o GZh@s8ddlmZmZmZddlmZddlmZmZddl m Z m Z m Z m Z ddlmZmZmZddlmZddlmZmZmZddlmZmZmZdd lmZmZmZdd l m!Z!m"Z"m#Z#dd l$m%Z%dd l&m'Z'dd l(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3ddl4m5Z5ddZ6eddZ7eddZ8eddZ9GdddeZ:ddZ;Gddde:ZGd!d"d"e:Z?Gd#d$d$e:Z@Gd%d&d&e@ZAGd'd(d(e@ZBGd)d*d*eZCGd+d,d,eCZDGd-d.d.eCZEGd/d0d0eCZFGd1d2d2eCZGGd3d4d4eCZHGd5d6d6eCZId7S)8)Ssympifycacheit)Add)DefinedFunctionArgumentIndexError)fuzzy_or fuzzy_and fuzzy_not FuzzyBool)IpiRational)Dummy)binomial factorialRisingFactorial) bernoullieulernC)Absimre)explogmatch_real_imag)floor)sqrt) acosacotasinatancoscotcscsecsintan_imaginary_unit_as_coefficient)symmetric_polycCs|dd|tDS)NcSsi|]}||tqS)rewriter).0hr*r*T/var/www/auris/lib/python3.10/site-packages/sympy/functions/elementary/hyperbolic.py sz/_rewrite_hyperbolics_as_exp..)ZxreplaceZatomsHyperbolicFunction)exprr*r*r._rewrite_hyperbolics_as_exps r2c Csitttdtdt tt dtdtjtdtddttddtddtdtd dttdddtdtddtdttddtddtdtd dttddtddtdttdd tdd tdttd d tdtddtdtdtd dttd dtdtddttddtdtd dttdddtddtdtd dtd dtdttd d tdddtdtdd dttddiS) N  )r rrrHalfr rr*r*r*r. _acosh_tablesN        $r?cCsBtt dttdtdt dtdtdt dtdtdtdt dtdt dttddtdt dttdt dttddd tdtdtd t d tdtdtdd tdttddtdd tdttdtdd tdtdt tdtddi S) Nr4r8r;r3r9 r:r7r5)r r rrrr*r*r*r. _acsch_table3s   rDcCstitttd tdtdt ttdtdtdtdtdtdtdtddtdtddtdtdtddtd dtddtdtdtd d tdtdd td dtd tdd td dtdtddtddtdd tdtdtd td d td tddtdd tdtddtd d tdtdtd td dtd tddtdd td tddtd dtd dtddtddtdd tdtdtddtdtd tdd tdttjt tdttjttdi S)Nr4r3r8r;r=r9r@ r:rBr<r5r7r6)r r rrrInfinityNegativeInfinityr*r*r*r. _asech_tableFsZ      rHc@seZdZdZdZdS)r0ze Base class for hyperbolic functions. See Also ======== sinh, cosh, tanh, coth TN)__name__ __module__ __qualname____doc__Z unbranchedr*r*r*r.r0js r0cCsztt}t|D]}||krtj}n|jr&|\}}||kr&|jr&nq |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) ) r r rZ make_argsrOneis_Mul as_two_termsZ is_RationalZeror>)argZipiaKpm1m2r*r*r. _peeloff_ipiws   rWc@seZdZdZd3ddZd3ddZeddZee d d Z d d Z d4ddZ d4ddZ d4ddZd5ddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2ZdS)6sinha ``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 r3cC |dkr t|jdSt||)z@ Returns the first derivative of this function. r3r)coshargsrselfargindexr*r*r.fdiffs z sinh.fdiffcCtSz7 Returns the inverse of this function. asinhr\r*r*r.inversez sinh.inversecCs|jr,|tjur tjS|tjurtjS|tjurtjS|jr!tjS|jr*||  SdS|tjur4tjSt |}|durBt t |S| rL||  S|j rmt|\}}|rm|tt }t|t|t|t|S|jrstjS|jtkr}|jdS|jtkr|jd}t|dt|dS|jtkr|jd}|td|dS|jtkr|jd}dt|dt|dSdSNrr3r4) is_NumberrNaNrFrGis_zerorP is_negativeComplexInfinityr(r r&could_extract_minus_signis_AddrWr rXrZfuncrcr[acoshratanhacoth)clsrQi_coeffxmr*r*r.evalsL                  z sinh.evalcGs^|dks |ddkr tjSt|}t|dkr'|d}||d||dS||t|S)zG Returns the next term in the Taylor series expansion. rr4rBr3rrPrlenrnrtprevious_termsrTr*r*r. taylor_terms zsinh.taylor_termcC||jdSNrrnr[ conjugater]r*r*r._eval_conjugatezsinh._eval_conjugateTcK|jdjr|rd|d<|j|fi|tjfS|tjfS|r0|jdj|fi|\}}n |jd\}}t|t|t|t |fS)z@ Returns this function as a complex coordinate. rFcomplex r[is_extended_realexpandrrP as_real_imagrXr"rZr&r]deephintsrrr*r*r.rs  " zsinh.as_real_imagcK$|jdd|i|\}}||tSNrr*rr r]rrZre_partZim_partr*r*r._eval_expand_complex zsinh._eval_expand_complexcKs|r|jdj|fi|}n|jd}d}|jr |\}}n|jdd\}}|tjur=|jr=|tjur=|}|d|}|durUt|t |t|t |jddSt|SNrTZrationalr3)Ztrig) r[rrmrO as_coeff_MulrrM is_IntegerrXrZr]rrrQrtycoefftermsr*r*r._eval_expand_trig  (zsinh._eval_expand_trigNcKt|t| dSNr4rr]rQlimitvarkwargsr*r*r._eval_rewrite_as_tractable&zsinh._eval_rewrite_as_tractablecKrrrr]rQrr*r*r._eval_rewrite_as_exp)rzsinh._eval_rewrite_as_expcKst tt|SNr r&rr*r*r._eval_rewrite_as_sin,zsinh._eval_rewrite_as_sincKst tt|Srr r$rr*r*r._eval_rewrite_as_csc/rzsinh._eval_rewrite_as_csccKst t|ttdSrr rZr rr*r*r._eval_rewrite_as_cosh2szsinh._eval_rewrite_as_coshcKs"ttj|}d|d|dSNr4r3tanhrr>r]rQrZ tanh_halfr*r*r._eval_rewrite_as_tanh5zsinh._eval_rewrite_as_tanhcKs"ttj|}d||ddSrcothrr>r]rQrZ coth_halfr*r*r._eval_rewrite_as_coth9rzsinh._eval_rewrite_as_cothcK dt|SNr3cschrr*r*r._eval_rewrite_as_csch= zsinh._eval_rewrite_as_cschcCsd|jdj|||d}||d}|tjur#|j|d|jrdndd}|jr(|S|jr0| |S|SNrlogxcdir-+)dir) r[as_leading_termsubsrrhlimitrjri is_finiternr]rtrrrQarg0r*r*r._eval_as_leading_term@s   zsinh._eval_as_leading_termcCs*|jd}|jr dS|\}}|tjSNrTr[is_realrr rir]rQrrr*r*r. _eval_is_realMs   zsinh._eval_is_realcC|jdjrdSdSrr[rrr*r*r._eval_is_extended_realW zsinh._eval_is_extended_realcC|jdjr |jdjSdSr~r[r is_positiverr*r*r._eval_is_positive[  zsinh._eval_is_positivecCrr~r[rrjrr*r*r._eval_is_negative_rzsinh._eval_is_negativecC|jd}|jSr~r[rr]rQr*r*r._eval_is_finitec zsinh._eval_is_finitecCs"t|jd\}}|jr|jSdSr~)rWr[ri is_integerr]restZipi_multr*r*r. _eval_is_zerogszsinh._eval_is_zeror3Tr)rIrJrKrLr_rd classmethodrv staticmethodrr|rrrrrrrrrrrrrrrrrrrr*r*r*r.rXs8   0        rXc@seZdZdZd/ddZeddZeeddZ d d Z d0d d Z d0ddZ d0ddZ d1ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.ZdS)2rZa" ``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 r3cCrYNr3rrXr[rr\r*r*r.r_s z cosh.fdiffcCsvddlm}|jr1|tjurtjS|tjurtjS|tjur!tjS|jr'tjS|j r/|| SdS|tj ur9tjSt |}|durE||S| rN|| S|j rot|\}}|ro|tt}t|t|t|t|S|jrutjS|jtkrtd|jddS|jtkr|jdS|jtkrdtd|jddS|jtkr|jd}|t|dt|dSdS)Nr)r"r3r4)(sympy.functions.elementary.trigonometricr"rgrrhrFrGrirMrjrkr(rlrmrWr r rZrXrnrcrr[rorprq)rrrQr"rsrtrur*r*r.rvsJ                z cosh.evalcGs^|dks |ddkr tjSt|}t|dkr'|d}||d||dS||t|S)Nrr4r3rBrwryr*r*r.r|s zcosh.taylor_termcCr}r~rrr*r*r.rrzcosh._eval_conjugateTcKr)NrFr) r[rrrrPrrZr"rXr&rr*r*r.rs  " zcosh.as_real_imagcKrrrrr*r*r.rrzcosh._eval_expand_complexcKs|r|jdj|fi|}n|jd}d}|jr |\}}n|jdd\}}|tjur=|jr=|tjur=|}|d|}|durUt|t|t |t |jddSt|Sr) r[rrmrOrrrMrrZrXrr*r*r.rrzcosh._eval_expand_trigNcKt|t| dSrrrr*r*r.rrzcosh._eval_rewrite_as_tractablecKrrrrr*r*r.rrzcosh._eval_rewrite_as_expcKtt|ddSNFevaluater"r rr*r*r._eval_rewrite_as_coszcosh._eval_rewrite_as_coscKdtt|ddSNr3Frr%r rr*r*r._eval_rewrite_as_secrzcosh._eval_rewrite_as_seccKst t|ttdddSNr4Frr rXr rr*r*r._eval_rewrite_as_sinhszcosh._eval_rewrite_as_sinhcKs"ttj|d}d|d|Srrrr*r*r.rzcosh._eval_rewrite_as_tanhcKs"ttj|d}|d|dSrrrr*r*r.rrzcosh._eval_rewrite_as_cothcKrrsechrr*r*r._eval_rewrite_as_sechrzcosh._eval_rewrite_as_sechcCsf|jdj|||d}||d}|tjur#|j|d|jrdndd}|jr)tjS|j r1| |S|Sr) r[rrrrhrrjrirMrrnrr*r*r.rs   zcosh._eval_as_leading_termcCs0|jd}|js |jr dS|\}}|tjSr)r[r is_imaginaryrr rirr*r*r.rs    zcosh._eval_is_realc Csr|jd}|\}}|dt}|j}|rdS|j}|dur!|St|t|t|tdk|dtdkgggSNrr4TFr5r[rr rirr r]zrtrZymodZyzeroZxzeror*r*r.rs   zcosh._eval_is_positivec Csr|jd}|\}}|dt}|j}|rdS|j}|dur!|St|t|t|tdk|dtdkgggSrrrr*r*r._eval_is_nonnegative?s   zcosh._eval_is_nonnegativecCrr~rrr*r*r.rYrzcosh._eval_is_finitecCs0t|jd\}}|r|jr|tjjSdSdSr~)rWr[rirr>rrr*r*r.r]s  zcosh._eval_is_zerorrr)rIrJrKrLr_rrvrrr|rrrrrrrrrrrrrrrrrrr*r*r*r.rZms4  /         rZc@seZdZdZd/ddZd/ddZeddZee d d Z d d Z d0ddZ ddZ d1ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.ZdS)2ra' ``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 r3cCs*|dkrtjt|jddSt||Nr3rr4)rrMrr[rr\r*r*r.r_ws z tanh.fdiffcCr`rarpr\r*r*r.rd}rez tanh.inversecCs|jr,|tjur tjS|tjurtjS|tjurtjS|jr!tjS|j r*||  SdS|tj ur4tjSt |}|durN| rHt t| St t|S| rX||  S|jrxt|\}}|rxt|tt }|tj urtt|St|S|jr~tjS|jtkr|jd}|td|dS|jtkr|jd}t|dt|d|S|jtkr|jdS|jtkrd|jdSdSrf)rgrrhrFrMrG NegativeOnerirPrjrkr(rlr r'rmrWrr rrnrcr[rrorprq)rrrQrsrtruZtanhmr*r*r.rvsR                z tanh.evalcGsb|dks |ddkr tjSt|}d|d}t|d}t|d}||d||||SNrr4r3)rrPrrr)rzrtr{rRBFr*r*r.r|s   ztanh.taylor_termcCr}r~rrr*r*r.rrztanh._eval_conjugateTcKs|jdjr|rd|d<|j|fi|tjfS|tjfS|r0|jdj|fi|\}}n |jd\}}t|dt|d}t|t||t |t||fS)NrFrr4r)r]rrrrdenomr*r*r.rs  "(ztanh.as_real_imagc  s|jd}|jr7t|j}dd|jD}ddg}t|dD]}||dt||7<q|d|dS|jrs|\}jrsdkrst|fddtdddD}fddtdddD}t |t |St|S)NrcSg|] }t|ddqSFr)rrr,rtr*r*r. sz*tanh._eval_expand_trig..r3r4c"g|] }tt||qSr*rranger,kTrr*r.r"crr*rrrr*r.rr) r[rmrxrr)rNrrrr) r]rrQrzZTXrTirdr*rr.rs$     ztanh._eval_expand_trigNcKs$t| t|}}||||Srrr]rQrrneg_exppos_expr*r*r.rztanh._eval_rewrite_as_tractablecKs$t| t|}}||||Srrr]rQrrrr*r*r.rrztanh._eval_rewrite_as_expcKt tt|ddSr)r r'rr*r*r._eval_rewrite_as_tanrztanh._eval_rewrite_as_tancKst tt|ddSr)r r#rr*r*r._eval_rewrite_as_cotrztanh._eval_rewrite_as_cotcKs$tt|tttd|ddSrrrr*r*r.r$ztanh._eval_rewrite_as_sinhcKs$ttttd|ddt|Srrrr*r*r.rr#ztanh._eval_rewrite_as_coshcKrrrrr*r*r.rrztanh._eval_rewrite_as_cothcCsDddlm}|jd|}||jvr|d||r|S||SNr)Orderr3sympy.series.orderr&r[rZ free_symbolscontainsrnr]rtrrr&rQr*r*r.rs  ztanh._eval_as_leading_termcCsJ|jd}|jr dS|\}}|dkr|ttdkrdS|tdjS)NrTr4rrr*r*r.r s  ztanh._eval_is_realcCrrrrr*r*r.rrztanh._eval_is_extended_realcCrr~rrr*r*r.rrztanh._eval_is_positivecCrr~rrr*r*r.r"rztanh._eval_is_negativecCsR|jd}|\}}t|dt|d}|dkrdS|jr"dS|jr'dSdS)Nrr4FT)r[rr"rX is_numberr)r]rQrrr r*r*r.r&s  ztanh._eval_is_finitecCs|jd}|jr dSdSrr[rirr*r*r.r2s ztanh._eval_is_zerorrr)rIrJrKrLr_rdrrvrrr|rrrrrr!r"rrrrrrrrrrr*r*r*r.rcs4   4     rc@seZdZdZd#ddZd#ddZeddZee d d Z d d Z d$ddZ d%ddZ ddZddZddZddZddZddZdd Zd!d"ZdS)&ra+ ``coth(x)`` is the hyperbolic cotangent of ``x``. The hyperbolic cotangent function is $\frac{\cosh(x)}{\sinh(x)}$. Examples ======== >>> from sympy import coth >>> from sympy.abc import x >>> coth(x) coth(x) See Also ======== sinh, cosh, acoth r3cCs(|dkrdt|jddSt||)Nr3r6rr4rr\r*r*r.r_L z coth.fdiffcCr`ra)rqr\r*r*r.rdRrez coth.inversecCs|jr,|tjur tjS|tjurtjS|tjurtjS|jr!tjS|j r*||  SdS|tjur4tjSt |}|durN| rGt t | St t |S| rX||  S|jrxt|\}}|rxt|tt }|tjurtt|St|S|jr~tjS|jtkr|jd}td|d|S|jtkr|jd}|t|dt|dS|jtkrd|jdS|jtkr|jdSdSrf)rgrrhrFrMrGrrirkrjr(rlr r#rmrWrr rrnrcr[rrorprq)rrrQrsrtruZcothmr*r*r.rvXsR               z coth.evalcGsj|dkr dt|S|dks|ddkrtjSt|}t|d}t|d}d|d||||SrfrrrPrrrzrtr{r r r*r*r.r|s   zcoth.taylor_termcCr}r~rrr*r*r.rrzcoth._eval_conjugateTcKsddlm}m}|jdjr%|r d|d<|j|fi|tjfS|tjfS|r8|jdj|fi|\}}n |jd\}}t |d||d}t |t |||| |||fS)Nr)r"r&Frr4) rr"r&r[rrrrPrrXrZ)r]rrr"r&rrr r*r*r.rs  "*zcoth.as_real_imagNcKs$t| t|}}||||Srrrr*r*r.rrzcoth._eval_rewrite_as_tractablecKs$t| t|}}||||Srrrr*r*r.rrzcoth._eval_rewrite_as_expcKs&t tttd|ddt|Srrrr*r*r.r&zcoth._eval_rewrite_as_sinhcKs&t t|tttd|ddSrrrr*r*r.rr0zcoth._eval_rewrite_as_coshcKrrrrr*r*r.rrzcoth._eval_rewrite_as_tanhcCrr~rrr*r*r.rrzcoth._eval_is_positivecCrr~rrr*r*r.rrzcoth._eval_is_negativecCsHddlm}|jd|}||jvr|d||rd|S||Sr%r'r*r*r*r.rs  zcoth._eval_as_leading_termc Ks|jd}|jr.r6r4r3TrFr) r[rmrxrappendr)rrNrrrr) r]rrQZCXrTrzrrrtcr*r*r.rs"   &zcoth._eval_expand_trigrrr)rIrJrKrLr_rdrrvrrr|rrrrrrrrrrrr*r*r*r.r8s(   4    rc@seZdZUdZdZdZeed<dZeed<e ddZ ddZ d d Z d d Z d dZd$ddZddZddZd%ddZddZd%ddZddZddZd d!Zd"d#ZdS)&ReciprocalHyperbolicFunctionz=Base class for reciprocal functions of hyperbolic functions. N_is_even_is_oddcCsj|r|jr || S|jr||  S|j|}t|dr+||kr+|jdS|dur3d|S|S)Nrdrr3)rlr5r6_reciprocal_ofrvhasattrrdr[)rrrQtr*r*r.rvs    z!ReciprocalHyperbolicFunction.evalcOs$||jd}t|||i|Sr~)r7r[getattr)r] method_namer[ror*r*r._call_reciprocalsz-ReciprocalHyperbolicFunction._call_reciprocalcOs,|j|g|Ri|}|durd|S|Sr)r=)r]r;r[rr9r*r*r._calculate_reciprocalsz2ReciprocalHyperbolicFunction._calculate_reciprocalcCs2|||}|dur|||krd|SdSdSr)r=r7)r]r;rQr9r*r*r._rewrite_reciprocals z0ReciprocalHyperbolicFunction._rewrite_reciprocalcK |d|S)Nrr?rr*r*r.r rz1ReciprocalHyperbolicFunction._eval_rewrite_as_expcKr@)NrrArr*r*r.rrz7ReciprocalHyperbolicFunction._eval_rewrite_as_tractablecKr@)NrrArr*r*r.rrz2ReciprocalHyperbolicFunction._eval_rewrite_as_tanhcKr@)NrrArr*r*r.rrz2ReciprocalHyperbolicFunction._eval_rewrite_as_cothTcKs"d||jdj|fi|Sr)r7r[r)r]rrr*r*r.rs"z)ReciprocalHyperbolicFunction.as_real_imagcCr}r~rrr*r*r.rrz,ReciprocalHyperbolicFunction._eval_conjugatecKs$|jdddi|\}}|t|S)NrTr*rrr*r*r.rrz1ReciprocalHyperbolicFunction._eval_expand_complexcKs|jdi|S)Nr)r)r>)r]rr*r*r.r!rz.ReciprocalHyperbolicFunction._eval_expand_trigcCs d||jdj|||dS)Nr3rr)r7r[r)r]rtrrr*r*r.r$ z2ReciprocalHyperbolicFunction._eval_as_leading_termcCs||jdjSr~)r7r[rrr*r*r.r'rz3ReciprocalHyperbolicFunction._eval_is_extended_realcCsd||jdjSr)r7r[rrr*r*r.r*rz,ReciprocalHyperbolicFunction._eval_is_finiterr)rIrJrKrLr7r5r __annotations__r6rrvr=r>r?rrrrrrrrrrrr*r*r*r.r4s*         r4c@sbeZdZdZeZdZdddZee ddZ dd Z d d Z d d Z ddZddZddZdS)ra8 ``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 Tr3cC0|dkrt|jd t|jdSt||)z? Returns the first derivative of this function r3r)rr[rrr\r*r*r.r_Es z csch.fdiffcGsn|dkr dt|S|dks|ddkrtjSt|}t|d}t|d}ddd|||||S)zF Returns the next term in the Taylor series expansion rr3r4r.r/r*r*r.r|Ns    zcsch.taylor_termcKsttt|ddSrrrr*r*r.r`rzcsch._eval_rewrite_as_sincKsttt|ddSrrrr*r*r.rcrzcsch._eval_rewrite_as_csccKtt|ttdddSrrrr*r*r.rfzcsch._eval_rewrite_as_coshcKrrrXrr*r*r.rirzcsch._eval_rewrite_as_sinhcCrr~rrr*r*r.rlrzcsch._eval_is_positivecCrr~rrr*r*r.rprzcsch._eval_is_negativeNr)rIrJrKrLrXr7r6r_rrr|rrrrrrr*r*r*r.r.s    rc@sZeZdZdZeZdZdddZee ddZ dd Z d d Z d d Z ddZddZdS)ra: ``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 Tr3cCrDr)rr[rrr\r*r*r.r_s z sech.fdiffcGs:|dks |ddkr tjSt|}t|t|||Sr )rrPrrrrzrtr{r*r*r.r|szsech.taylor_termcKrrrrr*r*r.rrzsech._eval_rewrite_as_coscKrrrrr*r*r.rrzsech._eval_rewrite_as_seccKrErrrr*r*r.rrFzsech._eval_rewrite_as_sinhcKrrrZrr*r*r.rrzsech._eval_rewrite_as_coshcCrrrrr*r*r.rrzsech._eval_is_positiveNr)rIrJrKrLrZr7r5r_rrr|rrrrrr*r*r*r.rus   rc@seZdZdZdS)InverseHyperbolicFunctionz,Base class for inverse hyperbolic functions.N)rIrJrKrLr*r*r*r.rJsrJceZdZdZd ddZeddZeeddZ d d Z d!fd d Z ddZ e Z ddZddZddZddZd ddZddZddZddZZS)"rcaM ``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 r3cCs,|dkrdt|jdddSt||r)rr[rr\r*r*r.r_s z asinh.fdiffc Cst|jrE|tjur tjS|tjurtjS|tjurtjS|jr!tjS|tjur.tt ddS|tj ur;tt ddS|j rD||  Sn&|tj urMtj S|jrStjSt |}|duratt|S|rk||  St|tr|jdjr|jd}|jr|St|\}}|dur|durt|tdt}|tt|}|j}|dur|S|dur| SdSdSdSdSdS)Nr4r3rTF)rgrrhrFrGrirPrMrrrrjrkr(r r rl isinstancerXr[r+rrrr is_even) rrrQrsrrrfruevenr*r*r.rvsR           z asinh.evalcGs|dks |ddkr tjSt|}t|dkr2|dkr2|d}| |dd||d|dS|dd}ttj|}t|}tj||||||SNrr4rBr3)rrPrrxrr>rrrzrtr{rTrRr r*r*r.r|s&  zasinh.taylor_termcCs|jd}||d}|jr||S|tjur)|||}|jr'|S|S|t t tj fvr=| t j |||dSd|djr|||rK|nd}t|jrct|jrb|| t tSn t|jrxt|jrw|| t tSn | t j |||dS||SNrrr3r4)r[rcancelrirrrhrnrr rkr+rrrjrrrrr r]rtrrrQZx0r1ndirr*r*r.rs.        zasinh._eval_as_leading_termrc s|jd}||d}|tt fvr|tj||||dStj|||d}|tjur.|Sd|dj rq| ||r<|nd}t |j rRt |j rP| ttS|St |j ret |j rc| ttS|S|tj||||dS|SNrrrzrr3r4)r[rr r+r _eval_nseriessuperrrkrjrrrrr r]rtrzrrrQrresrW __class__r*r.rZ-s&      zasinh._eval_nseriescKst|t|ddSrrrr]rtrr*r*r._eval_rewrite_as_logFzasinh._eval_rewrite_as_logcKst|td|dSNr3r4)rprrar*r*r._eval_rewrite_as_atanhKrczasinh._eval_rewrite_as_atanhcKs4t|}ttd|t|dt|tdSrd)r rror )r]rtrZixr*r*r._eval_rewrite_as_acoshNs,zasinh._eval_rewrite_as_acoshcKr r)r r rar*r*r._eval_rewrite_as_asinRrzasinh._eval_rewrite_as_asincKs ttt|ddttdS)NFrr4)r rr rar*r*r._eval_rewrite_as_acosUrBzasinh._eval_rewrite_as_acoscCr`rarGr\r*r*r.rdXrez asinh.inversecC |jdjSr~r,rr*r*r.r^rzasinh._eval_is_zerocCrir~rrr*r*r.rarzasinh._eval_is_extended_realcCrir~rrr*r*r.rdrzasinh._eval_is_finiterr)rIrJrKrLr_rrvrrr|rrZrbrrerfrgrhrdrrr __classcell__r*r*r^r.rcs(  -  rccrK)"roaM ``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 - 1)*sqrt(x + 1)) >>> acosh(1) 0 See Also ======== asinh, atanh, cosh r3cCs8|dkr|jd}dt|dt|dSt||rr[rr)r]r^rQr*r*r.r_~s  z acosh.fdiffc Cs|jr5|tjur tjS|tjurtjS|tjurtjS|jr$ttdS|tjur,tj S|tj ur5ttS|j rLt }||vrL|j rH||tS||S|tjurTtjS|ttjkrdtjttdS|t tjkrutjttdS|jrtttjSt|tr|jdj r|jd}|jrt|St|\}}|dur|durt|t}|tt|}|j}|dur|jr|S|jr| SdS|dur|tt8}|jr| S|jr|SdSdSdSdSdSdS)Nr4rTF)rgrrhrFrGrir r rMrPrr+r?rrkr>rLrZr[rrrrrMis_nonnegativerjZis_nonpositiver) rrrQ cst_tablerrNrrOrurPr*r*r.rvsh             z acosh.evalcGs|dkr ttdS|dks|ddkrtjSt|}t|dkr;|dkr;|d}||dd||d|dS|dd}ttj|}t|}| |t|||SrQ) r r rrPrrxrr>rrRr*r*r.r|s $  zacosh.taylor_termcCs|jd}||d}|tj tjtjtjfvr%|tj |||dS|tj ur9| | |}|j r7|S|S|djrs|||rE|nd}t|jrc|djr]| |dttS| | St|jss|tj |||dS| |SrT)r[rrUrrMrPrkr+rrrhrnrrrjrrr r rrVr*r*r.rs$        zacosh._eval_as_leading_termrc s|jd}||d}|tjtjfvr|tj||||dStj|||d}|tj ur/|S|dj rd| ||r;|nd}t |j rS|dj rP|dt tS| St |jsd|tj||||dS|SrXr[rrrMrr+rrZr[rkrjrrr r rr\r^r*r.rZs       zacosh._eval_nseriescKs t|t|dt|dSrr`rar*r*r.rbrBzacosh._eval_rewrite_as_logcKs t|dtd|t|Sr)rrrar*r*r.rhrBzacosh._eval_rewrite_as_acoscKs(t|dtd|tdt|Srd)rr r rar*r*r.rg(zacosh._eval_rewrite_as_asincKs4t|dtd|tdttt|ddSNr3r4Fr)rr r rcrar*r*r._eval_rewrite_as_asinh 4zacosh._eval_rewrite_as_asinhcKspt|d}td|}t|dd}td||d|td|d|t|d|t||Srd)rr rp)r]rtrZsxm1Zs1mxZsx2m1r*r*r.re s  &zacosh._eval_rewrite_as_atanhcCr`rarIr\r*r*r.rdrez acosh.inversecCs|jddjr dSdS)Nrr3Tr,rr*r*r.rszacosh._eval_is_zerocCs t|jdj|jddjgSNrr3)r r[ris_extended_nonnegativerr*r*r.rrBzacosh._eval_is_extended_realcCrir~rrr*r*r.r rzacosh._eval_is_finiterrj)rIrJrKrLr_rrvrrr|rrZrbrrhrgrrrerdrrrrkr*r*r^r.rohs(  6  rocseZdZdZdddZeddZeeddZ d d Z dfd d Z ddZ e Z ddZddZddZddZddZdddZZS)rpa) ``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 r3cC(|dkrdd|jddSt||rr[rr\r*r*r.r_8r-z atanh.fdiffc Cs|jrC|tjur tjS|jrtjS|tjurtjS|tjur!tjS|tjur-t t |S|tjur9t t | S|j rB||  Sn/|tj urZddl m}t |t dtdSt|}|durht t |S|rr||  S|jrxtjSt|tr|jdjr|jd}|jr|St|\}}|dur|durtd|t}|j}|t |td} |dur| S|dur| t tdSdSdSdSdSdS)Nr AccumBoundsr4TF)rgrrhrirPrMrFrrGr r!rjrk!sympy.calculus.accumulationboundsryr r(rlrLrr[r+rrrrM) rrrQryrsrrNrrOrPrur*r*r.rv>sT            z atanh.evalcGs.|dks |ddkr tjSt|}|||SNrr4)rrPrrHr*r*r.r|ms zatanh.taylor_termcCs|jd}||d}|jr||S|tjur)|||}|jr'|S|S|tj tj tj fvr?| t j |||dSd|djr|||rM|nd}t|jrb|jra||ttSnt|jrt|jrs||ttSn | t j |||dS||SrT)r[rrUrirrrhrnrrMrkr+rrrjrrr r rrVr*r*r.rvs.      zatanh._eval_as_leading_termrc s|jd}||d}|tjtjfvr|tj||||dStj|||d}|tj ur/|Sd|dj rl| ||r=|nd}t |j rP|j rN|t tS|St |jr`|jr^|t tS|S|tj||||dS|SrXror\r^r*r.rZs&       zatanh._eval_nseriescKstd|td|dSrdrrar*r*r.rbrFzatanh._eval_rewrite_as_logcKs\td|dd}t|dt|d t| td|dt||t|Srd)rr rc)r]rtrrOr*r*r.rrs,zatanh._eval_rewrite_as_asinhcCrrr,rr*r*r.rrzatanh._eval_is_zerocCs.t|jdjd|jdj|jddjgSrtr r[rrmrr*r*r.rs.zatanh._eval_is_extended_realcC(tt|jddj|jddjgSrtr rr[rirr*r*r.rrpzatanh._eval_is_finitecCrir~)r[rrr*r*r._eval_is_imaginaryrzatanh._eval_is_imaginarycCr`rar1r\r*r*r.rdrez atanh.inverserrj)rIrJrKrLr_rrvrrr|rrZrbrrrrrrrrdrkr*r*r^r.rp$s$  . rpcseZdZdZdddZeddZeeddZ d d Z dfd d Z ddZ e Z ddZddZdddZddZddZZS)rqa- ``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 r3cCrvrrwr\r*r*r.r_r-z acoth.fdiffcCs|jr>|tjur tjS|tjurtjS|tjurtjS|jr$ttdS|tj ur,tjS|tj ur4tjS|j r=||  Sn!|tj urFtjSt |}|durUt t|S|r_||  S|jritttjSdSr)rgrrhrFrPrGrir r rMrrjrkr(rrlr>)rrrQrsr*r*r.rvs4         z acoth.evalcGsD|dkr t tdS|dks|ddkrtjSt|}|||Sr{)r r rrPrrHr*r*r.r|s  zacoth.taylor_termcCs|jd}||d}|tjurd||S|tjur-|||}|jr+|S|S|tj tj tj fvrC| t j |||dS|jrd|djr|||rT|nd}t|jri|jrh||ttSnt|jr{|jrz||ttSn | t j |||dS||S)Nrr3rr4)r[rrUrrkrrhrnrrMrPr+rrrrrrrjr r rVr*r*r.rs.      zacoth._eval_as_leading_termrc s|jd}||d}|tjtjfvr|tj||||dStj|||d}|tj ur/|S|j rod|dj ro| ||r@|nd}t |jrS|j rQ|ttS|St |j rc|jra|ttS|S|tj||||dS|SrX)r[rrrMrr+rrZr[rkrrrrrjr r r\r^r*r.rZ*s&       zacoth._eval_nseriescKs$tdd|tdd|dSrdr|rar*r*r.rbCr#zacoth._eval_rewrite_as_logcK td|Srrrar*r*r.reHrzacoth._eval_rewrite_as_atanhcKsxttdt|d|t||dtdd|t||d|td|dttd|ddSr)r r rrcrar*r*r.rrKsJ*zacoth._eval_rewrite_as_asinhcCr`rar$r\r*r*r.rdOrez acoth.inversecCs4t|jdjt|jddj|jddjggSrt)r r[rrruZis_extended_nonpositiverr*r*r.rUrszacoth._eval_is_extended_realcCr~rtrrr*r*r.rXrpzacoth._eval_is_finiterrj)rIrJrKrLr_rrvrrr|rrZrbrrerrrdrrrkr*r*r^r.rqs"     rqceZdZdZdddZeddZeeddZ d d Z dfd d Z dddZ ddZ e ZddZddZddZddZddZddZZS) asecha ``asech(x)`` is the inverse hyperbolic secant of ``x``. The inverse hyperbolic secant function. Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(1 - x**2)) >>> asech(1).diff(x) 0 >>> asech(1) 0 >>> asech(S(2)) I*pi/3 >>> asech(-sqrt(2)) 3*I*pi/4 >>> asech((sqrt(6) - sqrt(2))) I*pi/12 See Also ======== asinh, atanh, cosh, acoth References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSech/ r3cCs4|dkr|jd}d|td|dSt||Nr3rr6r4rlr]r^rr*r*r.r_s  z asech.fdiffcCs|jr8|tjur tjS|tjurttdS|tjur!ttdS|jr'tjS|tjur/tj S|tj ur8ttS|j rOt }||vrO|j rK||tS||S|tjurfddlm}t|t dtdS|jrltjSdS)Nr4rrx)rgrrhrFr r rGrirMrPrr+rHrrkrzry)rrrQrnryr*r*r.rvs2          z asech.evalcGs|dkr td|S|dks|ddkrtjSt|}t|dkr?|dkr?|d}||d|d|dd|ddS|d}ttj||}t||d|d}d||||dS)Nrr4r3rBr7r6)rrrPrrxrr>rrRr*r*r.r|s ,zasech.taylor_termcCs|jd}||d}|tj tjtjtjfvr%|tj |||dS|tj ur9| | |}|j r7|S|S|jsAd|jry|||rH|nd}t|jri|jsX|djr^| | S| |dttSt|jsy|tj |||dS| |SrT)r[rrUrrMrPrkr+rrrhrnrrrjrrrr r rVr*r*r.rs$      zasech._eval_as_leading_termrcsddlm}|jd}||d}|tjurtddd}ttj|dt  |dd|} tj|jd} | |} | | | } | |dsV|dkrP|dS|t |St tj| j|||d} | t | }| ||||||S|tjurtddd}ttj|dt  |dd|} tj|jd} | |} | | | } | |ds|dkr|dStt|t |St tj| j|||d} | t | }| ||||||Stj|||d}|tjur|S|jsd|jrF|||r|nd}t|jr4|js)|djr,| S|dttSt|jsF|t j||||d S|S Nr)Or9T)Zpositiver4r3rYr)r(rr[rrrMrrr+rnseriesris_meromorphicrrZremoveOrpowsimprr r r[rkrjrrrr]rtrzrrrrQrr9ZserZarg1rOgZres1r]rWr^r*r.rZsJ     &   &  &   $&   zasech._eval_nseriescCr`rarr\r*r*r.rdrez asech.inversecKs,td|td|dtd|dSrr`rr*r*r.rbs,zasech._eval_rewrite_as_logcKrr)rorr*r*r.rf rzasech._eval_rewrite_as_acoshcKs>td|dtdd|ttt|ddttjSr)rr rcr rr>rr*r*r.rrs0zasech._eval_rewrite_as_asinhcKsttdt|td|tdt| t|tdt|dt|d td|dt|dttd|dSrd)r r rrprar*r*r.resZ.zasech._eval_rewrite_as_atanhcKs<td|dtdd|tdttt|ddSrq)rr r acschrar*r*r._eval_rewrite_as_acschs<zasech._eval_rewrite_as_acschcCs*t|jdj|jdjd|jdjgSrtr}rr*r*r.rs*zasech._eval_is_extended_realcCt|jdjSr~r r[rirr*r*r.rrzasech._eval_is_finiterrj)rIrJrKrLr_rrvrrr|rrZrdrbrrfrrrerrrrkr*r*r^r.r\s& %   -rcr) ra ``acsch(x)`` is the inverse hyperbolic cosecant of ``x``. The inverse hyperbolic cosecant function. Examples ======== >>> from sympy import acsch, sqrt, I >>> 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(I) -I*pi/2 >>> acsch(-2*I) I*pi/6 >>> acsch(I*(sqrt(6) - sqrt(2))) -5*I*pi/12 See Also ======== asinh References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsch/ r3cCs<|dkr|jd}d|dtdd|dSt||rrlrr*r*r.r_Fs   z acsch.fdiffcCs|jr<|tjur tjS|tjurtjS|tjurtjS|jr!tjS|tjur.t dt dS|tj urrrrRr*r*r.r|os ."zacsch.taylor_termcCs|jd}||d}|t ttjfvr!|tj|||dS|tj ur5| | |}|j r3|S|S|tj urAd| |S|jrd|djr|||rR|nd}t|jrjt|jri| | ttSn t|jrt|jr~| | ttSn |tj|||dS| |SrT)r[rrUr rrPr+rrrhrnrrrkrrrrrr rjrVr*r*r.rs.        zacsch._eval_as_leading_termrcsddlm}|jd}||d}|turtddd}tt|dt |dd|} t |jd} | |} | | | } | |ds[|dkrN|dSt t d|t |St tj| j|||d} | t | }| ||||||}|S|tjtkrtddd}tt |dt |dd|} t|jd} | |} | | | } | |ds|dkr|dStt d|t |St tj| j|||d} | t | }| ||||||Stj|||d}|tjur|S|jr^d|djr^|jd||r%|nd}t|jr=t|jr;| tt S|St|jrRt|jrP| tt S|S|tj||||d S|Sr)r(rr[rr rrr+rrrrr rrrMrZrrrrr[rkrrrrrrjrr^r*r.rZsR    $   *& &   (&    zacsch._eval_nseriescCr`rarr\r*r*r.rdrez acsch.inversecKs td|td|ddSrdr`rr*r*r.rbrBzacsch._eval_rewrite_as_logcKrrrbrr*r*r.rrrzacsch._eval_rewrite_as_asinhcKs>ttdt|tt|dtt|ddttjSr)r rror rr>rr*r*r.rfs zacsch._eval_rewrite_as_acoshcKsF|d}|d}t| |ttjt|d |tt|Sr)rr rr>rp)r]rQrZarg2Zarg2p1r*r*r.res zacsch._eval_rewrite_as_atanhcCrir~)r[rrr*r*r.rrzacsch._eval_is_zerocCrir~rrr*r*r.rrzacsch._eval_is_extended_realcCrr~rrr*r*r.rrzacsch._eval_is_finiterrj)rIrJrKrLr_rrvrrr|rrZrdrbrrrrfrerrrrkr*r*r^r.r s& % !  0rN)JZ sympy.corerrrZsympy.core.addrZsympy.core.functionrrZsympy.core.logicrr r r Zsympy.core.numbersr r rZsympy.core.symbolrZ(sympy.functions.combinatorial.factorialsrrrZ%sympy.functions.combinatorial.numbersrrrZ$sympy.functions.elementary.complexesrrrZ&sympy.functions.elementary.exponentialrrrZ#sympy.functions.elementary.integersrZ(sympy.functions.elementary.miscellaneousrrrrr r!r"r#r$r%r&r'r(Zsympy.polys.specialpolysr)r2r?rDrHr0rWrXrZrrr4rrrJrcrorprqrrr*r*r*r.s\    4    # "UwV-JG;3=&E