a d>> from sympy.functions.elementary.trigonometric import _peeloff_pi >>> from sympy import pi >>> from sympy.abc import x, y >>> _peeloff_pi(x + pi/2) (x, 1/2) >>> _peeloff_pi(x + 2*pi/3 + pi*y) (x + pi*y + pi/6, 1/2) F) rrDrZ make_argscoeffPir/appendHalf is_integeris_evenr)argr9Z rest_termsrSKm1m2r4r4r5 _peeloff_pi`s       rfc Cst|}|tjurtjS|s"tjS|jr |tj}|r|\}}|jrt |d}|dkrt t t |d  }d|}||}t |} | |krt| |}||}ntt |}||}|jr|d} | dkr|S| s|jdurtjStdS| |S|Sn|jrtjSdS)a6 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 >>> from sympy import pi, Dummy >>> from sympy.abc import x >>> _pi_coeff(3*x*pi) 3*x >>> _pi_coeff(11*pi/7) 11/7 >>> _pi_coeff(-11*pi/7) 3/7 >>> _pi_coeff(4*pi) 0 >>> _pi_coeff(5*pi) 1 >>> _pi_coeff(5.0*pi) 1 >>> _pi_coeff(5.5*pi) 3/2 >>> _pi_coeff(2 + pi) >>> _pi_coeff(2*Dummy(integer=True)*pi) 2 >>> _pi_coeff(2*Dummy(even=True)*pi) 0 rgrr[N)rrr]OnerDrKr\ as_coeff_MulZis_FloatrLintroundrZevalfrr`rarr0) rbZcyclescxcxrPpmcmic2r4r4r5r8sB%        r8c@seZdZdZd6ddZd7ddZedd Zee d d Z d8d dZ ddZ ddZ ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd9d(d)Zd*d+Zd:d,d-Zd.d/Zd0d1Zd2d3Zd4d5ZdS);sina The sine function. Returns the sine of x (measured in radians). Explanation =========== 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 .. 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Returns the cosine of x (measured in radians). Explanation =========== See :func:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import cos, pi >>> from sympy.abc import x >>> cos(x**2).diff(x) -2*x*sin(x**2) >>> cos(1).diff(x) 0 >>> cos(pi) -1 >>> cos(pi/2) 0 >>> cos(2*pi/3) -1/2 >>> cos(pi/12) sqrt(2)/4 + sqrt(6)/4 See Also ======== sin, csc, 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 .. 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[3] http://functions.wolfram.com/ElementaryFunctions/Tan NcCs |t|Srrvrwr4r4r5rxsz tan.periodrgcCs$|dkrtj|dSt||dSNrgr[)rrhrr{r4r4r5r}sz tan.fdiffcCstSz7 Returns the inverse of this function. rr{r4r4r5inversesz tan.inversec Csddlm}|jrL|tjur"tjS|jr.tjS|tjtjfvrL|tjtjS|tj ur\tjSt ||r|j |j }}t |tj}|tjur||tj}|tjur||tj}|||ttjdtjtddr|tjtjS|t|t|S|r||  S|tj}|dur6tjt|St|d}|dur|jrXtjS|js|tj}||kr|||SdS|jr|j} |j| } tddtddtddtdtddtddtddtdd} | dvr(d | | } | dkr d | } | |  S| | S|jds|tjd}t|t|tjd} }t | tst |ts|dkrtj Sd|| |Sd d d d ddddd}| |vr|| tj|| d|| tj|| d}}d||fvrdS||d||S|tjdtjtj}t|t|tjd} }t | tsxt |tsx| dkrptj S|| S||kr||S|jrt |\}}|rt|tj}|tj urt!| St|S|jrtjSt |t"r|j#dSt |t$r|j#\}}||St |t%r<|j#d}|td|dSt |t&rf|j#d}td|d|St |t'r|j#d}d|St |t(r|j#d}dtdd|d|St |t)r|j#d}tdd|d|SdS)Nrr~r[rrgr)rgr[rrrr r rrrr r r rrr)*rrrrrr0rDrrrYrrrrr]rr)rrrrr>r#r8r`rrror$rzr_rMrfrrr.rrrrrr)rrbrrrrrr9rrroZtable10rcresultsresultrrrrnrpZtanmrr4r4r5rs      (                 6                    ztan.evalcGsddlm}|dks |ddkr&tjSt|}|ddd|d}}||d}t|d}tj|||d||||SdS)Nr bernoullir[rg)%sympy.functions.combinatorial.numbersr^rrDrrr)rrnrr^rSrBFr4r4r5rs   ztan.taylor_termrcCsN|jd|ddtj}|r<|jr<|tj|||dStj||||dS)Nrr[rr) r.rrr]rrrzrrr2rnrrrrrr4r4r5rs ztan._eval_nseriescKsFt|trBtj}|jd}||| |||| ||SdSrrrr4r4r5rs  ztan._eval_rewrite_as_PowcCs||jdSrrrr4r4r5rsztan._eval_conjugateTcKsh|jfd|i|\}}|rTtd|td|}td||td||fS||tjfSdSNr<r[rHrzrrtr"r-rrDr2r<r?rFrGZdenomr4r4r5r=s  ztan.as_real_imagc sbddlm}m}|jd}d}|jrddlm}t|j}g}|jD]}t|dd } | | qDt dfddt |D} ddg} t |d D]2} | d | d || | d | d d 7<q| d| d  tt| |S|jrZ|jd d\} }| jrZ| d krZtj}tdd d}d ||| }|||| |t|fgSt|S)NrrGrFsymmetric_polyFrYcsg|] }tqSr4nextr"ZYgr4r5r&r'z)tan._eval_expand_trig..rgr[rrTrdummyreal)rrGrFr.rMsympy.polys.specialpolysrirrrr^r*rangerlistr2rKrirrr>rrC)r2r?rGrFrbrnrirZTXZtxrjrorrr\rUrrOPr4rmr5rs0     0   ztan._eval_expand_trigcKsZtj}t|ttfr*||jdt}t| |t||}}|||||Srrr2rbrrZneg_expZpos_expr4r4r5rs ztan._eval_rewrite_as_expcKsdt|dtd|Srurtr2rnrr4r4r5r!sztan._eval_rewrite_as_sincKst|tjdddt|Srrrwr4r4r5rsztan._eval_rewrite_as_coscKst|t|Srrrr4r4r5rsztan._eval_rewrite_as_sincoscKs dt|Srrrr4r4r5rsztan._eval_rewrite_as_cotcKs$t|t}t|t}||Sr)rtrrrz)r2rbrsin_in_sec_formcos_in_sec_formr4r4r5rsztan._eval_rewrite_as_seccKs$t|t}t|t}||Sr)rtrrrz)r2rbrsin_in_csc_formcos_in_csc_formr4r4r5rsztan._eval_rewrite_as_csccKs"|tt}|trdS|SrrrzrrJr2rbrrr4r4r5rs ztan._eval_rewrite_as_powcKs"|tt}|trdS|Srrrzr$rJr~r4r4r5rs ztan._eval_rewrite_as_sqrtcCsl|jd}||d}d|tj}|jrX||tjd|}|jrP|Sd|S|jrh| |S|S)Nrr[r r.rrrr]r`rrarr-r2rnrrrbrrrr4r4r5rs ztan._eval_as_leading_termcCs |jdjSrrrr4r4r5rsztan._eval_is_extended_realcCs,|jd}|jr(|ttjjdur(dSdSr7r.is_realrrr_r`rr4r4r5 _eval_is_reals ztan._eval_is_realcCs6|jd}|jr(|ttjjdur(dS|jr2dSdSr7)r.rrrr_r` is_imaginaryrr4r4r5rs  ztan._eval_is_finitecCs"t|jd\}}|jr|jSdSrrrr4r4r5rsztan._eval_is_zerocCs,|jd}|jr(|ttjjdur(dSdSr7rrr4r4r5rs ztan._eval_is_complex)N)rg)rg)r)T)Nr) rUrVrWrXrxr}rYrrrrrrrrr=rrr!rrrrrrrrrrrrrr4r4r4r5rs<%         rc@seZdZdZd:ddZd;ddZdddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd?d*d+Zd,d-Zd.d/Zd0d1Zd2d3Zd4d5Zd6d7Zd8d9Z dS)@ra The cotangent function. Returns the cotangent of x (measured in radians). Explanation =========== See :class:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import cot, pi >>> from sympy.abc import x >>> cot(x**2).diff(x) 2*x*(-cot(x**2)**2 - 1) >>> cot(1).diff(x) 0 >>> cot(pi/12) sqrt(3) + 2 See Also ======== sin, csc, cos, sec, tan asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Cot NcCs |t|Srrvrwr4r4r5rxCsz cot.periodrgcCs$|dkrtj|dSt||dSrV)rrrr{r4r4r5r}Fsz cot.fdiffcCstSrWrr{r4r4r5rYLsz cot.inversec Csddlm}|jr.|tjur"tjS|jr.tjS|tjur>tjSt||r\t|tj d S| rp||  S| tj }|durtj t |St|d}|durj|jrtjS|js|tj }||kr||SdS|jrj|jdvrttj d|S|jdkrf|jdsf|tj d}t|t|tj d}}t|tsft|tsfd|||Sdddd d d d d d}|j} |j| } | |vr|| tj || d|| tj || d} } d| | fvrdSd| | | | S|tjdtjtj }t|t|tj d}}t|tsXt|tsX|dkrPtjS||S||krj||S|jrt|\} }|rt|tj }|tjurt| St|  S|jrtjSt|tr|jdSt|tr|jd} d| St|tr|j\}} | |St|tr:|jd} td| d| St|trd|jd} | td| dSt|tr|jd} tdd| d| St|t r|jd} dtdd| d| SdS)Nrr~r[rZrgrrrr r r rrr)!rrrrrr0rYrrr]rrr>r r8r`rrrzror_rMrfrrr.rrrr$rrr)rrbrrr9rr[r\rrrorrrnrpZcotmrr4r4r5rRs              6                    zcot.evalcGsddlm}|dkr dt|S|dks4|ddkr:tjSt|}||d}t|d}tj|ddd|d||||SdSNrr]rgr[)r_r^rrrDrr)rrnrr^r`rar4r4r5rs    zcot.taylor_termrcCsN|jd|dtj}|r8|jr8|tj|||dS|tj|||dS)Nrrb) r.rrr]rrrzrrrcr4r4r5rs zcot._eval_nseriescCs||jdSrrrr4r4r5rszcot._eval_conjugateTcKsj|jfd|i|\}}|rVtd|td|}td| |td||fS||tjfSdSrdrerfr4r4r5r=s "zcot.as_real_imagcKsZtj}t|ttfr*||jdt}t| |t||}}|||||Srrrur4r4r5rs zcot._eval_rewrite_as_expcKsHt|trDtj}|jd}| || |||| ||SdSrrrr4r4r5rs  zcot._eval_rewrite_as_PowcKstd|dt|dSrurvrwr4r4r5r!szcot._eval_rewrite_as_sincKst|t|tjdddSrrrwr4r4r5rszcot._eval_rewrite_as_coscKst|t|Srrzrtrr4r4r5rszcot._eval_rewrite_as_sincoscKs dt|Srrrr4r4r5rszcot._eval_rewrite_as_tancKs$t|t}t|t}||Sr)rzrrrt)r2rbrrzryr4r4r5rszcot._eval_rewrite_as_seccKs$t|t}t|t}||Sr)rzrrrt)r2rbrr|r{r4r4r5rszcot._eval_rewrite_as_csccKs"|tt}|trdS|Srr}r~r4r4r5rs zcot._eval_rewrite_as_powcKs"|tt}|trdS|Srrr~r4r4r5r s zcot._eval_rewrite_as_sqrtcCsn|jd}||d}d|tj}|jrZ||tjd|}|jrTd|S| S|jrj| |S|SNrr[rgrrr4r4r5rs zcot._eval_as_leading_termcCs |jdjSrrrr4r4r5rszcot._eval_is_extended_realc sbddlm}m}|jd}d}|jrddlm}t|j}g}|jD]}t|dd } | | qDt dfddt |D} ddg} t |d d D]6} | || d || | d || d d 7<q| d| d  tt| |S|jrZ|jd d\} }| jrZ| d krZtj}tdd d}||| }|||| |t|fgSt|S)NrrgrhFrrjcsg|] }tqSr4rkr"rmr4r5r&)r'z)cot._eval_expand_trig..rr[rrgTrrnro)rrGrFr.rMrqrirrrr^r*rrrrsr2rKrirrr>rrC)r2r?rGrFrbrnrirZCXrlrjrorrr\rUrrOrtr4rmr5rs0     4   zcot._eval_expand_trigcCs0|jd}|jr"|tjdur"dS|jr,dSdSr7)r.rrr`rrr4r4r5r8s  zcot._eval_is_finitecCs&|jd}|jr"|tjdur"dSdSr7r.rrr`rr4r4r5r?s zcot._eval_is_realcCs&|jd}|jr"|tjdur"dSdSr7rrr4r4r5rDs zcot._eval_is_complexcCs,t|jd\}}|r(|jr(|tjjSdSr)rfr.r0rr_r`)r2rZpimultr4r4r5rIs zcot._eval_is_zerocCs8|jd}|||}||kr0|tjjr0tjSt|Sr)r.rrr]r`rYr)r2oldnewrbZargnewr4r4r5 _eval_subsNs   zcot._eval_subs)N)rg)rg)r)T)Nr)!rUrVrWrXrxr}rYrrrrrrrr=rrr!rrrrrrrrrrrrrrrr4r4r4r5rs<%    m    rc@seZdZdZdZejfZdZdZ e ddZ ddZ ddZ d d Zd d Zd.ddZddZddZddZddZddZddZddZddZd/d!d"Zd#d$Zd%d&Zd0d(d)Zd*d+Zd1d,d-ZdS)2ReciprocalTrigonometricFunctionz@Base class for reciprocal functions of trigonometric functions. NcCsJ|r*|jr|| S|jr*||  St|}|durd|js|jr|j}|jd|}||kr~|dtj }|| Sd||krd|tj }|jr||S|jr|| St |dr| |kr|j dS|j |}|dur|Stdd|| fDrd|tStdd|| fDr>d|tSd|SdS)Nr[rgrYrcss|]}t|tVqdSr)rrzr"r4r4r5 r'z7ReciprocalTrigonometricFunction.eval..css|]}t|tVqdSr)rrtr"r4r4r5rr')r_is_even_is_oddr8r`rrrorr]hasattrrYr._reciprocal_ofranyrrr)rrbr9rrortr4r4r5rds@       z$ReciprocalTrigonometricFunction.evalcOs$||jd}t|||i|Sr)rr.getattr)r2 method_namer.ror4r4r5_call_reciprocalsz0ReciprocalTrigonometricFunction._call_reciprocalcOs,|j|g|Ri|}|dur(d|S|Sr)r)r2rr.rrr4r4r5_calculate_reciprocalsz5ReciprocalTrigonometricFunction._calculate_reciprocalcCs.|||}|dur*|||kr*d|SdSr)rr)r2rrbrr4r4r5_rewrite_reciprocals z3ReciprocalTrigonometricFunction._rewrite_reciprocalcCst|jd}|||Sr)r r.rrx)r2rOrPr4r4r5rTsz'ReciprocalTrigonometricFunction._periodrgcCs|d| |dS)Nr}r[rr{r4r4r5r}sz%ReciprocalTrigonometricFunction.fdiffcKs |d|S)Nrrrr4r4r5rsz4ReciprocalTrigonometricFunction._eval_rewrite_as_expcKs |d|S)Nrrrr4r4r5rsz4ReciprocalTrigonometricFunction._eval_rewrite_as_PowcKs |d|S)Nr!rrr4r4r5r!sz4ReciprocalTrigonometricFunction._eval_rewrite_as_sincKs |d|S)Nrrrr4r4r5rsz4ReciprocalTrigonometricFunction._eval_rewrite_as_coscKs |d|S)Nrrrr4r4r5rsz4ReciprocalTrigonometricFunction._eval_rewrite_as_tancKs |d|S)Nrrrr4r4r5rsz4ReciprocalTrigonometricFunction._eval_rewrite_as_powcKs |d|S)Nrrrr4r4r5rsz5ReciprocalTrigonometricFunction._eval_rewrite_as_sqrtcCs||jdSrrrr4r4r5rsz/ReciprocalTrigonometricFunction._eval_conjugateTcKs"d||jdj|fi|Sry)rr.r=)r2r<r?r4r4r5r=sz,ReciprocalTrigonometricFunction.as_real_imagcKs|jdi|S)Nr)rr)r2r?r4r4r5rsz1ReciprocalTrigonometricFunction._eval_expand_trigcCs||jdSr)rr.rrr4r4r5rsz6ReciprocalTrigonometricFunction._eval_is_extended_realrcCsd||jd|Sry)rr.r)r2rnrrr4r4r5rsz5ReciprocalTrigonometricFunction._eval_as_leading_termcCsd||jdjSry)rr.rrr4r4r5rsz/ReciprocalTrigonometricFunction._eval_is_finitecCsd||jd|||Sry)rr.rr2rnrrrr4r4r5rsz-ReciprocalTrigonometricFunction._eval_nseries)rg)T)Nr)r)rUrVrWrXrrrYrZrrrrrrrrTr}rrr!rrrrrr=rrrrrr4r4r4r5rVs4 $   rc@s~eZdZdZeZdZdddZddZdd Z d d Z d d Z ddZ ddZ dddZddZeeddZdddZdS)ra The secant function. Returns the secant of x (measured in radians). Explanation =========== See :class:`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 TNcCs ||SrrTrwr4r4r5rxsz sec.periodcKs t|dd}|d|dSrrx)r2rbrZ cot_half_sqr4r4r5rszsec._eval_rewrite_as_cotcKs dt|Srrzrr4r4r5rszsec._eval_rewrite_as_coscKst|t|t|Srrrr4r4r5rszsec._eval_rewrite_as_sincoscKsdt|tSr)rzrrtrr4r4r5r!szsec._eval_rewrite_as_sincKsdt|tSr)rzrrrr4r4r5rszsec._eval_rewrite_as_tancKsttd|ddSr)rrrr4r4r5rszsec._eval_rewrite_as_cscrgcCs2|dkr$t|jdt|jdSt||dSry)rr.rrr{r4r4r5r} sz sec.fdiffcCs,|jd}|jr(|ttjjdur(dSdSr7)r.rrrr_r`rr4r4r5rs zsec._eval_is_complexcGshddlm}|dks |ddkr&tjSt|}|d}tj||d|td||d|SdS)Nr)eulerr[rg)r_rrrDrrr)rrnrrkr4r4r5rs  zsec.taylor_termrcCsj|jd}||d}|tjdtj}|jr`||tjtjd|}tj||S||Sr) r.rrrr]r`rrr-rr4r4r5r$s zsec._eval_as_leading_term)N)rg)Nr)rUrVrWrXrzrrrxrrrr!rrr}rrrrrr4r4r4r5rs #   rc@steZdZdZeZdZdddZddZdd Z d d Z d d Z ddZ ddZ dddZddZeeddZdS)ra The cosecant function. Returns the cosecant of x (measured in radians). Explanation =========== 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 TNcCs ||Srrrwr4r4r5rxUsz csc.periodcKs dt|Srrvrr4r4r5r!Xszcsc._eval_rewrite_as_sincKst|t|t|Srrrr4r4r5r[szcsc._eval_rewrite_as_sincoscKs t|d}d|dd|Srrxrr4r4r5r^s zcsc._eval_rewrite_as_cotcKsdt|tSr)rtrrzrr4r4r5rbszcsc._eval_rewrite_as_coscKsttd|ddSr)rrrr4r4r5reszcsc._eval_rewrite_as_seccKsdt|tSr)rtrrrr4r4r5rhszcsc._eval_rewrite_as_tanrgcCs4|dkr&t|jd t|jdSt||dSry)rr.rrr{r4r4r5r}ksz csc.fdiffcCs&|jd}|jr"|tjdur"dSdSr7rrr4r4r5rqs zcsc._eval_is_complexcGsddlm}|dkr dt|S|dks4|ddkr:tjSt|}|dd}tj|dddd|dd|d||d|dtd|SdSr)r_r^rrrDrr)rrnrr^rr4r4r5rvs   $  zcsc.taylor_term)N)rg)rUrVrWrXrtrrrxr!rrrrrr}rrrrr4r4r4r5r.s#  rc@s\eZdZdZejfZdddZeddZ ddd Z d d Z d d Z ddZ ddZeZdS)ra Represents an unnormalized sinc function: .. math:: \operatorname{sinc}(x) = \begin{cases} \frac{\sin x}{x} & \qquad x \neq 0 \\ 1 & \qquad x = 0 \end{cases} Examples ======== >>> from sympy import sinc, oo, jn >>> from sympy.abc import x >>> sinc(x) sinc(x) * Automated Evaluation >>> sinc(0) 1 >>> sinc(oo) 0 * Differentiation >>> sinc(x).diff() cos(x)/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) See also ======== sin References ========== .. [1] https://en.wikipedia.org/wiki/Sinc_function rgcCs<|jd}|dkr.t||t||dSt||dS)Nrrgr[)r.rzrtr)r2r|rnr4r4r5r}s z sinc.fdiffcCs|jr tjS|jr8|tjtjfvr(tjS|tjur8tjS|tjurHtjS| rZ|| St |}|dur|j rt |jrtjSnd|j rtj |tj|SdSru)r0rrhrrrrDrrYrr8r`r rr_)rrbr9r4r4r5rs$     z sinc.evalrcCs |jd}t|||||Sr)r.rtrrr4r4r5rs zsinc._eval_nseriescKsddlm}|d|S)Nr)jn)Zsympy.functions.special.besselr)r2rbrrr4r4r5_eval_rewrite_as_jns zsinc._eval_rewrite_as_jncKs&tt||t|tjftjtjfSr)r'rtrrrDrhtruerr4r4r5r!szsinc._eval_rewrite_as_sincCsL|jdjrdSt|jd\}}|jr8t|j|jgS|jrH|jrHdSdS)NrTF)r. is_infiniterfr0rr`Z is_nonzerorrr4r4r5rs  zsinc._eval_is_zerocCs |jdjs|jdjrdSdSr)r.rBrrr4r4r5rszsinc._eval_is_realN)rg)r)rUrVrWrXrrYrZr}rrrrr!rrrr4r4r4r5rs4    rc@sHeZdZdZejejejejfZ e ddZ e ddZ e ddZ dS) InverseTrigonometricFunctionz/Base class for inverse trigonometric functions.c-CsFtddtjdtddtjddtdtjdtdtddtjdtdtdtddtjdtdtddtjtddtdtdtddtjtddtjtjdtdtddtjdttjtddtjdtdtddtjtddttjtddtjtddtdddtjddtddtj dtdddtjtddtddtddtjd td dtddtj d tddtdtjd dtdtdtj d tddtddtjtdd dtdtdtjtdd iS) Nrr[rrgrrr r r)r$rr]rr_r4r4r4r5 _asin_tables," (  ""$ z(InverseTrigonometricFunction._asin_tablecCs.tddtjddtdtjdtdtjdtddtjddtdtj ddtdtjtddtddtdtjdtddtdtjtddtddtddtjdtddtddtjtdddtdtjdd tdtj ddtdtjtddi S) Nrr rgr[rrr rrr$rr]rr4r4r4r5 _atan_table!s $z(InverseTrigonometricFunction._atan_tablec%Csdtddtjdtdtjdtddtddtjddttddtddtjdtddtddtjtdddttddtddtjtdddtjdtddtdtjddtdtdtjdtddtdtjtdddtdtdtjtdddtdtjdtddtjtddtdd tjtd dtdtdtjd tdtdtjtdd tdtd tjtd d iS) Nr[rrrrgrr r rrr4r4r4r5 _acsc_table5s$$$*   z(InverseTrigonometricFunction._acsc_tableN)rUrVrWrXrrhrrDrYrZrrrrr4r4r4r5rs  rc@seZdZdZd%ddZddZddZd d Zed d Z e e d dZ d&ddZ d'ddZddZddZddZddZddZdd Zd!d"Zd(d#d$ZdS))rab The inverse sine function. Returns the arcsine of x in radians. Explanation =========== ``asin(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the ``eval`` class method). A purely imaginary argument will lead to an asinh expression. Examples ======== >>> from sympy import asin, oo >>> asin(1) pi/2 >>> asin(-1) -pi/2 >>> asin(-oo) oo*I >>> asin(oo) -oo*I 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 rgcCs0|dkr"dtd|jddSt||dSNrgrr[r$r.rr{r4r4r5r}ysz asin.fdiffcCs4|j|j}|j|jkr*|jdjr0dSn|jSdSr,r-r.r/r1r4r4r5r6s    zasin._eval_is_rationalcCs|o|jdjSr)rr. is_positiverr4r4r5_eval_is_positiveszasin._eval_is_positivecCs|o|jdjSr)rr.rrr4r4r5_eval_is_negativeszasin._eval_is_negativecCs|jrx|tjurtjS|tjur,tjtjS|tjurBtjtjS|jrNtjS|tjurbtj dS|tj urxtj dS|tj urtj S| r||  S|j r|}||vr||S|tj}|durtjt|S|jrtjSt|trX|jd}|jrX|dt;}|tkr$t|}|tdkr:t|}|t dkrTt |}|St|tr|jd}|jrtdt|SdSNr[r)rrrrrr>r0rDrhr]rrYr is_numberrrrrrtr. is_comparablerrzrrrbZ asin_tablerangr4r4r5rsR                   z asin.evalcGs|dks|ddkrtjSt|}t|dkrb|dkrb|d}||dd||d|dS|dd}ttj|}t|}|||||SdSr)rrDrrrr_rrrnrrorRrar4r4r5rs$  zasin.taylor_termNrcCsddlm}|jd}||d}|jr6||S|tjurTtj t ||S|dkrh| ||}||dkr|j r|tj krtj ||S||dkr|j r|tjkrtj||S||SNrrG)rrGr.rrr0rrrYr>rrrrr]r-rhr2rnrrrGrbrr4r4r5rs     zasin._eval_as_leading_termcCsddlm}ddlm}|jd|d}|tjurtddd}t tj|d t  |dd|} tj|jd} | |} | | | } | |ds|dkr|dStjd|t|Sttj| j|||d } | t| }| ||||||S|tjurtddd}t tj|d t  |dd|} tj|jd} | |} | | | } | |ds|dkr|dStj d|t|Sttj| j|||d } | t| }| ||||||Stj||||d }|tjur&|S|dkrB|jd||}||dkrp|jrp|tjkrptj |S||dkr|jr|tjkrtj|S|S NrrOrTZpositiver[rgrb)rrGsympy.series.orderrr.rrrhrrrrnseriesris_meromorphicr]r$rremoveOrCpowsimprrrYrrr2rnrrrrGrarg0rZserZarg1rPrQZres1resr4r4r5rsF    &   &&  &  *&  " " zasin._eval_nseriescKstjdt|Srurr]rrwr4r4r5_eval_rewrite_as_acos szasin._eval_rewrite_as_acoscKs dt|dtd|dSr)rr$rwr4r4r5_eval_rewrite_as_atan szasin._eval_rewrite_as_atancKs&tj ttj|td|dSrVrr>rr$rwr4r4r5_eval_rewrite_as_log szasin._eval_rewrite_as_logcKs dtdtd|d|Sr)rr$rr4r4r5_eval_rewrite_as_acot szasin._eval_rewrite_as_acotcKstjdtd|Srrr]rrr4r4r5_eval_rewrite_as_asec szasin._eval_rewrite_as_aseccKs td|Sr)rrr4r4r5_eval_rewrite_as_acsc szasin._eval_rewrite_as_acsccCs|jd}|jodt|jSNrrgr.rBrLis_nonnegativer2rnr4r4r5r s zasin._eval_is_extended_realcCstSrWrvr{r4r4r5rY sz asin.inverse)rg)Nr)r)rg)rUrVrWrXr}r6rrrrrrrrrrrrrrrrrYr4r4r4r5rNs(*  5   'rc@seZdZdZd%ddZddZeddZee d d Z d&d dZ ddZ ddZ d'ddZddZddZddZd(ddZddZdd Zd!d"Zd#d$Zd S))ra The inverse cosine function. Returns the arc cosine of x (measured in radians). Examples ======== ``acos(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). ``acos(zoo)`` evaluates to ``zoo`` (see note in :class:`sympy.functions.elementary.trigonometric.asec`) A purely imaginary argument will be rewritten to asinh. Examples ======== >>> from sympy import acos, oo >>> 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 rgcCs0|dkr"dtd|jddSt||dSNrgrrr[rr{r4r4r5r}S sz acos.fdiffcCs4|j|j}|j|jkr*|jdjr0dSn|jSdSr,rr1r4r4r5r6Y s    zacos._eval_is_rationalcCs^|jrr|tjurtjS|tjur,tjtjS|tjurBtjtjS|jrRtjdS|tjurbtj S|tj urrtjS|tj urtj S|j r| }||vrtd||S| |vrtd|| S|tj}|durtdt|St|tr,|jd}|jr,|dt;}|tkr(dt|}|St|trZ|jd}|jrZtdt|SdSr)rrrrr>rr0r]rhrDrrYrrrrrrrzr.rrtrr4r4r5ra sF                  z acos.evalcGs|dkrtjdS|dks&|ddkr,tjSt|}t|dkrt|dkrt|d}||dd||d|dS|dd}ttj|}t|}| ||||SdSr)rr]rDrrrr_rrr4r4r5r s $  zacos.taylor_termNrcCsddlm}|jd}||d}|dkrJtdttj||S|tj urhtj t ||S|dkr|| ||}||dkr|j r|tjkrdtj||S||dkr|j r|tjkr|| S||SNrrrgr[)rrGr.rrr$rrhrrYr>rrrrr]r-rr4r4r5r s     zacos._eval_as_leading_termcCs|jd}|jodt|jSrrrr4r4r5r s zacos._eval_is_extended_realcCs|Sr)rrr4r4r5_eval_is_nonnegative szacos._eval_is_nonnegativecCsddlm}ddlm}|jd|d}|tjur tddd}t tj|d t  |dd|} tj|jd} | |} | | | } | |ds|dkr|dS|t|Sttj| j|||d } | t| }| ||||||S|tjurtddd}t tj|d t  |dd|} tj|jd} | |} | | | } | |ds|dkr|dStj|t|Sttj| j|||d } | t| }| ||||||Stj||||d }|tjur|S|dkr2|jd||}||dkrb|jrb|tjkrbdtj|S||dkr|jr|tjkr| S|Sr)rrGrrr.rrrhrrrrrrrr$rrrCrrr]rrYrrrr4r4r5r sF    &   &  &  $&  ""zacos._eval_nseriescKs.tjdtjttj|td|dSrrr]r>rr$rwr4r4r5r s zacos._eval_rewrite_as_logcKstjdt|Srurr]rrwr4r4r5_eval_rewrite_as_asin szacos._eval_rewrite_as_asincKs:ttd|d|tjdd|td|dSrV)rr$rr]rwr4r4r5r szacos._eval_rewrite_as_atancCstSrWrr{r4r4r5rY sz acos.inversecKs*tjddtdtd|d|Sr)rr]rr$rr4r4r5r szacos._eval_rewrite_as_acotcKs td|Sr)rrr4r4r5r szacos._eval_rewrite_as_aseccKstjdtd|Srrr]rrr4r4r5r szacos._eval_rewrite_as_acsccCsN|jd}||jd}|jdur,|S|jrJ|djrJ|djrJ|SdSNrFrg)r.r-rrBrZis_nonpositive)r2rOr/r4r4r5r s   zacos._eval_conjugate)rg)Nr)r)rg)rUrVrWrXr}r6rrrrrrrrrrrrrYrrrrr4r4r4r5r' s(+  +   ' rcseZdZUdZeeed<ejej fZ d*ddZ ddZ dd Z d d Z d d ZddZeddZeeddZd+ddZd,ddZddZfddZd-ddZd d!Zd"d#Zd$d%Zd&d'Zd(d)ZZS).ra  The inverse tangent function. Returns the arc tangent of x (measured in radians). Explanation =========== ``atan(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 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 atan, oo >>> 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 r.rgcCs,|dkrdd|jddSt||dSrr.rr{r4r4r5r}* sz atan.fdiffcCs4|j|j}|j|jkr*|jdjr0dSn|jSdSr,rr1r4r4r5r60 s    zatan._eval_is_rationalcCs |jdjSr)r.Zis_extended_positiverr4r4r5r8 szatan._eval_is_positivecCs |jdjSr)r.Zis_extended_nonnegativerr4r4r5r; szatan._eval_is_nonnegativecCs |jdjSr)r.r0rr4r4r5r> szatan._eval_is_zerocCs |jdjSrrrr4r4r5rA szatan._eval_is_realcCs|jrv|tjurtjS|tjur*tjdS|tjur@tj dS|jrLtjS|tjur`tjdS|tj urvtj dS|tj urddl m }|tj dtjdS| r||  S|jr|}||vr||S|tj}|durtjt|S|jrtjSt|trH|jd}|jrH|t;}|tdkrD|t8}|St|tr|jd}|jrtdt|}|tdkr|t8}|SdS)Nr[rrr~)rrrrr]rr0rDrhrrYrrrrrrr>rrrr.rrrr)rrbr atan_tablerrr4r4r5rD sR                  z atan.evalcGsD|dks|ddkrtjSt|}tj|dd|||SdSr)rrDrrrrnrr4r4r5rx szatan.taylor_termNrcCsddlm}m}|jd}||d}|jr:||S|tj urZt d|j ||dS|dkrn| ||}||dkr||jr||tj kr||tjS||dkr||jr||tjkr||tjS||S)Nrrgrg)r)rrGrFr.rrr0rrrYrrrrhr-r]rr2rnrrrGrFrbrr4r4r5r s    $$zatan._eval_as_leading_termc Csddlm}m}|jd|d}tj||||d}|dkrN|jd||}|tj urr||dkrn|tj S|S||dkr||j r||tj kr|tj S||dkr||j r||tj kr|tj S|SNrrgrb)rrGrFr.rrrrrrYr]r0rhr r2rnrrrrGrFrrr4r4r5r s   $ $ zatan._eval_nseriescKs2tjdttjtj|ttjtj|Sru)rr>rrhrwr4r4r5r szatan._eval_rewrite_as_logcs|dtjur4tjdtd|jd|||S|dtjurjtj dtd|jd|||St||||SdSr) rrr]rr.rrsuper _eval_aseriesr2rZargs0rnr __class__r4r5r s &(zatan._eval_aseriescCstSrWrr{r4r4r5rY sz atan.inversecKs2t|d|tjdtdtd|dSrr$rr]rrr4r4r5r szatan._eval_rewrite_as_asincKs(t|d|tdtd|dSrr$rrr4r4r5r szatan._eval_rewrite_as_acoscKs td|Srrrr4r4r5r szatan._eval_rewrite_as_acotcKs$t|d|ttd|dSrr$rrr4r4r5r szatan._eval_rewrite_as_aseccKs.t|d|tjdttd|dSrr$rr]rrr4r4r5r szatan._eval_rewrite_as_acsc)rg)Nr)r)rg) rUrVrWrXtTupler__annotations__rr>rZr}r6rrrrrrrrrrrrrrYrrrrr __classcell__r4r4rr5r s0 &   3     rcseZdZdZejej fZd'ddZddZddZ d d Z d d Z e d dZ eeddZd(ddZd)ddZfddZddZd*ddZddZdd Zd!d"Zd#d$Zd%d&ZZS)+ra The inverse cotangent function. Returns the arc cotangent of x (measured in radians). Explanation =========== ``acot(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). A purely imaginary argument will lead to an ``acoth`` expression. ``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$. Examples ======== >>> from sympy import acot, sqrt >>> acot(0) pi/2 >>> acot(1) pi/4 >>> acot(sqrt(3) - 2) -5*pi/12 See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, atan2 References ========== .. [1] http://dlmf.nist.gov/4.23 .. [2] http://functions.wolfram.com/ElementaryFunctions/ArcCot rgcCs,|dkrdd|jddSt||dSrrr{r4r4r5r} sz acot.fdiffcCs4|j|j}|j|jkr*|jdjr0dSn|jSdSr,rr1r4r4r5r6 s    zacot._eval_is_rationalcCs |jdjSr)r.rrr4r4r5r szacot._eval_is_positivecCs |jdjSr)r.rrr4r4r5r szacot._eval_is_negativecCs |jdjSrrrr4r4r5r szacot._eval_is_extended_realcCs|jrp|tjurtjS|tjur&tjS|tjur6tjS|jrFtjdS|tjurZtjdS|tj urptj dS|tj urtjS| r||  S|j r| }||vrtd||}|tdkr|t8}|S|tj}|durtj t|S|jr tjtjSt|trL|jd}|jrL|t;}|tdkrH|t8}|St|tr|jd}|jrtdt|}|tdkr|t8}|SdS)Nr[rr)rrrrrDrr0r]rhrrYrrrrrr>rr_rrr.rrr)rrbrrrr4r4r5r sV                  z acot.evalcGsV|dkrtjdS|dks&|ddkr,tjSt|}tj|dd|||SdSr)rr]rDrrrr4r4r5r< s  zacot.taylor_termNrcCsddlm}m}|jd}||d}|tjurBd||S|dkrV| ||}|j r||dkrx| |tj S| |S||dkr||j r||tj kr||tjkr| |tj S||dkr||j r||tj kr||tjkr| |tj S| |S)Nrrgrg)rrGrFr.rrrrYrrr0r-r]rDrhrrr4r4r5rG s      2:zacot._eval_as_leading_termc Csddlm}m}|jd|d}tj||||d}|tjurB|S|dkr\|jd ||}|j r|||dkrx|tj S|S||dkr||j r||tj kr||tj kr|tj S||dkr||j r||tj kr||tjkr|tj S|Sr)rrGrFr.rrrrrYrr0r]rDrhrrr4r4r5rY s    2 2 zacot._eval_nseriescs|dtjur4tjdtd|jd|||S|dtjurntjtddtd|jd|||Stt | ||||SdS)Nrr[rgr) rrr]rr.rrrrrrrrr4r5rk s &,zacot._eval_aseriescKs.tjdtdtj|tdtj|Sr)rr>rrwr4r4r5rs szacot._eval_rewrite_as_logcCstSrWrxr{r4r4r5rYw sz acot.inversecKsB|td|dtjdtt|d t|d dSrVrrr4r4r5r} s,zacot._eval_rewrite_as_asincKs8|td|dtt|d t|d dSrVrrr4r4r5r szacot._eval_rewrite_as_acoscKs td|SrrXrr4r4r5r szacot._eval_rewrite_as_atancKs0|td|dttd|d|dSrVrrr4r4r5r szacot._eval_rewrite_as_aseccKs:|td|dtjdttd|d|dSrVrrr4r4r5r szacot._eval_rewrite_as_acsc)rg)Nr)r)rg)rUrVrWrXrr>rZr}r6rrrrrrrrrrrrrYrrrrrrr4r4rr5r s,*  4    rc@s|eZdZdZeddZdddZdddZdd d Zd d dZ ddZ ddZ ddZ ddZ ddZddZddZd S)!ra The inverse secant function. Returns the arc secant of x (measured in radians). Explanation =========== ``asec(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). ``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments, it can be defined [4]_ as .. math:: \operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{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\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z} simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which ultimately evaluates to ``zoo``. As ``acos(x) = asec(1/x)``, a similar argument can be given for ``acos(x)``. Examples ======== >>> from sympy import asec, oo >>> asec(1) 0 >>> asec(-1) pi >>> asec(0) zoo >>> asec(-oo) pi/2 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 cCs|jr tjS|jrB|tjur"tjS|tjur2tjS|tjurBtjS|tj tj tjfvr`tjdS|j r| }||vrt d||S| |vrt d|| St|tr|jd}|jr|dt ;}|t krdt |}|St|tr|jd}|jrt dt|SdSr)r0rrYrrrhrDrr]rrrrrrrr.rrrrrbZ acsc_tablerr4r4r5r s8           z asec.evalrgcCsB|dkr4d|jddtdd|jddSt||dSrr.r$rr{r4r4r5r} s,z asec.fdiffcCstSrWrTr{r4r4r5rY sz asec.inverseNrcCsddlm}|jd}||d}|dkrJtdt|tj|S|j rdtj t ||S|dkrx| ||}||dkr|j r|tjkr|tjkr|| S||dkr|j r|tjkr|tjkrdtj||S||Sr)rrGr.rrr$rrhrr0r>rrrrDr-rr]rr4r4r5r s   & &zasec._eval_as_leading_termcCsJddlm}ddlm}|jd|d}|tjurtddd}t tj|d t  |dd|} tj |jd} | |} | | | } ttj| j|||d} | t| }| ||||||S|tj urtddd}t tj |d t  |dd|} tj |jd} | |} | | | } ttj| j|||d} | t| }| ||||||Stj||||d}|tjur|S|dkr|jd||}||dkr |jr |tjkr |tjkr | S||dkrF|jrF|tjkrF|tj krFdtj|S|S NrrrrTrr[rb)rrGrrr.rrrhrrrrrrrr$rrrCrrrYrrrDr]rr4r4r5r s>    &  &  &  &  ..zasec._eval_nseriescCs2|jd}|jdurdSt|dj| djfSr)r.rBr rrr4r4r5r, s  zasec._eval_is_extended_realc Ks2tjdtjttj|tdd|dSrrrr4r4r5r2 szasec._eval_rewrite_as_logcKstjdtd|Srrrr4r4r5r5 szasec._eval_rewrite_as_asincKs td|Sr)rrr4r4r5r8 szasec._eval_rewrite_as_acoscKs8t|d|tj ddt|t|ddSrr$rr]rrr4r4r5r; szasec._eval_rewrite_as_atancKs8t|d|tj ddt|t|ddSrr$rr]rrr4r4r5r> szasec._eval_rewrite_as_acotcKstjdt|Srurrr4r4r5rA szasec._eval_rewrite_as_acsc)rg)rg)Nr)r)rUrVrWrXrrr}rYrrrrrrrrrr4r4r4r5r s; "    #rc@steZdZdZeddZdddZdddZdd d Zdd dZ ddZ ddZ ddZ ddZ ddZddZd S)raT The inverse cosecant function. Returns the arc cosecant of x (measured in radians). Explanation =========== ``acsc(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the result is a rational multiple of $\pi$ (see the ``eval`` class method). Examples ======== >>> from sympy import acsc, oo >>> acsc(1) pi/2 >>> acsc(-1) -pi/2 >>> acsc(oo) 0 >>> acsc(-oo) == acsc(oo) True >>> acsc(0) zoo 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 cCs6|jr tjS|jrL|tjur"tjS|tjur6tjdS|tjurLtj dS|tjtj tjfvrftj S| rz||  S|j r| }||vr||St|tr|jd}|jr|dt;}|tkrt|}|tdkrt|}|t dkrt |}|St|tr2|jd}|jr2tdt|SdSr)r0rrYrrrhr]rrrrDrrrrrr.rrrrrr4r4r5rp s@             z acsc.evalrgcCsB|dkr4d|jddtdd|jddSt||dSrrr{r4r4r5r} s,z acsc.fdiffcCstSrWrr{r4r4r5rY sz acsc.inverseNrcCsddlm}|jd}||d}|jr@tjt| |S|tj urT| |S|dkrh| ||}||dkr|j r|tj kr|tjkrtj||S||dkr|j r|tj kr|tjkrtj ||S||Sr)rrGr.rrr0rr>rrrYrrrDrhr]r-rrr4r4r5r s     &&zacsc._eval_as_leading_termcCsLddlm}ddlm}|jd|d}|tjurtddd}t tj|d t  |dd|} tj |jd} | |} | | | } ttj| j|||d} | t| }| ||||||S|tj urtddd}t tj |d t  |dd|} tj |jd} | |} | | | } ttj| j|||d} | t| }| ||||||Stj||||d}|tjur|S|dkr|jd||}||dkr|jr|tjkr|tjkrtj|S||dkrH|jrH|tjkrH|tj krHtj |S|Sr)rrGrrr.rrrhrrrrrrrr$rrrCrrrYrrrDr]rr4r4r5r s>    &  &  &  &  . . zacsc._eval_nseriescKs*tj ttj|tdd|dSrVrrr4r4r5r szacsc._eval_rewrite_as_logcKs td|Sr)rrr4r4r5r szacsc._eval_rewrite_as_asincKstjdtd|Srrrr4r4r5r szacsc._eval_rewrite_as_acoscKs.t|d|tjdtt|ddSrrrr4r4r5r szacsc._eval_rewrite_as_atancKs2t|d|tjdtdt|ddSrrrr4r4r5r szacsc._eval_rewrite_as_acotcKstjdt|Srurrr4r4r5r szacsc._eval_rewrite_as_asec)rg)rg)Nr)r)rUrVrWrXrrr}rYrrrrrrrrr4r4r4r5rE s* )    #rcs\eZdZdZeddZddZddZdd Zd d Z d d Z ddZ fddZ Z S)ra 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. If either `x` or `y` is complex: .. math:: \operatorname{atan2}(y, x) = -i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right) 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) Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True)) 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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