|}||jkrQ|d|j|rQ|S|j|SdS(Ni(Rii(RRR%RRBRR$(R)RkRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs 'cK stjdt|S(Ni(RRUR(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_acosscK s$dt|dtd|dS(Nii(RR(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_atanscK s+tj ttj|td|dS(Nii(RR4RR(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_logscK s$dtdtd|d|S(Nii(RR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_acotscK stjdtd|S(Nii(RRUR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_asecscK std|S(Ni(R(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_acscscC s'|jd}|jo&dt|jS(Nii(R%R:RFtis_nonnegative(R)Rk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs cC stS(s7 Returns the inverse of this function. (Rp(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs(RPRQRRRtR+RTRVRRRRRRR\R]R^R_R`RaRR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRJs""    G       RcB seZdZddZdZedZeedZ dZ dZ dZ d Z d Zd Zd Zdd ZdZdZdZdZRS(s  The inverse cosine function. Returns the arc cosine of x (measured in radians). Notes ===== ``acos(x)`` will evaluate automatically in the cases ``oo``, ``-oo``, ``0``, ``1``, ``-1``. ``acos(zoo)`` evaluates to ``zoo`` (see note in :py:class`sympy.functions.elementary.trigonometric.asec`) Examples ======== >>> from sympy import acos, oo, pi >>> acos(1) 0 >>> acos(0) pi/2 >>> acos(oo) oo*I See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos icC s<|dkr)dtd|jddSt||dS(Niiii(RR%R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRt's cC sF|j|j}|j|jkr;|jdjrBtSn|jSdS(Ni(R$R%R&R((R)R*((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR+-s cC s8|jr|tjkrtjS|tjkr<tjtjS|tjkrYtjtjS|tjkrstjdS|tjkrtjS|tj krtjSn|tj krtj S|j ritjdtj 6dtjdtj 6tjdt dd6dtjdt d d6tjddt d6dtjddt d6tjdt dd6dtjdt d d6}||kr||Snt|tr|jd}|jr|dt;}|tkrdt|}n|Snt|tr4|jd}|jr4tdt|SndS( Niiiiiiii(RyRRzR{R4R|R<RURVRR}RWRYRR~RrR%RXR RpR(RRZRYRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR5sL            cG s|dkrtjdS|dks3|ddkr:tjSt|}t|dkr|dkr|d}||dd||d|dS|dd}ttj|}t|}| ||||SdS(Niiii(RRUR<RRRRYR(RRkRR\RNR[R ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRes    $ cC sbddlm}|jdj|}||jkrQ|d|j|rQ|S|j|SdS(Ni(Rii(RRR%RRBRR$(R)RkRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRws 'cC s'|jd}|jo&dt|jS(Nii(R%R:RFRb(R)Rk((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs cC s |jS(N(R(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_is_nonnegativescC s#|j|jdj|||S(Ni(R^R%R#(R)RkRR!((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR#scK s5tjdtjttj|td|dS(Nii(RRUR4RR(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR^scK stjdt|S(Ni(RRUR(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyt_eval_rewrite_as_asinscK sAttd|d|tjdd|td|dS(Nii(RRRRU(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR]scC stS(s7 Returns the inverse of this function. (Rr(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRscK s/tjddtdtd|d|S(Nii(RRURR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR_scK std|S(Ni(R(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR`scK stjdtd|S(Nii(RRUR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRascC sg|jd}|j|jdj}|jtkr<|S|jrc|djrc|djrc|SdS(Nii(R%R$RR:R(Rbtis_nonpositive(R)RR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs  #(RPRQRRRtR+RRRRRRRRcR#R^RdR]RR_R`RaR(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs$&  0          RcB seZdZddZdZdZdZedZe e dZ dZ d Z d Zd Zdd Zd ZdZdZdZdZRS(s The inverse tangent function. Returns the arc tangent of x (measured in radians). Notes ===== atan(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1. Examples ======== >>> from sympy import atan, oo, pi >>> atan(0) 0 >>> atan(1) pi/4 >>> atan(oo) pi/2 See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan icC s6|dkr#dd|jddSt||dS(Niii(R%R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRts cC sF|j|j}|j|jkr;|jdjrBtSn|jSdS(Ni(R$R%R&R((R)R*((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR+s cC s|jdjS(Ni(R%RS(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRTscC s|jdjS(Ni(R%Rb(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRcscC s-|jr|tjkrtjS|tjkr9tjdS|tjkrTtj dS|tjkrjtjS|tjkrtjdS|tjkrtj dSn|tj krddl m }|tj dtjdS|j r||  S|j rJidtdd6dtd d6ddtd6ddtd6dtd6d td 6td ddtd6td ddtd 6d tdd6d dtd6td dtd dtd 6td  dtd dtd  6d dtd6ddtd 6}||krJtj||Sn|jtj}|dk rytjt|St|tr|jd}|jr|t;}|tdkr|t8}n|Snt|tr)|jd}|jr)tdt|}|tdkr"|t8}n|SndS(Niii(Ruiiiiiiiii ii(RyRRzR{RUR|R<RVRR}RRuRRWRRR4R.RR~RR%RXR RR(RRZRuRYRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRsj        %'         cG sO|dks|ddkr#tjSt|}d|dd|||SdS(Niiii(RR<R(RRkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR& s cC sbddlm}|jdj|}||jkrQ|d|j|rQ|S|j|SdS(Ni(Rii(RRR%RRBRR$(R)RkRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR/ s 'cC s|jdjS(Ni(R%R:(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR8 scK sAtjdttdtj|ttdtj|S(Nii(RR4R(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR^; s$cC s|dtjkrBtjdtd|jdj|||S|dtjkrtj dtd|jdj|||Stt|j||||SdS(Niii( RR{RURR%R#R|tsupert _eval_aseries(R)Rtargs0RkR!((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRg? s /0cC stS(s7 Returns the inverse of this function. (R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRG scK s9t|d|tjdtdtd|dS(Nii(RRRUR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRdM scK s.t|d|tdtd|dS(Nii(RR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR\P scK std|S(Ni(R(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR_S scK s*t|d|ttd|dS(Nii(RR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR`V scK s5t|d|tjdttd|dS(Nii(RRRUR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRaY s(RPRQRRRtR+RTRcRRRRRRRR^RgRRdR\R_R`Ra(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRs$#    A        RcB seZdZddZdZdZdZdZedZ e e dZ d Z d Zd Zdd Zd ZdZdZdZdZRS(s The inverse cotangent function. Returns the arc cotangent of x (measured in radians). See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCot icC s6|dkr#dd|jddSt||dS(Niiii(R%R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRtq s cC sF|j|j}|j|jkr;|jdjrBtSn|jSdS(Ni(R$R%R&R((R)R*((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR+w s cC s|jdjS(Ni(R%Rb(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRT scC s|jdjS(Ni(R%RU(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRV scC s|jdjS(Ni(R%R:(R)((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR scC s(|jr|tjkrtjS|tjkr5tjS|tjkrKtjS|tjkretjdS|tjkrtjdS|tjkrtj dSn|tj krtjS|j r||  S|j rDidt dd6dt d d6ddt d6ddt d6dt d6dt d 6d dt d6d dt d 6td  ddt d6td dt dd6d t d dt d 6d t d dt d  6ddt d6ddt d 6tdd dt d6td d dt d 6}||krDtj||Sn|j tj}|dk rttj t|St|tr|jd}|jr|t;}|tdkr|t8}n|Snt|tr$|jd}|jr$tdt|}|tdkr|t8}n|SndS(Niiiiiiiiiii iii ii(RyRRzR{R<R|RURVRR}RRWRRR4R.RR~RR%RXR RR(RRZRYRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR sl                 cG sf|dkrtjdS|dks3|ddkr:tjSt|}d|dd|||SdS(Niiii(RRUR<R(RRkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR s    cC sbddlm}|jdj|}||jkrQ|d|j|rQ|S|j|SdS(Ni(Rii(RRR%RRBRR$(R)RkRRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR s 'cC s|dtjkrBtjdtd|jdj|||S|dtjkrdtjdtd|jdj|||Stt|j ||||SdS(Niiii( RR{RURR%R#R|RfRRg(R)RRhRkR!((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRg s /3cK s5tjdtdtj|tdtj|S(Nii(RR4R(R)RkR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR^ scC stS(s7 Returns the inverse of this function. (R(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR scK sI|td|dtjdtt|d t|d dS(Nii(RRRUR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRd scK s>|td|dtt|d t|d dS(Nii(RR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR\ scK std|S(Ni(R(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR] scK s6|td|dttd|d|dS(Nii(RR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR` scK sA|td|dtjdttd|d|dS(Nii(RRRUR(R)RZR((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRa s(RPRQRRRtR+RTRVRRRRRRRRgR^RRdR\R]R`Ra(((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR] s$     A       RcB s}eZdZedZddZddZdZdZdZ dZ d Z d Z d Z d ZRS( s The inverse secant function. Returns the arc secant of x (measured in radians). Notes ===== ``asec(x)`` will evaluate automatically in the cases ``oo``, ``-oo``, ``0``, ``1``, ``-1``. ``asec(x)`` has branch cut in the interval [-1, 1]. For complex arguments, it can be defined [4]_ as .. math:: sec^{-1}(z) = -i*(log(\sqrt{1 - z^2} + 1) / z) At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For negative branch cut, the limit .. math:: \lim_{z \to 0}-i*(log(-\sqrt{1 - z^2} + 1) / z) simplifies to :math:`-i*log(z/2 + O(z^3))` which ultimately evaluates to ``zoo``. As ``asex(x)`` = ``asec(1/x)``, a similar argument can be given for ``acos(x)``. Examples ======== >>> from sympy import asec, oo, pi >>> asec(1) 0 >>> asec(-1) pi See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec .. [4] http://reference.wolfram.com/language/ref/ArcSec.html cC s|jrtjS|jr^|tjkr/tjS|tjkrEtjS|tjkr^tjSn|tj tj tjgkrtjdSt |t r|j d}|jr|dt;}|tkrdt|}n|Snt |tr|j d}|jrtdt|SndS(Nii(R'RR}RyRzRVR<RRUR{R|R~RR%RXR RR(RRZRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR6 s,         icC sO|dkr<d|jddtdd|jddSt||dS(Niii(R%RR(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRtR s 0cC stS(s7 Returns the inverse of this function. 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Returns the arc cosecant of x (measured in radians). Notes ===== acsc(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1. Examples ======== >>> from sympy import acsc, oo, pi >>> acsc(1) pi/2 >>> acsc(-1) -pi/2 See Also ======== sin, csc, cos, sec, tan, cot asin, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc cC sI|jrW|tjkrtjS|tjkr9tjdS|tjkrWtj dSn|tjtjtjgkr|tj St |t r |j d}|j r |dt;}|tkrt|}n|tdkrt|}n|t dkrt |}n|Snt |trE|j d}|j rEtdt|SndS(Nii(RyRRzRVRURR{R|R}R<R~RR%RXR RR(RRZRZ((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyR s0         icC sO|dkr<d|jddtdd|jddSt||dS(Niiii(R%RR(R)Rs((sGlib/python2.7/site-packages/sympy/functions/elementary/trigonometric.pyRt s 0cC stS(s7 Returns the inverse of this function. 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RcB s\eZdZedZdZdZdZdZdZ dZ dZ RS( s The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking two arguments `y` and `x`. Signs of both `y` and `x` are considered to determine the appropriate quadrant of `\operatorname{atan}(y/x)`. The range is `(-\pi, \pi]`. The complete definition reads as follows: .. math:: \operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac y x\right) & \qquad x > 0 \\ \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\ \arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\ +\frac{\pi}{2} & \qquad y > 0 , x = 0 \\ -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases} Attention: Note the role reversal of both arguments. The `y`-coordinate is the first argument and the `x`-coordinate the second. Examples ======== Going counter-clock wise around the origin we find the following angles: >>> from sympy import atan2 >>> atan2(0, 1) 0 >>> atan2(1, 1) pi/4 >>> atan2(1, 0) pi/2 >>> atan2(1, -1) 3*pi/4 >>> atan2(0, -1) pi >>> atan2(-1, -1) -3*pi/4 >>> atan2(-1, 0) -pi/2 >>> atan2(-1, 1) -pi/4 which are all correct. Compare this to the results of the ordinary `\operatorname{atan}` function for the point `(x, y) = (-1, 1)` >>> from sympy import atan, S >>> atan(S(1) / -1) -pi/4 >>> atan2(1, -1) 3*pi/4 where only the `\operatorname{atan2}` function reurns what we expect. We can differentiate the function with respect to both arguments: >>> from sympy import diff >>> from sympy.abc import x, y >>> diff(atan2(y, x), x) -y/(x**2 + y**2) >>> diff(atan2(y, x), y) x/(x**2 + y**2) We can express the `\operatorname{atan2}` function in terms of complex logarithms: >>> from sympy import log >>> atan2(y, x).rewrite(log) -I*log((x + I*y)/sqrt(x**2 + y**2)) and in terms of `\operatorname(atan)`: >>> from sympy import atan >>> atan2(y, x).rewrite(atan) 2*atan(y/(x + sqrt(x**2 + y**2))) but note that this form is undefined on the negative real axis. See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, acot References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] https://en.wikipedia.org/wiki/Atan2 .. 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