U 9%eD@s4ddlmZmZmZddlmZddlmZmZddl m Z m Z m Z ddl mZmZmZddlmZddlmZmZmZddlmZmZmZdd lmZmZmZdd lm 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.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4ddZ5eddZ6eddZ7eddZ8GdddeZ9ddZ:Gddde9Z;Gddde9ZGd#d$d$e9Z?Gd%d&d&e?Z@Gd'd(d(e?ZAGd)d*d*eZBGd+d,d,eBZCGd-d.d.eBZDGd/d0d0eBZEGd1d2d2eBZFGd3d4d4eBZGGd5d6d6eBZHd7S)8)Ssympifycacheit)Add)FunctionArgumentIndexError)fuzzy_or fuzzy_and 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)d/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/functions/elementary/hyperbolic.py sz/_rewrite_hyperbolics_as_exp..)ZxreplaceZatomsHyperbolicFunction)exprr)r)r-_rewrite_hyperbolics_as_exps r1c+Cstttdtdt tt dtdtjtdtddttddtddtdtd dttdddtdtddtdttddtddtdtd dttddtddtdttdd tdd tdttd d tdtddtdtdtd dttd dtdtddttddtdtd dttdddtddtdtd dtd dtdttd d tdddtdtdd dttddiS) N  )r rrrHalfr r r)r)r)r- _acosh_tablesR              r>cCsBtt dttdtdt dtdtdt dtdtdtdt dtdt dttddtdt dttdt dttddd tdtdtd t d tdtdtdd tdttddtdd tdttdtdd tdtdt tdtddi S) Nr3r7r:r2r8 r9r6r4)r r rrrr)r)r)r- _acsch_table3s6     rCc5CsNtttd 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 tdttjttdiS)Nr3r2r7r:r<r8r? r9rAr;r4r6r5)r r rrrInfinityNegativeInfinityr)r)r)r- _asech_tableFsj                   rGc@seZdZdZdZdS)r/ze Base class for hyperbolic functions. See Also ======== sinh, cosh, tanh, coth TN)__name__ __module__ __qualname____doc__Z unbranchedr)r)r)r-r/js r/cCs|tt}t|D]<}||kr*tj}qZq|jr|\}}||kr|jrqZq|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   rVc@seZdZdZd4ddZd5ddZeddZee d d Z d d Z d6ddZ d7ddZ d8ddZd9ddZddZddZddZddZdd Zd!d"Zd#d$Zd:d&d'Zd(d)Zd*d+Zd,d-Zd.d/Zd0d1Zd2d3ZdS);sinha ``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 r2cCs$|dkrt|jdSt||dS)z@ Returns the first derivative of this function. r2rN)coshargsrselfargindexr)r)r-fdiffsz sinh.fdiffcCstSz7 Returns the inverse of this function. asinhrZr)r)r-inversesz sinh.inversecCs|jrX|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjS|jrT||  Sn.|tjkrhtjSt |}|dk rt t |S| r||  S|j rt|\}}|r|tt }t|t|t|t|S|jrtjS|jtkr|jdS|jtkr*|jd}t|dt|dS|jtkrT|jd}|td|dS|jtkr|jd}dt|dt|dSdSNrr2r3) is_NumberrNaNrErFis_zerorO is_negativeComplexInfinityr'r r%could_extract_minus_signis_AddrVr rWrXfuncr`rYacoshratanhacoth)clsrPi_coeffxmr)r)r-evalsH                 z sinh.evalcGsb|dks|ddkrtjSt|}t|dkrN|d}||d||dS||t|SdS)zG Returns the next term in the Taylor series expansion. rr3rAr2NrrOrlenrnrpprevious_termsrSr)r)r- taylor_terms zsinh.taylor_termcCs||jdSNrrjrY conjugater[r)r)r-_eval_conjugateszsinh._eval_conjugateTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}t|t|t|t |fS)z@ Returns this function as a complex coordinate. rFcomplex rYis_extended_realexpandrrO as_real_imagrWr!rXr%r[deephintsrrr)r)r-rs  zsinh.as_real_imagcKs$|jfd|i|\}}||tSNrrr r[rrZre_partZim_partr)r)r-_eval_expand_complexszsinh._eval_expand_complexcKs|r|jdj|f|}n |jd}d}|jr<|\}}n:|jdd\}}|tjk rv|jrv|tjk rv|}|d|}|dk rt|t |t|t |jddSt|SNrTZrationalr2)Ztrig) rYrrirN as_coeff_MulrrL is_IntegerrWrXr[rrrPrpycoefftermsr)r)r-_eval_expand_trigs  (zsinh._eval_expand_trigNcKst|t| dSNr3rr[rPlimitvarkwargsr)r)r-_eval_rewrite_as_tractable&szsinh._eval_rewrite_as_tractablecKst|t| dSrrr[rPrr)r)r-_eval_rewrite_as_exp)szsinh._eval_rewrite_as_expcKst tt|SNr r%rr)r)r-_eval_rewrite_as_sin,szsinh._eval_rewrite_as_sincKst tt|Srr r#rr)r)r-_eval_rewrite_as_csc/szsinh._eval_rewrite_as_csccKst t|ttdSrr rXr rr)r)r-_eval_rewrite_as_cosh2szsinh._eval_rewrite_as_coshcKs"ttj|}d|d|dSNr3r2tanhrr=r[rPrZ tanh_halfr)r)r-_eval_rewrite_as_tanh5szsinh._eval_rewrite_as_tanhcKs"ttj|}d||ddSrcothrr=r[rPrZ coth_halfr)r)r-_eval_rewrite_as_coth9szsinh._eval_rewrite_as_cothcKs dt|SNr2cschrr)r)r-_eval_rewrite_as_csch=szsinh._eval_rewrite_as_cschrcCsh|jdj|||d}||d}|tjkrF|j|d|jr>dndd}|jrP|S|jr`| |S|SdSNrlogxcdir-+)dir) rYas_leading_termsubsrrdlimitrfre is_finiterjr[rprrrParg0r)r)r-_eval_as_leading_term@s   zsinh._eval_as_leading_termcCs*|jd}|jrdS|\}}|tjSNrTrYis_realrr rer[rPrrr)r)r- _eval_is_realMs   zsinh._eval_is_realcCs|jdjrdSdSrrYrr|r)r)r-_eval_is_extended_realWs zsinh._eval_is_extended_realcCs|jdjr|jdjSdSryrYr is_positiver|r)r)r-_eval_is_positive[s zsinh._eval_is_positivecCs|jdjr|jdjSdSryrYrrfr|r)r)r-_eval_is_negative_s zsinh._eval_is_negativecCs|jd}|jSryrYrr[rPr)r)r-_eval_is_finitecs zsinh._eval_is_finitecCs"t|jd\}}|jr|jSdSry)rVrYre is_integerr[restZipi_multr)r)r- _eval_is_zerogszsinh._eval_is_zero)r2)r2)T)T)T)N)Nr)rHrIrJrKr]ra classmethodrr staticmethodrrxr}rrrrrrrrrrrrrrrrrrr)r)r)r-rWs6  0       rWc@seZdZdZd0ddZeddZeeddZ d d Z d1d d Z d2ddZ d3ddZ d4ddZddZddZddZddZddZdd Zd!d"Zd5d$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/ZdS)6rXa" ``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 r2cCs$|dkrt|jdSt||dSNr2rrWrYrrZr)r)r-r]sz cosh.fdiffcCs~ddlm}|jrb|tjkr"tjS|tjkr2tjS|tjkrBtjS|jrNtjS|j r^|| Sn|tj krrtjSt |}|dk r||S| r|| S|j rt|\}}|r|tt}t|t|t|t|S|jrtjS|jtkr td|jddS|jtkr"|jdS|jtkrHdtd|jddS|jtkrz|jd}|t|dt|dSdS)Nr)r!r2r3)(sympy.functions.elementary.trigonometricr!rcrrdrErFrerLrfrgr'rhrirVr r rXrWrjr`rrYrkrlrm)rnrPr!rorprqr)r)r-rrsF               z cosh.evalcGsb|dks|ddkrtjSt|}t|dkrN|d}||d||dS||t|SdS)Nrr3r2rArsrur)r)r-rxs zcosh.taylor_termcCs||jdSryrzr|r)r)r-r}szcosh._eval_conjugateTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}t|t|t|t |fS)NrFr~) rYrrrrOrrXr!rWr%rr)r)r-rs  zcosh.as_real_imagcKs$|jfd|i|\}}||tSrrrr)r)r-rszcosh._eval_expand_complexcKs|r|jdj|f|}n |jd}d}|jr<|\}}n:|jdd\}}|tjk rv|jrv|tjk rv|}|d|}|dk rt|t|t |t |jddSt|Sr) rYrrirNrrrLrrXrWrr)r)r-rs  (zcosh._eval_expand_trigNcKst|t| dSrrrr)r)r-rszcosh._eval_rewrite_as_tractablecKst|t| dSrrrr)r)r-rszcosh._eval_rewrite_as_expcKs tt|Srr!r rr)r)r-_eval_rewrite_as_cosszcosh._eval_rewrite_as_coscKsdtt|Srr$r rr)r)r-_eval_rewrite_as_secszcosh._eval_rewrite_as_seccKst t|ttdSrr rWr rr)r)r-_eval_rewrite_as_sinhszcosh._eval_rewrite_as_sinhcKs"ttj|d}d|d|Srrrr)r)r-rszcosh._eval_rewrite_as_tanhcKs"ttj|d}|d|dSrrrr)r)r-rszcosh._eval_rewrite_as_cothcKs dt|Srsechrr)r)r-_eval_rewrite_as_sechszcosh._eval_rewrite_as_sechrcCsj|jdj|||d}||d}|tjkrF|j|d|jr>dndd}|jrRtjS|j rb| |S|SdSr) rYrrrrdrrfrerLrrjrr)r)r-rs   zcosh._eval_as_leading_termcCs0|jd}|js|jrdS|\}}|tjSr)rYr is_imaginaryrr rerr)r)r-rs    zcosh._eval_is_realc Csr|jd}|\}}|dt}|j}|r0dS|j}|dkrB|St|t|t|tdk|dtdkgggSNrr3TFr4rYrr rerr r[zrprZymodZyzeroZxzeror)r)r-rs    zcosh._eval_is_positivec Csr|jd}|\}}|dt}|j}|r0dS|j}|dkrB|St|t|t|tdk|dtdkgggSrrrr)r)r-_eval_is_nonnegative?s    zcosh._eval_is_nonnegativecCs|jd}|jSryrrr)r)r-rYs zcosh._eval_is_finitecCs,t|jd\}}|r(|jr(|tjjSdSry)rVrYrerr=rrr)r)r-r]s zcosh._eval_is_zero)r2)T)T)T)N)Nr)rHrIrJrKr]rrrrrrxr}rrrrrrrrrrrrrrrrrr)r)r)r-rXms2  /        rXc@seZdZdZd0ddZd1ddZeddZee d d Z d d Z d2ddZ ddZ d3ddZddZddZddZddZddZdd Zd4d"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/ZdS)5ra' ``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 r2cCs.|dkr tjt|jddSt||dSNr2rr3)rrLrrYrrZr)r)r-r]wsz tanh.fdiffcCstSr^rlrZr)r)r-ra}sz tanh.inversecCs|jrX|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjS|j rT||  Sn2|tj krhtjSt |}|dk r| rt t| St t|S| r||  S|jrt|\}}|rt|tt }|tj krt|St|S|jrtjS|jtkr(|jd}|td|dS|jtkrZ|jd}t|dt|d|S|jtkrp|jdS|jtkrd|jdSdSrb)rcrrdrErLrF NegativeOnererOrfrgr'rhr r&rirVrr rrjr`rYrrkrlrm)rnrProrprqZtanhmr)r)r-rrsN               z tanh.evalcGsf|dks|ddkrtjSt|}d|d}t|d}t|d}||d||||SdSNrr3r2)rrOrrr)rvrprwrQBFr)r)r-rxs   ztanh.taylor_termcCs||jdSryrzr|r)r)r-r}sztanh._eval_conjugateTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}t|dt|d}t|t||t |t||fS)NrFr~r3r)r[rrrrdenomr)r)r-rs  ztanh.as_real_imagc  s|jd}|jrnt|j}dd|jD}ddg}t|dD]}||dt||7<q>|d|dS|jr|\}jrdkrt|fddtdddD}fddtdddD}t |t |St|S)NrcSsg|]}t|ddqSFevaluate)rrr+rpr)r)r- sz*tanh._eval_expand_trig..r2r3cs"g|]}tt||qSr)rranger+kTrr)r-rscs"g|]}tt||qSr)rrrr)r-rs) rYrirtrr(rMrrrr) r[rrPrvZTXrSirdr)rr-rs$     ztanh._eval_expand_trigNcKs$t| t|}}||||Srrr[rPrrneg_exppos_expr)r)r-rsztanh._eval_rewrite_as_tractablecKs$t| t|}}||||Srrr[rPrrrr)r)r-rsztanh._eval_rewrite_as_expcKst tt|Sr)r r&rr)r)r-_eval_rewrite_as_tansztanh._eval_rewrite_as_tancKst tt|Sr)r r"rr)r)r-_eval_rewrite_as_cotsztanh._eval_rewrite_as_cotcKs tt|tttd|Srrrr)r)r-rsztanh._eval_rewrite_as_sinhcKs ttttd|t|Srrrr)r)r-rsztanh._eval_rewrite_as_coshcKs dt|Srrrr)r)r-rsztanh._eval_rewrite_as_cothrcCsHddlm}|jd|}||jkr:|d||r:|S||SdSNr)Orderr2sympy.series.orderrrYrZ free_symbolscontainsrjr[rprrrrPr)r)r-rs  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 r2cCs,|dkrdt|jddSt||dS)Nr2r5rr3rrZr)r)r-r]Lsz coth.fdiffcCstSr^)rmrZr)r)r-raRsz coth.inversecCs|jrX|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjS|j rT||  Sn2|tjkrhtjSt |}|dk r| rt t | St t |S| r||  S|jrt|\}}|rt|tt }|tjkrt|St|S|jrtjS|jtkr(|jd}td|d|S|jtkrZ|jd}|t|dt|dS|jtkrtd|jdS|jtkr|jdSdSrb)rcrrdrErLrFrrergrfr'rhr r"rirVrr rrjr`rYrrkrlrm)rnrProrprqZcothmr)r)r-rrXsN             z coth.evalcGsn|dkrdt|S|dks(|ddkr.tjSt|}t|d}t|d}d|d||||SdSrbrrrOrrrvrprwrrr)r)r-rxs   zcoth.taylor_termcCs||jdSryrzr|r)r)r-r}szcoth._eval_conjugateTcKsddlm}m}|jdjrF|r|d||r>d|S||SdSrrrr)r)r-rs  zcoth._eval_as_leading_termc Ks |jd}|jrxdd|jD}ggg}t|j}t|ddD] }|||dt||q>t|dt|dS|jr|jdd\}}|j r|dkrt |d d } ggg}t|ddD](}|||dt ||| |qt|dt|dSt |S) NrcSsg|]}t|ddqSr)rrrr)r)r-rsz*coth._eval_expand_trig..r5r3r2TrFr) rYrirtrappendr(rrMrrrr) r[rrPZCXrSrvrrrpcr)r)r-rs"   &zcoth._eval_expand_trig)r2)r2)T)N)Nr)rHrIrJrKr]rarrrrrrxr}rrrrrrrrrrr)r)r)r-r8s&   4    rc@seZdZUdZdZdZeed<dZeed<e ddZ ddZ d d Z d d Z d dZd%ddZddZddZd&ddZddZd'ddZddZd(dd Zd!d"Zd#d$ZdS))ReciprocalHyperbolicFunctionz=Base class for reciprocal functions of hyperbolic functions. N_is_even_is_oddcCsj|r*|jr|| S|jr*||  S|j|}t|drV||krV|jdS|dk rfd|S|S)Nrarr2)rhr r_reciprocal_ofrrhasattrrarY)rnrPtr)r)r-rrs    z!ReciprocalHyperbolicFunction.evalcOs ||jd}t||||Sry)rrYgetattr)r[ method_namerYror)r)r-_call_reciprocalsz-ReciprocalHyperbolicFunction._call_reciprocalcOs&|j|f||}|dk r"d|S|Sr)r)r[rrYrrr)r)r-_calculate_reciprocalsz2ReciprocalHyperbolicFunction._calculate_reciprocalcCs.|||}|dk r*|||kr*d|SdSr)rr)r[rrPrr)r)r-_rewrite_reciprocals z0ReciprocalHyperbolicFunction._rewrite_reciprocalcKs |d|S)Nrrrr)r)r-r sz1ReciprocalHyperbolicFunction._eval_rewrite_as_expcKs |d|S)Nrrrr)r)r-rsz7ReciprocalHyperbolicFunction._eval_rewrite_as_tractablecKs |d|S)Nrrrr)r)r-rsz2ReciprocalHyperbolicFunction._eval_rewrite_as_tanhcKs |d|S)Nrrrr)r)r-rsz2ReciprocalHyperbolicFunction._eval_rewrite_as_cothTcKsd||jdj|f|Sr)rrYr)r[rrr)r)r-rsz)ReciprocalHyperbolicFunction.as_real_imagcCs||jdSryrzr|r)r)r-r}sz,ReciprocalHyperbolicFunction._eval_conjugatecKs$|jfddi|\}}|t|S)NrTrrr)r)r-rsz1ReciprocalHyperbolicFunction._eval_expand_complexcKs |jd|S)Nr)r)r)r[rr)r)r-r!sz.ReciprocalHyperbolicFunction._eval_expand_trigrcCsd||jd|Sr)rrYr)r[rprrr)r)r-r$sz2ReciprocalHyperbolicFunction._eval_as_leading_termcCs||jdjSry)rrYrr|r)r)r-r'sz3ReciprocalHyperbolicFunction._eval_is_extended_realcCsd||jdjSr)rrYrr|r)r)r-r*sz,ReciprocalHyperbolicFunction._eval_is_finite)N)T)T)Nr)rHrIrJrKrr r __annotations__rrrrrrrrrrrrr}rrrrrr)r)r)r-r s(        r c@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 Tr2cCs4|dkr&t|jd t|jdSt||dS)z? Returns the first derivative of this function r2rN)rrYrrrZr)r)r-r]Esz csch.fdiffcGsr|dkrdt|S|dks(|ddkr.tjSt|}t|d}t|d}ddd|||||SdS)zF Returns the next term in the Taylor series expansion rr2r3Nrrr)r)r-rxNs   zcsch.taylor_termcKsttt|Srrrr)r)r-r`szcsch._eval_rewrite_as_sincKsttt|Srrrr)r)r-rcszcsch._eval_rewrite_as_csccKstt|ttdSrrrr)r)r-rfszcsch._eval_rewrite_as_coshcKs dt|SrrWrr)r)r-riszcsch._eval_rewrite_as_sinhcCs|jdjr|jdjSdSryrr|r)r)r-rls zcsch._eval_is_positivecCs|jdjr|jdjSdSryrr|r)r)r-rps zcsch._eval_is_negativeN)r2)rHrIrJrKrWrrr]rrrxrrrrrrr)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 Tr2cCs4|dkr&t|jd t|jdSt||dSr)rrYrrrZr)r)r-r]sz sech.fdiffcGs>|dks|ddkrtjSt|}t|t|||SdSr)rrOrrrrvrprwr)r)r-rxszsech.taylor_termcKsdtt|Srrrr)r)r-rszsech._eval_rewrite_as_coscKs tt|Srrrr)r)r-rszsech._eval_rewrite_as_seccKstt|ttdSrrrr)r)r-rszsech._eval_rewrite_as_sinhcKs dt|SrrXrr)r)r-rszsech._eval_rewrite_as_coshcCs|jdjrdSdSrrr|r)r)r-rs zsech._eval_is_positiveN)r2)rHrIrJrKrXrr r]rrrxrrrrrr)r)r)r-rus  rc@seZdZdZdS)InverseHyperbolicFunctionz,Base class for inverse hyperbolic functions.N)rHrIrJrKr)r)r)r-rsrc@seZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZddZddZd ddZddZd S)!r`aM ``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 r2cCs0|dkr"dt|jdddSt||dSr)rrYrrZr)r)r-r]sz asinh.fdiffc Csr|jr|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjS|tjkr\tt ddS|tj krvtt ddS|j r||  SnL|tj krtj S|jrtjSt |}|dk rtt|S|r||  St|trn|jdjrn|jd}|jr|St|\}}|dk rn|dk rnt|tdt}|tt|}|j}|dkr^|S|dkrn| SdS)Nr3r2rTF)rcrrdrErFrerOrLrrrrfrgr'r rrh isinstancerWrYrrrrr is_even) rnrProrrrfrqevenr)r)r-rrsJ            z asinh.evalcGs|dks|ddkrtjSt|}t|dkrd|dkrd|d}| |dd||d|dS|dd}ttj|}t|}tj||||||SdSNrr3rAr2)rrOrrtrr=rrrvrprwrSrRrr)r)r-rxs&  zasinh.taylor_termNrcCs|jd}||d}|jr*||S|t ttjfkrR|t j |||dSd|dj r| ||rn|nd}t |jrt|j r|| ttSn@t |j rt|jr|| ttSn|t j |||dS||SNrrr2r3)rYrcancelrerr rrgr*rrrfrrrrrjr r[rprrrPZx0ndirr)r)r-rs       zasinh._eval_as_leading_termc Cs|jd}||d}|tt fkr<|tj||||dStj||||d}|tjkr\|Sd|dj r| ||rx|nd}t |j rt |j r| ttSnszasinh._eval_rewrite_as_logcKst|td|dSNr2r3)rlrr0r)r)r-_eval_rewrite_as_atanhCszasinh._eval_rewrite_as_atanhcKs4t|}ttd|t|dt|tdSr2)r rrkr )r[rprZixr)r)r-_eval_rewrite_as_acoshFszasinh._eval_rewrite_as_acoshcKst tt|Sr)r rr0r)r)r-_eval_rewrite_as_asinJszasinh._eval_rewrite_as_asincKsttt|ttdSr)r rr r0r)r)r-_eval_rewrite_as_acosMszasinh._eval_rewrite_as_acoscCstSr^rrZr)r)r-raPsz asinh.inversecCs |jdjSryrr|r)r)r-rVszasinh._eval_is_zero)r2)Nr)r)r2)rHrIrJrKr]rrrrrrxrr,r1rr3r4r5r6rarr)r)r)r-r`s"  -    r`c@seZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZddZddZd ddZddZd S)!rkaM ``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 r2cCs<|dkr.|jd}dt|dt|dSt||dSrrYrr)r[r\rPr)r)r-r]ps z acosh.fdiffc Cs|jrj|tjkrtjS|tjkr&tjS|tjkr6tjS|jrHttdS|tjkrXtj S|tj krjttS|j rt }||kr|j r||tS||S|tjkrtjS|ttjkrtjttdS|t tjkrtjttdS|jrtttjSt|tr|jdj r|jd}|jr4t|St|\}}|dk r|dk rt|t}|tt|}|j}|dkr|jr|S|jr| Sn0|dkr|tt8}|jr| S|jr|SdS)Nr3rTF)rcrrdrErFrer r rLrOrrr>rrgr=rrXrYrrrrrZis_nonnegativerfZis_nonpositiver) rnrP cst_tablerr rr!rqr"r)r)r-rrws^              z acosh.evalcGs|dkrttdS|dks(|ddkr.tjSt|}t|dkrv|dkrv|d}||dd||d|dS|dd}ttj|}t|}| |t|||SdSr#) r r rrOrrtrr=rr$r)r)r-rxs $  zacosh.taylor_termNrcCs|jd}||d}|tj tjtjtjfkrJ|tj |||dS|dj r| ||rb|nd}t |j r|dj r| |dttS| | St |js|tj |||dS| |Sr&)rYrr'rrLrOrgr*rrrfrrrjr r rr(r)r)r-rs      zacosh._eval_as_leading_termc Cs|jd}||d}|tjtjfkr>|tj||||dStj||||d}|tj kr^|S|dj r| ||rv|nd}t |j r|dj r|dt tS| St |js|tj||||dS|Sr*rYrrrLrr*rr,rrgrfrrr r rr-r)r)r-r,s        zacosh._eval_nseriescKs t|t|dt|dSrr/r0r)r)r-r1szacosh._eval_rewrite_as_logcKs t|dtd|t|Sr)rrr0r)r)r-r6szacosh._eval_rewrite_as_acoscKs(t|dtd|tdt|Sr2)rr rr0r)r)r-r5szacosh._eval_rewrite_as_asincKs0t|dtd|tdttt|Sr2)rr r r`r0r)r)r-_eval_rewrite_as_asinhszacosh._eval_rewrite_as_asinhcKspt|d}td|}t|dd}td||d|td|d|t|d|t||Sr2)rr rl)r[rprZsxm1Zs1mxZsx2m1r)r)r-r3s   &zacosh._eval_rewrite_as_atanhcCstSr^rrZr)r)r-rasz acosh.inversecCs|jddjrdSdS)Nrr2Trr|r)r)r-rszacosh._eval_is_zero)r2)Nr)r)r2)rHrIrJrKr]rrrrrrxrr,r1rr6r5r:r3rarr)r)r)r-rkZs"  6    rkc@sxeZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZddZdddZd S)rla) ``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 r2cCs,|dkrdd|jddSt||dSrrYrrZr)r)r-r]sz atanh.fdiffc Cs|jr|tjkrtjS|jr"tjS|tjkr2tjS|tjkrBtjS|tjkrZt t |S|tjkrrt t | S|j r||  Sn^|tj krddl m}t |t dtdSt|}|dk rt t |S|r||  S|jrtjSt|tr|jdjr|jd}|jr |St|\}}|dk r|dk rtd|t}|j}|t |td} |dkrx| S|dkr| t tdSdS)Nr AccumBoundsr3TF)rcrrdrerOrLrErrFr r rfrg!sympy.calculus.accumulationboundsr=r r'rhrrrYrrrrr) rnrPr=rorr rr!r"rqr)r)r-rr"sL             z atanh.evalcGs2|dks|ddkrtjSt|}|||SdSNrr3)rrOrrr)r)r-rxQszatanh.taylor_termNrcCs|jd}||d}|jr*||S|tj tjtjfkrV|t j |||dSd|dj r| ||rr|nd}t |j r|j r||ttSn:t |jr|jr||ttSn|t j |||dS||Sr&)rYrr'rerrrLrgr*rrrfrrrjr r rr(r)r)r-rZs     zatanh._eval_as_leading_termc Cs|jd}||d}|tjtjfkr>|tj||||dStj||||d}|tj kr^|Sd|dj r| ||rz|nd}t |j r|j r|t tSn6t |jr|jr|t tSn|tj||||dS|Sr*r9r-r)r)r-r,os"     zatanh._eval_nseriescKstd|td|dSr2rr0r)r)r-r1szatanh._eval_rewrite_as_logcKs\td|dd}t|dt|d t| td|dt||t|Sr2)rr r`)r[rprr!r)r)r-r:s,zatanh._eval_rewrite_as_asinhcCs|jdjrdSdSrrr|r)r)r-rs zatanh._eval_is_zerocCs |jdjSry)rYrr|r)r)r-_eval_is_imaginaryszatanh._eval_is_imaginarycCstSr^r rZr)r)r-rasz atanh.inverse)r2)Nr)r)r2)rHrIrJrKr]rrrrrrxrr,r1rr:rrArar)r)r)r-rls  .   rlc@speZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZdddZd S)rma- ``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 r2cCs,|dkrdd|jddSt||dSrr;rZr)r)r-r]sz acoth.fdiffcCs|jr||tjkrtjS|tjkr&tjS|tjkr6tjS|jrHttdS|tj krXtjS|tj krhtjS|j r||  SnB|tj krtjSt |}|dk rt t|S|r||  S|jrtttjSdSr)rcrrdrErOrFrer r rLrrfrgr'rrhr=)rnrPror)r)r-rrs0        z acoth.evalcGsH|dkrt tdS|dks*|ddkr0tjSt|}|||SdSr?)r r rrOrrr)r)r-rxs zacoth.taylor_termNrcCs|jd}||d}|tjkr2d||S|tj tjtjfkr^|t j |||dS|j rd|dj r| ||r|nd}t|jr|j r||ttSn:t|j r|jr||ttSn|t j |||dS||S)Nrr2rr3)rYrr'rrgrrLrOr*rrrrrrrfrjr r r(r)r)r-rs     zacoth._eval_as_leading_termc Cs|jd}||d}|tjtjfkr>|tj||||dStj||||d}|tj kr^|S|j rd|dj r| ||r|nd}t |jr|j r|ttSn6t |j r|jr|ttSn|tj||||dS|Sr*)rYrrrLrr*rr,rrgrrrrrfr r r-r)r)r-r,s"     zacoth._eval_nseriescKs$tdd|tdd|dSr2r@r0r)r)r-r1szacoth._eval_rewrite_as_logcKs td|Srrr0r)r)r-r3szacoth._eval_rewrite_as_atanhcKsxttdt|d|t||dtdd|t||d|td|dttd|ddSr)r r rr`r0r)r)r-r:sJ*zacoth._eval_rewrite_as_asinhcCstSr^rrZr)r)r-rasz acoth.inverse)r2)Nr)r)r2)rHrIrJrKr]rrrrrrxrr,r1rr3r:rar)r)r)r-rms    rmc@seZdZdZdddZeddZeeddZ dd d Z dd dZ dddZ ddZ e ZddZddZddZddZd S)asecha ``asech(x)`` is the inverse hyperbolic secant of ``x``. The inverse hyperbolic secant function. Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(1 - x**2)) >>> asech(1).diff(x) 0 >>> asech(1) 0 >>> asech(S(2)) I*pi/3 >>> asech(-sqrt(2)) 3*I*pi/4 >>> asech((sqrt(6) - sqrt(2))) I*pi/12 See Also ======== asinh, atanh, cosh, acoth References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSech/ r2cCs8|dkr*|jd}d|td|dSt||dSNr2rr5r3r7r[r\rr)r)r-r]Ls z asech.fdiffcCs|jrp|tjkrtjS|tjkr,ttdS|tjkrBttdS|jrNtjS|tjkr^tj S|tj krpttS|j rt }||kr|j r||tS||S|tjkrddlm}t|t dtdS|jrtjSdS)Nr3rr<)rcrrdrEr r rFrerLrOrrrGrrgr>r=)rnrPr8r=r)r)r-rrSs0          z asech.evalcGs|dkrtd|S|dks(|ddkr.tjSt|}t|dkr~|dkr~|d}||d|d|dd|ddS|d}ttj||}t||d|d}d||||dSdS)Nrr3r2rAr6r5)rrrOrrtrr=rr$r)r)r-rxrs ,zasech.taylor_termNrcCs|jd}||d}|tj tjtjtjfkrJ|tj |||dS|j sZd|j r| ||rh|nd}t |j r|j s|dj r|| S||dttSt |j s|tj |||dS||Sr&)rYrr'rrLrOrgr*rrrfrrrrjr r r(r)r)r-rs    zasech._eval_as_leading_termcCsddlm}|jd}||d}|tjkrtddd}ttj|dt  |dd|} tj|jd} | |} | | | } | |ds|dkr|dS|t |St tj| j|||d} | t | }| ||||||S|tjkrtddd}ttj|dt  |dd|} tj|jd} | |} | | | } | |ds|dkr|dStt|t |St tj| j|||d} | t | }| ||||||Stj||||d}|tjkr|S|js$d|jr|||r4|nd}t|jrp|jsZ|djr`| S|dttSt|js|t j||||d S|S Nr)OrT)Zpositiver3r2r+r)rrFrYrrrLrrBr*rnseriesris_meromorphicrr,removeOrpowsimprr r rrgrfrrrr[rprvrrrFrPrrZserZarg1r!gZres1r.r)r)r)r-r,sJ     &   &  &  &&   zasech._eval_nseriescCstSr^rrZr)r)r-rasz asech.inversecKs,td|td|dtd|dSrr/rr)r)r-r1szasech._eval_rewrite_as_logcKs td|Sr)rkrr)r)r-r4szasech._eval_rewrite_as_acoshcKs:td|dtdd|ttt|ttjSr)rr r`r rr=rr)r)r-r:s,zasech._eval_rewrite_as_asinhcKsttdt|td|tdt| t|tdt|dt|d td|dt|dttd|dSr2)r r rrlr0r)r)r-r3sZ.zasech._eval_rewrite_as_atanhcKs8td|dtdd|tdttt|Sr2)rr r acschr0r)r)r-_eval_rewrite_as_acschszasech._eval_rewrite_as_acsch)r2)Nr)r)r2)rHrIrJrKr]rrrrrrxrr,rar1rr4r:r3rNr)r)r)r-rB&s %     - rBc@seZdZdZdddZeddZeeddZ dd d Z dd dZ dddZ ddZ e ZddZddZddZddZd S)rMa ``acsch(x)`` is the inverse hyperbolic cosecant of ``x``. The inverse hyperbolic cosecant function. Examples ======== >>> from sympy import acsch, sqrt, I >>> from sympy.abc import x >>> acsch(x).diff(x) -1/(x**2*sqrt(1 + x**(-2))) >>> acsch(1).diff(x) 0 >>> acsch(1) log(1 + sqrt(2)) >>> acsch(I) -I*pi/2 >>> acsch(-2*I) I*pi/6 >>> acsch(I*(sqrt(6) - sqrt(2))) -5*I*pi/12 See Also ======== asinh References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsch/ r2cCs@|dkr2|jd}d|dtdd|dSt||dSrCr7rDr)r)r-r]s  z acsch.fdiffcCs|jrx|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjS|tjkr\t dt dS|tj krxt dt d S|j rt }||kr||tS|tjkrtjS|jrtjS|jrtjS|r||  SdSr2)rcrrdrErOrFrergrLrrrrrCr is_infiniterh)rnrPr8r)r)r-rr s2       z acsch.evalcGs|dkrtd|S|dks(|ddkr.tjSt|}t|dkr|dkr|d}| |d|d|dd|ddS|d}ttj||}t||d|d}tj|d||||dSdS)Nrr3r2rAr6) rrrOrrtrr=rrr$r)r)r-rx+s .zacsch.taylor_termNrcCs|jd}||d}|t ttjfkrB|tj|||dS|tj krZd| |S|j rd|dj r| ||r||nd}t|j rt|j r|| ttSn@t|jrt|jr|| ttSn|tj|||dS||Sr&)rYrr'r rrOr*rrrgrrrrrrrjr rfr(r)r)r-r=s       zacsch._eval_as_leading_termcCsddlm}|jd}||d}|tkrtddd}tt|dt |dd|} t |jd} | |} | | | } | |ds|dkr|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}|tjkr$|S|jrd|djr|jd||rR|nd}t|jrt|jr| tt Sn@t|jrt|jr| tt Sn|tj||||d S|SrE)rrFrYrr rrMr*rrGrrHr rrrLr,rIrrJrrrgrrrrrrfrKr)r)r-r,RsN     $   *& &  *&     zacsch._eval_nseriescCstSr^rrZr)r)r-rasz acsch.inversecKs td|td|ddSr2r/rr)r)r-r1szacsch._eval_rewrite_as_logcKs td|Srr_rr)r)r-r:szacsch._eval_rewrite_as_asinhcKs:ttdt|tt|dtt|ttjSr)r rrkr rr=rr)r)r-r4s   zacsch._eval_rewrite_as_acoshcKsF|d}|d}t| |ttjt|d |tt|Sr)rr rr=rl)r[rPrZarg2Zarg2p1r)r)r-r3s zacsch._eval_rewrite_as_atanhcCs |jdjSry)rYrOr|r)r)r-rszacsch._eval_is_zero)r2)Nr)r)r2)rHrIrJrKr]rrrrrrxrr,rar1rr:r4r3rr)r)r)r-rMs %  !   0 rMN)IZ sympy.corerrrZsympy.core.addrZsympy.core.functionrrZsympy.core.logicrr r Zsympy.core.numbersr r r Zsympy.core.symbolrZ(sympy.functions.combinatorial.factorialsrrrZ%sympy.functions.combinatorial.numbersrrrZ$sympy.functions.elementary.complexesrrrZ&sympy.functions.elementary.exponentialrrrZ#sympy.functions.elementary.integersrZ(sympy.functions.elementary.miscellaneousrrrrrr r!r"r#r$r%r&r'Zsympy.polys.specialpolysr(r1r>rCrGr/rVrWrXrrr rrrr`rkrlrmrBrMr)r)r)r-sZ    4    # "UwV-JG;%/7