a sz/_rewrite_hyperbolics_as_exp..)rZxreplaceZatomsHyperbolicFunction)exprrrr_rewrite_hyperbolics_as_exp s rc@seZdZdZdZdS)rze Base class for hyperbolic functions. See Also ======== sinh, cosh, tanh, coth TN)__name__ __module__ __qualname____doc__Z unbranchedrrrrrs rcCstjtj}t|D]<}||kr.tj}q^q|jr|\}}||kr|jrq^q|tj fS|tj }||}||||fS)a Split ARG into two parts, a "rest" and a multiple of $I\pi$. This assumes ARG to be an ``Add``. The multiple of $I\pi$ returned in the second position is always a ``Rational``. Examples ======== >>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel >>> from sympy import pi, I >>> from sympy.abc import x, y >>> peel(x + I*pi/2) (x, 1/2) >>> peel(x + I*2*pi/3 + I*pi*y) (x + I*pi*y + I*pi/6, 1/2) ) rPi ImaginaryUnitr Z make_argsOneis_Mul as_two_termsZ is_RationalZeroHalf)argZipiaKpm1m2rrr _peeloff_ipi%s    r.c@seZdZdZd.ddZd/ddZeddZee d d Z d d Z d0ddZ d1ddZ d2ddZd3ddZddZddZddZddZd4d d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-ZdS)5sinha ``sinh(x)`` is the hyperbolic sine of ``x``. The hyperbolic sine function is $\frac{e^x - e^{-x}}{2}$. Examples ======== >>> from sympy import sinh >>> from sympy.abc import x >>> sinh(x) sinh(x) See Also ======== cosh, tanh, asinh cCs$|dkrt|jdSt||dS)z@ Returns the first derivative of this function. r0rN)coshargsr selfargindexrrrfdiff[sz sinh.fdiffcCstSz7 Returns the inverse of this function. )asinhr3rrrinversedsz sinh.inversecCsddlm}t|}|jrl|tjur*tjS|tjur:tjS|tjurJtjS|jrVtj S|j rh||  Sn>|tj ur|tjS| tj }|durtj ||S|r||  S|jrt|\}}|r|tjtj }t|t|t|t|S|jr tj S|jtkr |jdS|jtkrN|jd}t|dt|dS|jtkrx|jd}|td|dS|jtkr|jd}dt|dt|dSdS)Nr)sinr0)(sympy.functions.elementary.trigonometricr:r is_NumberrNaNInfinityNegativeInfinityis_zeror& is_negativeComplexInfinityas_coefficientr"could_extract_minus_signis_Addr.r!r/r1funcr8r2acoshratanhacoth)clsr(r:i_coeffxmrrrevaljsL                 z sinh.evalcGsb|dks|ddkrtjSt|}t|dkrN|d}||d||dS||t|SdS)zG Returns the next term in the Taylor series expansion. rr;r0Nrr&rlenr nrMprevious_termsr+rrr taylor_terms zsinh.taylor_termcCs||jdSNrrGr2 conjugater4rrr_eval_conjugateszsinh._eval_conjugateTcKsddlm}m}|jdjrJ|r@d|d<|j|fi|tjfS|tjfS|rp|jdj|fi|\}}n|jd\}}t |||t |||fS)z@ Returns this function as a complex coordinate. rcosr:Fcomplex r<r]r:r2is_extended_realexpandrr& as_real_imagr/r1r4deephintsr]r:reimrrrrbs  "zsinh.as_real_imagcKs&|jfd|i|\}}||tjSNrdrbrr"r4rdreZre_partZim_partrrr_eval_expand_complexszsinh._eval_expand_complexcKs|r|jdj|fi|}n |jd}d}|jr@|\}}n:|jdd\}}|tjurz|jrz|tjurz|}|d|}|durt|t |t|t |jddSt|SNrTZrationalr0)Ztrig) r2rarFr% as_coeff_Mulrr# is_Integerr/r1r4rdrer(rMycoefftermsrrr_eval_expand_trigs  (zsinh._eval_expand_trigNcKst|t| dSNr;rr4r(limitvarkwargsrrr_eval_rewrite_as_tractableszsinh._eval_rewrite_as_tractablecKst|t| dSrurvr4r(ryrrr_eval_rewrite_as_expszsinh._eval_rewrite_as_expcKs tj t|tjtjdSrurr"r1r!r{rrr_eval_rewrite_as_coshszsinh._eval_rewrite_as_coshcKs"ttj|}d|d|dSNr;r0tanhrr'r4r(ryZ tanh_halfrrr_eval_rewrite_as_tanhszsinh._eval_rewrite_as_tanhcKs"ttj|}d||ddSrcothrr'r4r(ryZ coth_halfrrr_eval_rewrite_as_cothszsinh._eval_rewrite_as_cothrcCsh|jdj|||d}||d}|tjurF|j|d|jr>dndd}|jrP|S|jr`| |S|SdSNr)logxcdir-+)dir) r2as_leading_termsubsrr>limitrBrA is_finiterGr4rMrrr(Zarg0rrr_eval_as_leading_terms   zsinh._eval_as_leading_termcCs*|jd}|jrdS|\}}|tjSNrTr2is_realrbrrAr4r(rfrgrrr _eval_is_reals   zsinh._eval_is_realcCs|jdjrdSdSrr2r`rZrrr_eval_is_extended_reals zsinh._eval_is_extended_realcCs|jdjr|jdjSdSrWr2r` is_positiverZrrr_eval_is_positives zsinh._eval_is_positivecCs|jdjr|jdjSdSrWr2r`rBrZrrr_eval_is_negative s zsinh._eval_is_negativecCs|jd}|jSrWr2rr4r(rrr_eval_is_finite s zsinh._eval_is_finitecCs"t|jd\}}|jr|jSdSrW)r.r2rA is_integerr4restZipi_multrrr _eval_is_zeroszsinh._eval_is_zero)r0)r0)T)T)T)N)Nr)rrrr r6r9 classmethodrO staticmethodrrVr[rbrkrtrzr|r~rrrrrrrrrrrrrr/Gs0  4       r/c@seZdZdZd*ddZeddZeeddZ d d Z d+d d Z d,ddZ d-ddZ d.ddZddZddZddZddZd/ddZd d!Zd"d#Zd$d%Zd&d'Zd(d)ZdS)0r1a" ``cosh(x)`` is the hyperbolic cosine of ``x``. The hyperbolic cosine function is $\frac{e^x + e^{-x}}{2}$. Examples ======== >>> from sympy import cosh >>> from sympy.abc import x >>> cosh(x) cosh(x) See Also ======== sinh, tanh, acosh r0cCs$|dkrt|jdSt||dSNr0rr/r2r r3rrrr6+sz cosh.fdiffcCsddlm}t|}|jrj|tjur*tjS|tjur:tjS|tjurJtjS|jrVtj S|j rf|| Sn"|tj urztjS| tj }|dur||S|r|| S|jrt|\}}|r|tjtj }t|t|t|t|S|jrtj S|jtkrtd|jddS|jtkr4|jdS|jtkrZdtd|jddS|jtkr|jd}|t|dt|dSdS)Nrr]r0r;)r<r]rr=rr>r?r@rAr#rBrCrDr"rErFr.r!r1r/rGr8rr2rHrIrJ)rKr(r]rLrMrNrrrrO1sH               z cosh.evalcGsb|dks|ddkrtjSt|}t|dkrN|d}||d||dS||t|SdS)Nrr;r0rPrQrSrrrrVcs zcosh.taylor_termcCs||jdSrWrXrZrrrr[qszcosh._eval_conjugateTcKsddlm}m}|jdjrJ|r@d|d<|j|fi|tjfS|tjfS|rp|jdj|fi|\}}n|jd\}}t |||t |||fS)Nrr\Fr^) r<r]r:r2r`rarr&rbr1r/rcrrrrbts  "zcosh.as_real_imagcKs&|jfd|i|\}}||tjSrhrirjrrrrkszcosh._eval_expand_complexcKs|r|jdj|fi|}n |jd}d}|jr@|\}}n:|jdd\}}|tjurz|jrz|tjurz|}|d|}|durt|t|t |t |jddSt|Srl) r2rarFr%rnrr#ror1r/rprrrrts  (zcosh._eval_expand_trigNcKst|t| dSrurvrwrrrrzszcosh._eval_rewrite_as_tractablecKst|t| dSrurvr{rrrr|szcosh._eval_rewrite_as_expcKs tj t|tjtjdSrurr"r/r!r{rrr_eval_rewrite_as_sinhszcosh._eval_rewrite_as_sinhcKs"ttj|d}d|d|Srrrrrrrszcosh._eval_rewrite_as_tanhcKs"ttj|d}|d|dSrrrrrrrszcosh._eval_rewrite_as_cothrcCsj|jdj|||d}||d}|tjurF|j|d|jr>dndd}|jrRtjS|j rb| |S|SdSr) r2rrrr>rrBrAr#rrGrrrrrs   zcosh._eval_as_leading_termcCs0|jd}|js|jrdS|\}}|tjSr)r2r is_imaginaryrbrrArrrrrs    zcosh._eval_is_realc Csr|jd}|\}}|dt}|j}|r0dS|j}|durB|St|t|t|tdk|dtdkgggSNrr;TFr2rbrrArrr4zrMrqZymodZyzeroZxzerorrrrs    zcosh._eval_is_positivec Csr|jd}|\}}|dt}|j}|r0dS|j}|durB|St|t|t|tdk|dtdkgggSrrrrrr_eval_is_nonnegatives    zcosh._eval_is_nonnegativecCs|jd}|jSrWrrrrrrs zcosh._eval_is_finitecCs,t|jd\}}|r(|jr(|tjjSdSrW)r.r2rArr'rrrrrrs zcosh._eval_is_zero)r0)T)T)T)N)Nr)rrrr r6rrOrrrVr[rbrkrtrzr|rrrrrrrrrrrrrr1s,  1        r1c@seZdZdZd,ddZd-ddZeddZee d d Z d d Z d.ddZ ddZ d/ddZddZddZddZddZd0ddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+ZdS)1ra' ``tanh(x)`` is the hyperbolic tangent of ``x``. The hyperbolic tangent function is $\frac{\sinh(x)}{\cosh(x)}$. Examples ======== >>> from sympy import tanh >>> from sympy.abc import x >>> tanh(x) tanh(x) See Also ======== sinh, cosh, atanh r0cCs.|dkr tjt|jddSt||dSNr0rr;)rr#rr2r r3rrrr6sz tanh.fdiffcCstSr7)rIr3rrrr9!sz tanh.inversecCsddlm}t|}|jrl|tjur*tjS|tjur:tjS|tjurJtj S|j rVtj S|j rh||  SnD|tj ur|tjS|tj}|dur|rtj || Stj||S|r||  S|jrt|\}}|rt|tjtj}|tj urt|St|S|j r$tj S|jtkrN|jd}|td|dS|jtkr|jd}t|dt|d|S|jtkr|jdS|jtkrd|jdSdS)Nr)tanr0r;)r<rrr=rr>r?r#r@ NegativeOnerAr&rBrCrDr"rErFr.rr!rrGr8r2rrHrIrJ)rKr(rrLrMrNZtanhmrrrrO'sR                z tanh.evalcGsrddlm}|dks |ddkr&tjSt|}d|d}||d}t|d}||d||||SdS)Nr bernoullir;r0)%sympy.functions.combinatorial.numbersrrr&rr )rTrMrUrr)BFrrrrV_s    ztanh.taylor_termcCs||jdSrWrXrZrrrr[osztanh._eval_conjugateTcKsddlm}m}|jdjrJ|r@d|d<|j|fi|tjfS|tjfS|rp|jdj|fi|\}}n|jd\}}t |d||d}t |t |||||||fSNrr\Fr^r;r_r4rdrer]r:rfrgdenomrrrrbrs  "ztanh.as_real_imagcKs:|jd}|jrzddlm}t|j}dd|jD}ddg}t|dD]}||d|||7<qJ|d|dS|jr2ddlm}| \} } | j r2| dkr2g}g} t | } td| ddD] } | |t| | | | qtd| ddD]"} | |t| | | | qt |t | St |S)Nrsymmetric_polycSsg|]}t|ddqSFevaluate)rrtrrMrrr sz*tanh._eval_expand_trig..r0r;)nC)r2rFsympy.polys.specialpolysrrRranger$rrrnrorappendr )r4rer(rrTZTXr+irrrrsdTkrrrrts0      ztanh._eval_expand_trigNcKs$t| t|}}||||SNrvr4r(rxryneg_exppos_exprrrrzsztanh._eval_rewrite_as_tractablecKs$t| t|}}||||Srrvr4r(ryrrrrrr|sztanh._eval_rewrite_as_expcKs&tjt|ttjtjd|Srurr{rrrrsztanh._eval_rewrite_as_sinhcKs&tjttjtjd|t|Srur}r{rrrr~sztanh._eval_rewrite_as_coshcKs dt|SNr0rr{rrrrsztanh._eval_rewrite_as_cothrcCsHddlm}|jd|}||jvr:|d||r:|S||SdSNrOrderr0sympy.series.orderrr2r free_symbolscontainsrGr4rMrrrr(rrrrs  ztanh._eval_as_leading_termcCsJ|jd}|jrdS|\}}|dkr<|ttdkr>> from sympy import coth >>> from sympy.abc import x >>> coth(x) coth(x) See Also ======== sinh, cosh, acoth r0cCs,|dkrdt|jddSt||dS)Nr0rr;rr3rrrr6sz coth.fdiffcCstSr7)rJr3rrrr9sz coth.inversecCsddlm}t|}|jrl|tjur*tjS|tjur:tjS|tjurJtj S|j rVtj S|j rh||  SnD|tj ur|tjS| tj}|dur|rtj|| Stj ||S|r||  S|jrt|\}}|rt|tjtj}|tj urt|St|S|j r$tj S|jtkrN|jd}td|d|S|jtkr|jd}|t|dt|dS|jtkrd|jdS|jtkr|jdSdS)Nr)cotr0r;)r<rrr=rr>r?r#r@rrArCrBrDr"rErFr.rr!rrGr8r2rrHrIrJ)rKr(rrLrMrNZcothmrrrrOsR               z coth.evalcGszddlm}|dkr dt|S|dks4|ddkr:tjSt|}||d}t|d}d|d||||SdS)Nrrr0r;rrrrr&r rTrMrUrrrrrrrV:s    zcoth.taylor_termcCs||jdSrWrXrZrrrr[Jszcoth._eval_conjugateTcKsddlm}m}|jdjrJ|r@d|d<|j|fi|tjfS|tjfS|rp|jdj|fi|\}}n|jd\}}t |d||d}t |t |||| |||fSrr_rrrrrbMs  "zcoth.as_real_imagNcKs$t| t|}}||||Srrvrrrrrz\szcoth._eval_rewrite_as_tractablecKs$t| t|}}||||Srrvrrrrr|`szcoth._eval_rewrite_as_expcKs(tj ttjtjd|t|Srurr{rrrrdszcoth._eval_rewrite_as_sinhcKs(tj t|ttjtjd|Srur}r{rrrr~gszcoth._eval_rewrite_as_coshcKs dt|Srrr{rrrrjszcoth._eval_rewrite_as_tanhcCs|jdjr|jdjSdSrWrrZrrrrms zcoth._eval_is_positivecCs|jdjr|jdjSdSrWrrZrrrrqs zcoth._eval_is_negativercCsLddlm}|jd|}||jvr>|d||r>d|S||SdSrrrrrrrus  zcoth._eval_as_leading_termc Ks$|jd}|jrddlm}dd|jD}ggg}t|j}t|ddD] }|||d|||qJt|dt|dS|jrddl m }|j d d \} } | j r| dkrt | d d } ggg}t| ddD](}|| |d|| || |qt|dt|dSt |S) NrrcSsg|]}t|ddqSr)rrtrrrrrz*coth._eval_expand_trig..rr;r0)binomialTrmFr)r2rFrrrRrrr r$(sympy.functions.combinatorial.factorialsrrnror) r4rer(rZCXr+rTrrrrrMcrrrrt~s&     &zcoth._eval_expand_trig)r0)r0)T)N)Nr)rrrr r6r9rrOrrrVr[rbrzr|rr~rrrrrtrrrrrs&   7    rc@seZdZdZdZdZdZeddZddZ ddZ d d Z d d Z d#d dZ ddZddZd$ddZddZd%ddZddZd&ddZdd Zd!d"ZdS)'ReciprocalHyperbolicFunctionz=Base class for reciprocal functions of hyperbolic functions. NcCsj|r*|jr|| S|jr*||  S|j|}t|drV||krV|jdS|durfd|S|S)Nr9rr0)rE_is_even_is_odd_reciprocal_ofrOhasattrr9r2)rKr(trrrrOs    z!ReciprocalHyperbolicFunction.evalcOs$||jd}t|||i|SrW)rr2getattr)r4 method_namer2ryorrr_call_reciprocalsz-ReciprocalHyperbolicFunction._call_reciprocalcOs,|j|g|Ri|}|dur(d|S|Sr)r)r4rr2ryrrrr_calculate_reciprocalsz2ReciprocalHyperbolicFunction._calculate_reciprocalcCs.|||}|dur*|||kr*d|SdSr)rr)r4rr(rrrr_rewrite_reciprocals z0ReciprocalHyperbolicFunction._rewrite_reciprocalcKs |d|S)Nr|rr{rrrr|sz1ReciprocalHyperbolicFunction._eval_rewrite_as_expcKs |d|S)Nrzrrwrrrrzsz7ReciprocalHyperbolicFunction._eval_rewrite_as_tractablecKs |d|S)Nrrr{rrrrsz2ReciprocalHyperbolicFunction._eval_rewrite_as_tanhcKs |d|S)Nrrr{rrrrsz2ReciprocalHyperbolicFunction._eval_rewrite_as_cothTcKs"d||jdj|fi|Sr)rr2rb)r4rdrerrrrbsz)ReciprocalHyperbolicFunction.as_real_imagcCs||jdSrWrXrZrrrr[sz,ReciprocalHyperbolicFunction._eval_conjugatecKs&|jfddi|\}}|tj|S)NrdTrirjrrrrksz1ReciprocalHyperbolicFunction._eval_expand_complexcKs|jdi|S)Nrt)rt)r)r4rerrrrtsz.ReciprocalHyperbolicFunction._eval_expand_trigrcCsd||jd|Sr)rr2r)r4rMrrrrrrsz2ReciprocalHyperbolicFunction._eval_as_leading_termcCs||jdjSrW)rr2r`rZrrrrsz3ReciprocalHyperbolicFunction._eval_is_extended_realcCsd||jdjSr)rr2rrZrrrrsz,ReciprocalHyperbolicFunction._eval_is_finite)N)T)T)Nr)rrrr rrrrrOrrrr|rzrrrbr[rkrtrrrrrrrrs(     rc@sJeZdZdZeZdZdddZee ddZ dd Z d d Z d d Z dS)cscha8 ``csch(x)`` is the hyperbolic cosecant of ``x``. The hyperbolic cosecant function is $\frac{2}{e^x - e^{-x}}$ Examples ======== >>> from sympy import csch >>> from sympy.abc import x >>> csch(x) csch(x) See Also ======== sinh, cosh, tanh, sech, asinh, acosh Tr0cCs4|dkr&t|jd t|jdSt||dS)z? Returns the first derivative of this function r0rN)rr2rr r3rrrr6sz csch.fdiffcGs~ddlm}|dkr dt|S|dks4|ddkr:tjSt|}||d}t|d}ddd|||||SdS)zF Returns the next term in the Taylor series expansion rrr0r;NrrrrrrVs    zcsch.taylor_termcKstjt|tjtjdSrur}r{rrrr~szcsch._eval_rewrite_as_coshcCs|jdjr|jdjSdSrWrrZrrrrs zcsch._eval_is_positivecCs|jdjr|jdjSdSrWrrZrrrrs zcsch._eval_is_negativeN)r0)rrrr r/rrr6rrrVr~rrrrrrrs  rc@sBeZdZdZeZdZd ddZee ddZ dd Z d d Z d S)secha: ``sech(x)`` is the hyperbolic secant of ``x``. The hyperbolic secant function is $\frac{2}{e^x + e^{-x}}$ Examples ======== >>> from sympy import sech >>> from sympy.abc import x >>> sech(x) sech(x) See Also ======== sinh, cosh, tanh, coth, csch, asinh, acosh Tr0cCs4|dkr&t|jd t|jdSt||dSr)rr2rr r3rrrr64sz sech.fdiffcGsJddlm}|dks |ddkr&tjSt|}||t|||SdS)Nr)eulerr;r0)rrrr&rr )rTrMrUrrrrrV:s  zsech.taylor_termcKstjt|tjtjdSrurr{rrrrDszsech._eval_rewrite_as_sinhcCs|jdjrdSdSrrrZrrrrGs zsech._eval_is_positiveN)r0) rrrr r1rrr6rrrVrrrrrrrs  rc@seZdZdZdS)InverseHyperbolicFunctionz,Base class for inverse hyperbolic functions.N)rrrr rrrrrPsrc@sZeZdZdZdddZeddZeeddZ dd d Z d dZ dddZ ddZ d S)r8aM ``asinh(x)`` is the inverse hyperbolic sine of ``x``. The inverse hyperbolic sine function. Examples ======== >>> from sympy import asinh >>> from sympy.abc import x >>> asinh(x).diff(x) 1/sqrt(x**2 + 1) >>> asinh(1) log(1 + sqrt(2)) See Also ======== acosh, atanh, sinh r0cCs0|dkr"dt|jdddSt||dSrrr2r r3rrrr6lsz asinh.fdiffc Csddlm}t|}|jr|tjur*tjS|tjur:tjS|tjurJtjS|jrVtj S|tj urpt t ddS|tj urt t ddS|jr||  SnR|tjurtjS|jrtj S|tj}|durtj||S|r||  St|tr|jdjr|jd}|jr |St|\}}|dur|durt|tdt}|tt|}|j} | durx|S| dur| SdS)Nr)asinr;r0TF)r<rrr=rr>r?r@rAr&r#rrrrBrCrDr"rE isinstancer/r2rrrrrris_even) rKr(rrLrrrfrNevenrrrrOrsN             z asinh.evalcGs|dks|ddkrtjSt|}t|dkrd|dkrd|d}| |dd||d|dS|dd}ttj|}t|}tj||||||SdSNrr;rPr0)rr&rrRr r'r rrTrMrUr+rRrrrrrVs&  zasinh.taylor_termNrcCsHddlm}|jd|}||jvr:|d||r:|S||SdSrrrrrrrs  zasinh._eval_as_leading_termcKst|t|ddSrrrr4rMryrrr_eval_rewrite_as_logszasinh._eval_rewrite_as_logcCstSr7)r/r3rrrr9sz asinh.inversecCs |jdjSrWrrZrrrrszasinh._eval_is_zero)r0)Nr)r0rrrr r6rrOrrrVrrr9rrrrrr8Vs  0   r8c@sZeZdZdZdddZeddZeeddZ dd d Z d dZ dddZ ddZ d S)rHaB ``acosh(x)`` is the inverse hyperbolic cosine of ``x``. The inverse hyperbolic cosine function. Examples ======== >>> from sympy import acosh >>> from sympy.abc import x >>> acosh(x).diff(x) 1/sqrt(x**2 - 1) >>> acosh(1) 0 See Also ======== asinh, atanh, cosh r0cCs0|dkr"dt|jdddSt||dSrrr3rrrr6sz acosh.fdiffc +Cst|}|jrz|tjurtjS|tjur.tjS|tjur>tjS|jrTtjtjdS|tj urdtj S|tj urztjtjS|j rtjt tjdtdtj t tj dtdtjtjdtddtjtddtddtjdtd dtjtdddtdtjddtdtjtddtddtjdtd dtjtddtddtdtjtdd tdd tdtjtd d tdtddtjdtdtd dtjtd dtdtddtjtddtdtd dtjtdddtddtdtjd dtd dtdtjtd d tdddtjdtdd dtjtddi}||vr|jr||tjS||S|tjurtjS|tjtjkrtjtjtjdS|tj tjkr"tjtjtjdS|jrr?r@rAr!r"r#r&rrrrr'rr`rCrr1r2rZ$sympy.functions.elementary.complexesrrrrrrZis_nonnegativerBZis_nonpositiver) rKr( cst_tablerrrrrrNrrrrrOs        "" "&         z acosh.evalcGs|dkrtjtjdS|dks,|ddkr2tjSt|}t|dkrz|dkrz|d}||dd||d|dS|dd}ttj|}t|}| |tj|||SdSr) rr!r"r&rrRr r'r rrrrrV4s$  zacosh.taylor_termNrcCsTddlm}|jd|}||jvrF|d||rFtjtjdS| |SdSNrrr0r; rrr2rrrrr"r!rGrrrrrFs  zacosh._eval_as_leading_termcKs t|t|dt|dSrrrrrrrOszacosh._eval_rewrite_as_logcCstSr7)r1r3rrrr9Rsz acosh.inversecCs|jddjrdSdS)Nrr0TrrZrrrrXszacosh._eval_is_zero)r0)Nr)r0rrrrrrHs  N   rHc@sbeZdZdZdddZeddZeeddZ dd d Z d dZ ddZ ddZ dddZd S)rIa) ``atanh(x)`` is the inverse hyperbolic tangent of ``x``. The inverse hyperbolic tangent function. Examples ======== >>> from sympy import atanh >>> from sympy.abc import x >>> atanh(x).diff(x) 1/(1 - x**2) See Also ======== asinh, acosh, tanh r0cCs,|dkrdd|jddSt||dSrr2r r3rrrr6qsz atanh.fdiffc Csddlm}t|}|jr|tjur*tjS|jr6tjS|tjurFtj S|tj urVtj S|tj urptj ||S|tj urtj || S|j r||  Snn|tjurddlm}tj |tj dtjdS|tj }|durtj ||S|r ||  S|jrtjSt|tr|jdjr|jd}|jrJ|St|\}}|dur|durtd|t}|j} |t|td} | dur| S| dur| ttdSdS)Nr)atan AccumBoundsr;TF)r<rrr=rr>rAr&r#r?rr@r"rBrC!sympy.calculus.accumulationboundsrr!rDrErrr2rrrrrrr) rKr(rrrLrrrrrrNrrrrOwsP                z atanh.evalcGs2|dks|ddkrtjSt|}|||SdSNrr;)rr&rrTrMrUrrrrVszatanh.taylor_termNrcCsHddlm}|jd|}||jvr:|d||r:|S||SdSrrrrrrrs  zatanh._eval_as_leading_termcKstd|td|dSNr0r;rrrrrrszatanh._eval_rewrite_as_logcCs|jdjrdSdSrrrZrrrrs zatanh._eval_is_zerocCs |jdjSrW)r2rrZrrr_eval_is_imaginaryszatanh._eval_is_imaginarycCstSr7rr3rrrr9sz atanh.inverse)r0)Nr)r0)rrrr r6rrOrrrVrrrrr9rrrrrI]s  1  rIc@sReZdZdZdddZeddZeeddZ dd d Z d dZ dddZ d S)rJa- ``acoth(x)`` is the inverse hyperbolic cotangent of ``x``. The inverse hyperbolic cotangent function. Examples ======== >>> from sympy import acoth >>> from sympy.abc import x >>> acoth(x).diff(x) 1/(1 - x**2) See Also ======== asinh, acosh, coth r0cCs,|dkrdd|jddSt||dSrrr3rrrr6sz acoth.fdiffcCsddlm}t|}|jr|tjur*tjS|tjur:tjS|tjurJtjS|j r`tj tj dS|tj urptjS|tj urtjS|jr||  SnH|tjurtjS|tj }|durtj ||S|r||  S|j rtj tj tjSdS)Nr)acotr;)r<rrr=rr>r?r&r@rAr!r"r#rrBrCrDrEr')rKr(rrLrrrrOs4         z acoth.evalcGsJ|dkrtjtjdS|dks,|ddkr2tjSt|}|||SdSr)rr!r"r&rrrrrrV s zacoth.taylor_termNrcCsTddlm}|jd|}||jvrF|d||rFtjtjdS| |SdSrrrrrrrs  zacoth._eval_as_leading_termcKs$tdd|tdd|dSrrrrrrrszacoth._eval_rewrite_as_logcCstSr7rr3rrrr9 sz acoth.inverse)r0)Nr)r0) rrrr r6rrOrrrVrrr9rrrrrJs  " rJc@sHeZdZdZdddZeddZeeddZ dd d Z d d Z d S)asecha ``asech(x)`` is the inverse hyperbolic secant of ``x``. The inverse hyperbolic secant function. Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(1 - x**2)) >>> asech(1).diff(x) 0 >>> asech(1) 0 >>> asech(S(2)) I*pi/3 >>> asech(-sqrt(2)) 3*I*pi/4 >>> asech((sqrt(6) - sqrt(2))) I*pi/12 See Also ======== asinh, atanh, cosh, acoth References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] http://dlmf.nist.gov/4.37 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/ r0cCs8|dkr*|jd}d|td|dSt||dSNr0rrr;r2rr r4r5rrrrr6Ms z asech.fdiffc1Cs\t|}|jr|tjurtjS|tjur8tjtjdS|tjurRtjtjdS|jr^tjS|tj urntj S|tj urtjtjS|j rtjtjtjd t dtdtj tjtjdt dtdtdtdtjdtdtddtjdtddtdtjdtddtd dtjddtdtdtjd d tdtdd tjd dtd tjdd td dtjdtddtjddtdd tjdtdtjd td d tjd tddtdd tjdtddtd d tjdtdtjd td dtjd tddtdd tjd tddtd dtjd dtddtjddtdd tjdtdtddtjdtd tdd tjdi}||vr|jr ||tjS||S|tjurJddlm}tj|tj dtjdS|jrXtjSdS)Nr;r0r r rr  r rPr rrrrr)rr=rr>r?r!r"r@rAr#r&rrrrr`rCrr)rKr(rrrrrrOTsb      $$      z asech.evalcGs|dkrtd|S|dks(|ddkr.tjSt|}t|dkrz|dkrz|d}||dd|dd|ddS|d}ttj||}t||d|d}d||||dSdS)Nrr;r0rPrr)rrr&rrRr r'r rrrrexpansion_terms (zasech.expansion_termcCstSr7)rr3rrrr9sz asech.inversecKs,td|td|dtd|dSrrr{rrrrszasech._eval_rewrite_as_logN)r0)r0) rrrr r6rrOrrr$r9rrrrrr's%  9  rc@s@eZdZdZdddZeddZdddZd d Zd d Z d S)acscha ``acsch(x)`` is the inverse hyperbolic cosecant of ``x``. The inverse hyperbolic cosecant function. Examples ======== >>> from sympy import acsch, sqrt, S >>> from sympy.abc import x >>> acsch(x).diff(x) -1/(x**2*sqrt(1 + x**(-2))) >>> acsch(1).diff(x) 0 >>> acsch(1) log(1 + sqrt(2)) >>> acsch(S.ImaginaryUnit) -I*pi/2 >>> acsch(-2*S.ImaginaryUnit) I*pi/6 >>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2))) -5*I*pi/12 See Also ======== asinh References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] http://dlmf.nist.gov/4.37 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/ r0cCs@|dkr2|jd}d|dtdd|dSt||dSrr r!rrrr6s  z acsch.fdiffcCsNt|}|jr|tjurtjS|tjur.tjS|tjur>tjS|jrJtjS|tj urdt dt dS|tj urt dt d S|j rtjtj dtjt dt dtj dtjdt dtj dtjdt dt dtj dtjdtj dtjt ddt dtj dtjt dtj dtjt ddd tjdtjdt d tj d tjdt dt dd tjdtjt ddt dd tjdtjt dt dd tjdtdtj t dt ddi }||vr||tjS|tjur&tjS|jr4tjS|rJ||  SdS) Nr0r;r r r r"r rrrP)rr=rr>r?r&r@rArCr#rrrrr"r!rE)rKr(rrrrrOsJ     ""$$     z acsch.evalcCstSr7)rr3rrrr9sz acsch.inversecKs td|td|ddSrrr{rrrr szacsch._eval_rewrite_as_logcCs |jdjSrW)r2 is_infiniterZrrrrszacsch._eval_is_zeroN)r0)r0) rrrr r6rrOr9rrrrrrr%s%  . r%N),Zsympy.core.logicrZ sympy.corerrrrrrZsympy.core.addr Zsympy.core.functionr r rr r Z&sympy.functions.elementary.exponentialrrrZ(sympy.functions.elementary.miscellaneousrZ#sympy.functions.elementary.integersrrrrrr.r/r1rrrrrrr8rHrIrJrr%rrrrs>       "Qq\3J?3so[