U ;qLeD@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)|/home/p21-0144/sympy/latex2sympy2solve-back-end/sympyEq/lib/python3.8/site-packages/sympy/functions/elementary/hyperbolic.py sz/_rewrite_hyperbolics_as_exp..)xreplaceatomsHyperbolicFunction)exprr)r)r-_rewrite_hyperbolics_as_exps r3c+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) Nr5r9r<r4r: r;r8r6)r r rrrr)r)r)r- _acsch_table3s6     rEc5CsNtttd 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)Nr5r4r9r<r>r:rA r;rCr=r6r8r7)r r rrrInfinityNegativeInfinityr)r)r)r- _asech_tableFsj                   rIc@seZdZdZdZdS)r1ze Base class for hyperbolic functions. See Also ======== sinh, cosh, tanh, coth TN)__name__ __module__ __qualname____doc__ unbranchedr)r)r)r-r1js r1cCs|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 r make_argsrOneis_Mul as_two_terms is_RationalZeror?)argipiaKpm1m2r)r)r- _peeloff_ipiws   r\c@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 r4cCs$|dkrt|jdSt||dS)z@ Returns the first derivative of this function. r4rN)coshargsrselfargindexr)r)r-fdiffsz sinh.fdiffcCstSz7 Returns the inverse of this function. asinhr`r)r)r-inversesz sinh.inversecCs|jrX|tjkrtjS|tjkr&tjS|tjkr6tjS|jrBtjS|jrT||  Sn.|tjkrhtjSt |}|dk rt t |S| r||  S|j rt|\}}|r|tt }t|t|t|t|S|jrtjS|jtkr|jdS|jtkr*|jd}t|dt|dS|jtkrT|jd}|td|dS|jtkr|jd}dt|dt|dSdSNrr4r5) is_NumberrNaNrGrHis_zerorT is_negativeComplexInfinityr'r r%could_extract_minus_signis_Addr\r r]r^funcrfr_acoshratanhacoth)clsrUi_coeffxmr)r)r-evalsH                 z sinh.evalcGsb|dks|ddkrtjSt|}t|dkrN|d}||d||dS||t|SdS)zG Returns the next term in the Taylor series expansion. rr5rCr4NrrTrlenrnrvprevious_termsrYr)r)r- taylor_terms zsinh.taylor_termcCs||jdSNrrpr_ conjugaterar)r)r-_eval_conjugateszsinh._eval_conjugateTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}t|t|t|t |fS)z@ Returns this function as a complex coordinate. rFcomplex r_is_extended_realexpandrrT as_real_imagr]r!r^r%radeephintsrrr)r)r-rs  zsinh.as_real_imagcKs$|jfd|i|\}}||tSNrrr rarrre_partim_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|SNrTrationalr4)trig) r_rrorR as_coeff_MulrrP is_Integerr]r^rarrrUrvycoefftermsr)r)r-_eval_expand_trigs  (zsinh._eval_expand_trigNcKst|t| dSNr5rrarUlimitvarkwargsr)r)r-_eval_rewrite_as_tractable&szsinh._eval_rewrite_as_tractablecKst|t| dSrrrarUrr)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 r^r rr)r)r-_eval_rewrite_as_cosh2szsinh._eval_rewrite_as_coshcKs"ttj|}d|d|dSNr5r4tanhrr?rarUrZ tanh_halfr)r)r-_eval_rewrite_as_tanh5szsinh._eval_rewrite_as_tanhcKs"ttj|}d||ddSrcothrr?rarUrZ coth_halfr)r)r-_eval_rewrite_as_coth9szsinh._eval_rewrite_as_cothcKs dt|SNr4cschrr)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) r_as_leading_termsubsrrjlimitrlrk is_finiterprarvrrrUarg0r)r)r-_eval_as_leading_term@s   zsinh._eval_as_leading_termcCs*|jd}|jrdS|\}}|tjSNrTr_is_realrr rkrarUrrr)r)r- _eval_is_realMs   zsinh._eval_is_realcCs|jdjrdSdSrr_rrr)r)r-_eval_is_extended_realWs zsinh._eval_is_extended_realcCs|jdjr|jdjSdSrr_r is_positiverr)r)r-_eval_is_positive[s zsinh._eval_is_positivecCs|jdjr|jdjSdSrr_rrlrr)r)r-_eval_is_negative_s zsinh._eval_is_negativecCs|jd}|jSrr_rrarUr)r)r-_eval_is_finitecs zsinh._eval_is_finitecCs"t|jd\}}|jr|jSdSr)r\r_rk is_integerrarestZipi_multr)r)r- _eval_is_zerogszsinh._eval_is_zero)r4)r4)T)T)T)N)Nr)rJrKrLrMrcrg classmethodrx staticmethodrr~rrrrrrrrrrrrrrrrrrrr)r)r)r-r]s6  0       r]c@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)6r^a" ``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 r4cCs$|dkrt|jdSt||dSNr4rr]r_rr`r)r)r-rcsz cosh.fdiffcCs~ddlm}|jrb|tjkr"tjS|tjkr2tjS|tjkrBtjS|jrNtjS|j r^|| Sn|tj krrtjSt |}|dk r||S| r|| S|j rt|\}}|r|tt}t|t|t|t|S|jrtjS|jtkr td|jddS|jtkr"|jdS|jtkrHdtd|jddS|jtkrz|jd}|t|dt|dSdS)Nr)r!r4r5)(sympy.functions.elementary.trigonometricr!rirrjrGrHrkrPrlrmr'rnror\r r r^r]rprfrr_rqrrrs)rtrUr!rurvrwr)r)r-rxsF               z cosh.evalcGsb|dks|ddkrtjSt|}t|dkrN|d}||d||dS||t|SdS)Nrr5r4rCryr{r)r)r-r~s zcosh.taylor_termcCs||jdSrrrr)r)r-rszcosh._eval_conjugateTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}t|t|t|t |fS)NrFr) r_rrrrTrr^r!r]r%rr)r)r-rs  zcosh.as_real_imagcKs$|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) r_rrorRrrrPrr^r]rr)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$r rr)r)r-_eval_rewrite_as_secszcosh._eval_rewrite_as_seccKst t|ttdSrr r]r rr)r)r-_eval_rewrite_as_sinhszcosh._eval_rewrite_as_sinhcKs"ttj|d}d|d|Srrrr)r)r-rszcosh._eval_rewrite_as_tanhcKs"ttj|d}|d|dSrrrr)r)r-rszcosh._eval_rewrite_as_cothcKs dt|Srsechrr)r)r-_eval_rewrite_as_sechszcosh._eval_rewrite_as_sechrcCsj|jdj|||d}||d}|tjkrF|j|d|jr>dndd}|jrRtjS|j rb| |S|SdSr) r_rrrrjrrlrkrPrrprr)r)r-rs   zcosh._eval_as_leading_termcCs0|jd}|js|jrdS|\}}|tjSr)r_r is_imaginaryrr rkrr)r)r-rs    zcosh._eval_is_realc Csr|jd}|\}}|dt}|j}|r0dS|j}|dkrB|St|t|t|tdk|dtdkgggSNrr5TFr6r_rr rkrr razrvrZymodZyzeroZxzeror)r)r-rs    zcosh._eval_is_positivec Csr|jd}|\}}|dt}|j}|r0dS|j}|dkrB|St|t|t|tdk|dtdkgggSrrrr)r)r-_eval_is_nonnegative?s    zcosh._eval_is_nonnegativecCs|jd}|jSrrrr)r)r-rYs zcosh._eval_is_finitecCs,t|jd\}}|r(|jr(|tjjSdSr)r\r_rkrr?rrr)r)r-r]s zcosh._eval_is_zero)r4)T)T)T)N)Nr)rJrKrLrMrcrrxrrr~rrrrrrrrrrrrrrrrrrr)r)r)r-r^ms2  /        r^c@seZdZdZd0ddZd1ddZeddZee d d Z d d Z d2ddZ ddZ d3ddZddZddZddZddZddZdd Zd4d"d#Zd$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 r4cCs.|dkr tjt|jddSt||dSNr4rr5)rrPrr_rr`r)r)r-rcwsz tanh.fdiffcCstSrdrrr`r)r)r-rg}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|jdSdSrh)rirrjrGrPrH NegativeOnerkrTrlrmr'rnr r&ror\rr rrprfr_rrqrrrs)rtrUrurvrwZtanhmr)r)r-rxsN               z tanh.evalcGsf|dks|ddkrtjSt|}d|d}t|d}t|d}||d||||SdSNrr5r4)rrTrrr)r|rvr}rWBFr)r)r-r~s   ztanh.taylor_termcCs||jdSrrrr)r)r-rsztanh._eval_conjugateTcKs|jdjr6|r,d|d<|j|f|tjfS|tjfS|rX|jdj|f|\}}n|jd\}}t|dt|d}t|t||t |t||fS)NrFrr5r)rarrrrdenomr)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+rvr)r)r- sz*tanh._eval_expand_trig..r4r5cs"g|]}tt||qSr)rranger+kTrr)r-rscs"g|]}tt||qSr)rrrr)r-rs) r_rorzrr(rQrrrr) rarrUr|TXrYirdr)rr-rs$     ztanh._eval_expand_trigNcKs$t| t|}}||||SrrrarUrrneg_exppos_expr)r)r-rsztanh._eval_rewrite_as_tractablecKs$t| t|}}||||SrrrarUrrrr)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)Orderr4sympy.series.orderr r_r free_symbolscontainsrprarvrrr rUr)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 r4cCs,|dkrdt|jddSt||dS)Nr4r7rr5rr`r)r)r-rcLsz coth.fdiffcCstSrd)rsr`r)r)r-rgRsz 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|jdSdSrh)rirrjrGrPrHrrkrmrlr'rnr r"ror\rr rrprfr_rrqrrrs)rtrUrurvrwZcothmr)r)r-rxXsN             z coth.evalcGsn|dkrdt|S|dks(|ddkr.tjSt|}t|d}t|d}d|d||||SdSrhrrrTrrr|rvr}rrr)r)r-r~s   zcoth.taylor_termcCs||jdSrrrr)r)r-rszcoth._eval_conjugateTcKsddlm}m}|jdjrF|r|d||r>d|S||SdSr r rr)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..r7r5r4TrFr) r_rorzrappendr(rrQrrrr) rarrUCXrYr|rrrvcr)r)r-rs"   &zcoth._eval_expand_trig)r4)r4)T)N)Nr)rJrKrLrMrcrgrrxrrr~rrrrrrrrrrrr)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$ZdS))ReciprocalHyperbolicFunctionz=Base class for reciprocal functions of hyperbolic functions. N_is_even_is_oddcCsj|r*|jr|| S|jr*||  S|j|}t|drV||krV|jdS|dk rfd|S|S)Nrgrr4)rnrr_reciprocal_ofrxhasattrrgr_)rtrUtr)r)r-rxs    z!ReciprocalHyperbolicFunction.evalcOs ||jd}t||||Sr)rr_getattr)ra method_namer_ror)r)r-_call_reciprocalsz-ReciprocalHyperbolicFunction._call_reciprocalcOs&|j|f||}|dk r"d|S|Sr)r")rar r_rrr)r)r-_calculate_reciprocalsz2ReciprocalHyperbolicFunction._calculate_reciprocalcCs.|||}|dk r*|||kr*d|SdSr)r"r)rar rUrr)r)r-_rewrite_reciprocals z0ReciprocalHyperbolicFunction._rewrite_reciprocalcKs |d|S)Nrr$rr)r)r-r sz1ReciprocalHyperbolicFunction._eval_rewrite_as_expcKs |d|S)Nrr%rr)r)r-rsz7ReciprocalHyperbolicFunction._eval_rewrite_as_tractablecKs |d|S)Nrr%rr)r)r-rsz2ReciprocalHyperbolicFunction._eval_rewrite_as_tanhcKs |d|S)Nrr%rr)r)r-rsz2ReciprocalHyperbolicFunction._eval_rewrite_as_cothTcKsd||jdj|f|Sr)rr_r)rarrr)r)r-rsz)ReciprocalHyperbolicFunction.as_real_imagcCs||jdSrrrr)r)r-rsz,ReciprocalHyperbolicFunction._eval_conjugatecKs$|jfddi|\}}|t|S)NrTrrr)r)r-rsz1ReciprocalHyperbolicFunction._eval_expand_complexcKs |jd|S)Nr)r)r#)rarr)r)r-r!sz.ReciprocalHyperbolicFunction._eval_expand_trigrcCsd||jd|Sr)rr_r)rarvrrr)r)r-r$sz2ReciprocalHyperbolicFunction._eval_as_leading_termcCs||jdjSr)rr_rrr)r)r-r'sz3ReciprocalHyperbolicFunction._eval_is_extended_realcCsd||jdjSr)rr_rrr)r)r-r*sz,ReciprocalHyperbolicFunction._eval_is_finite)N)T)T)Nr)rJrKrLrMrrr __annotations__rrrxr"r#r$rrrrrrrrrrrr)r)r)r-rs(        rc@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 Tr4cCs4|dkr&t|jd t|jdSt||dS)z? Returns the first derivative of this function r4rN)rr_rrr`r)r)r-rcEsz csch.fdiffcGsr|dkrdt|S|dks(|ddkr.tjSt|}t|d}t|d}ddd|||||SdS)zF Returns the next term in the Taylor series expansion rr4r5Nrrr)r)r-r~Ns   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|Srr]rr)r)r-riszcsch._eval_rewrite_as_sinhcCs|jdjr|jdjSdSrrrr)r)r-rls zcsch._eval_is_positivecCs|jdjr|jdjSdSrrrr)r)r-rps zcsch._eval_is_negativeN)r4)rJrKrLrMr]rrrcrrr~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 Tr4cCs4|dkr&t|jd t|jdSt||dSr)rr_rrr`r)r)r-rcsz sech.fdiffcGs>|dks|ddkrtjSt|}t|t|||SdSr)rrTrrrr|rvr}r)r)r-r~szsech.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|Srr^rr)r)r-rszsech._eval_rewrite_as_coshcCs|jdjrdSdSrrrr)r)r-rs zsech._eval_is_positiveN)r4)rJrKrLrMr^rrrcrrr~rrrrrr)r)r)r-rus  rc@seZdZdZdS)InverseHyperbolicFunctionz,Base class for inverse hyperbolic functions.N)rJrKrLrMr)r)r)r-r*sr*c@seZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZddZddZd ddZddZd S)!rfaM ``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 r4cCs0|dkr"dt|jdddSt||dSr)rr_rr`r)r)r-rcsz 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)Nr5r4rTF)rirrjrGrHrkrTrPrrrrlrmr'r rrn isinstancer]r_rrrrr is_even) rtrUrurrrfrwevenr)r)r-rxsJ            z asinh.evalcGs|dks|ddkrtjSt|}t|dkrd|dkrd|d}| |dd||d|dS|dd}ttj|}t|}tj||||||SdSNrr5rCr4)rrTrrzrr?rrr|rvr}rYrRrr)r)r-r~s&  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||SNrrr4r5)r_rcancelrkrr rrmr*rrrlrrrrrpr rarvrrrUx0ndirr)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|dSNr4r5)rrrr>r)r)r-_eval_rewrite_as_atanhCszasinh._eval_rewrite_as_atanhcKs4t|}ttd|t|dt|tdSr@)r rrqr )rarvrixr)r)r-_eval_rewrite_as_acoshFszasinh._eval_rewrite_as_acoshcKst tt|Sr)r rr>r)r)r-_eval_rewrite_as_asinJszasinh._eval_rewrite_as_asincKsttt|ttdSr)r rr r>r)r)r-_eval_rewrite_as_acosMszasinh._eval_rewrite_as_acoscCstSrdr'r`r)r)r-rgPsz asinh.inversecCs |jdjSrrrr)r)r-rVszasinh._eval_is_zero)r4)Nr)r)r4)rJrKrLrMrcrrxrrr~rr:r?rrArCrDrErgrr)r)r)r-rfs"  -    rfc@seZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZddZddZd ddZddZd S)!rqaM ``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 r4cCs<|dkr.|jd}dt|dt|dSt||dSrr_rr)rarbrUr)r)r-rcps 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)Nr5rTF)rirrjrGrHrkr r rPrTrrr@rrmr?r+r^r_rrrrr,is_nonnegativerlis_nonpositiver) rtrU cst_tablerr-rr.rwr/r)r)r-rxws^              z acosh.evalcGs|dkrttdS|dks(|ddkr.tjSt|}t|dkrv|dkrv|d}||dd||d|dS|dd}ttj|}t|}| |t|||SdSr0) r r rrTrrzrr?rr1r)r)r-r~s $  zacosh.taylor_termNrcCs|jd}||d}|tj tjtjtjfkrJ|tj |||dS|dj r| ||rb|nd}t |j r|dj r| |dttS| | St |js|tj |||dS| |Sr3)r_rr4rrPrTrmr*rrrlrrrpr r rr5r)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|Sr8r_rrrPrr*rr:rrmrlrrr r rr;r)r)r-r:s        zacosh._eval_nseriescKs t|t|dt|dSrr=r>r)r)r-r?szacosh._eval_rewrite_as_logcKs t|dtd|t|Sr)rrr>r)r)r-rEszacosh._eval_rewrite_as_acoscKs(t|dtd|tdt|Sr@)rr rr>r)r)r-rDszacosh._eval_rewrite_as_asincKs0t|dtd|tdttt|Sr@)rr r rfr>r)r)r-_eval_rewrite_as_asinhszacosh._eval_rewrite_as_asinhcKspt|d}td|}t|dd}td||d|td|d|t|d|t||Sr@)rr rr)rarvrZsxm1Zs1mxZsx2m1r)r)r-rAs   &zacosh._eval_rewrite_as_atanhcCstSrdr)r`r)r)r-rgsz acosh.inversecCs|jddjrdSdS)Nrr4Trrr)r)r-rszacosh._eval_is_zero)r4)Nr)r)r4)rJrKrLrMrcrrxrrr~rr:r?rrErDrKrArgrr)r)r)r-rqZs"  6    rqc@sxeZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZddZdddZd S)rra) ``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 r4cCs,|dkrdd|jddSt||dSrr_rr`r)r)r-rcsz 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 AccumBoundsr5TF)rirrjrkrTrPrGrrHr r rlrm!sympy.calculus.accumulationboundsrNr r'rnr+rr_rrrrr,) rtrUrNrurr-rr.r/rwr)r)r-rx"sL             z atanh.evalcGs2|dks|ddkrtjSt|}|||SdSNrr5)rrTrr(r)r)r-r~Qszatanh.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||Sr3)r_rr4rkrrrPrmr*rrrlrrrpr r rr5r)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|Sr8rJr;r)r)r-r:os"     zatanh._eval_nseriescKstd|td|dSr@rr>r)r)r-r?szatanh._eval_rewrite_as_logcKs\td|dd}t|dt|d t| td|dt||t|Sr@)rr rf)rarvrr.r)r)r-rKs,zatanh._eval_rewrite_as_asinhcCs|jdjrdSdSrrrr)r)r-rs zatanh._eval_is_zerocCs |jdjSr)r_rrr)r)r-_eval_is_imaginaryszatanh._eval_is_imaginarycCstSrdrr`r)r)r-rgsz atanh.inverse)r4)Nr)r)r4)rJrKrLrMrcrrxrrr~rr:r?rrKrrRrgr)r)r)r-rrs  .   rrc@speZdZdZdddZeddZeeddZ dd d Z dd dZ ddZ e Z ddZddZdddZd S)rsa- ``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 r4cCs,|dkrdd|jddSt||dSrrLr`r)r)r-rcsz acoth.fdiffcCs|jr||tjkrtjS|tjkr&tjS|tjkr6tjS|jrHttdS|tj krXtjS|tj krhtjS|j r||  SnB|tj krtjSt |}|dk rt t|S|r||  S|jrtttjSdSr)rirrjrGrTrHrkr r rPrrlrmr'rrnr?)rtrUrur)r)r-rxs0        z acoth.evalcGsH|dkrt tdS|dks*|ddkr0tjSt|}|||SdSrP)r r rrTrr(r)r)r-r~s 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)Nrr4rr5)r_rr4rrmrrPrTr*rrrrrrrlrpr r r5r)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|Sr8)r_rrrPrr*rr:rrmrrrrrlr r r;r)r)r-r:s"     zacoth._eval_nseriescKs$tdd|tdd|dSr@rQr>r)r)r-r?szacoth._eval_rewrite_as_logcKs td|Srrr>r)r)r-rAszacoth._eval_rewrite_as_atanhcKsxttdt|d|t||dtdd|t||d|td|dttd|ddSr)r r rrfr>r)r)r-rKsJ*zacoth._eval_rewrite_as_asinhcCstSrdr r`r)r)r-rgsz acoth.inverse)r4)Nr)r)r4)rJrKrLrMrcrrxrrr~rr:r?rrArKrgr)r)r)r-rss    rsc@seZdZdZdddZeddZeeddZ dd d Z dd dZ dddZ ddZ e ZddZddZddZddZd S)asecha ``asech(x)`` is the inverse hyperbolic secant of ``x``. The inverse hyperbolic secant function. Examples ======== >>> from sympy import asech, sqrt, S >>> from sympy.abc import x >>> asech(x).diff(x) -1/(x*sqrt(1 - x**2)) >>> asech(1).diff(x) 0 >>> asech(1) 0 >>> asech(S(2)) I*pi/3 >>> asech(-sqrt(2)) 3*I*pi/4 >>> asech((sqrt(6) - sqrt(2))) I*pi/12 See Also ======== asinh, atanh, cosh, acoth References ========== .. [1] https://en.wikipedia.org/wiki/Hyperbolic_function .. [2] https://dlmf.nist.gov/4.37 .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSech/ r4cCs8|dkr*|jd}d|td|dSt||dSNr4rr7r5rFrarbrr)r)r-rcLs z asech.fdiffcCs|jrp|tjkrtjS|tjkr,ttdS|tjkrBttdS|jrNtjS|tjkr^tj S|tj krpttS|j rt }||kr|j r||tS||S|tjkrddlm}t|t dtdS|jrtjSdS)Nr5rrM)rirrjrGr r rHrkrPrTrrrIrrmrOrN)rtrUrIrNr)r)r-rxSs0          z asech.evalcGs|dkrtd|S|dks(|ddkr.tjSt|}t|dkr~|dkr~|d}||d|d|dd|ddS|d}ttj||}t||d|d}d||||dSdS)Nrr5r4rCr8r7)rrrTrrzrr?rr1r)r)r-r~rs ,zasech.taylor_termNrcCs|jd}||d}|tj tjtjtjfkrJ|tj |||dS|j sZd|j r| ||rh|nd}t |j r|j s|dj r|| S||dttSt |j s|tj |||dS||Sr3)r_rr4rrPrTrmr*rrrlrrrrpr r r5r)r)r-rs    zasech._eval_as_leading_termcCsddlm}|jd}||d}|tjkrtddd}ttj|dt  |dd|} tj|jd} | |} | | | } | |ds|dkr|dS|t |St tj| j|||d} | t | }| ||||||S|tjkrtddd}ttj|dt  |dd|} tj|jd} | |} | | | } | |ds|dkr|dStt|t |St tj| j|||d} | t | }| ||||||Stj||||d}|tjkr|S|js$d|jr|||r4|nd}t|jrp|jsZ|djr`| S|dttSt|js|t j||||d S|S Nr)OrT)positiver5r4r9r)r rWr_rrrPrrSr*rnseriesris_meromorphicrr:removeOrpowsimprr r rrmrlrrrrarvr|rrrWrUrrserarg1r.gres1r<r7r)r)r-r:sJ     &   &  &  &&   zasech._eval_nseriescCstSrdrr`r)r)r-rgsz asech.inversecKs,td|td|dtd|dSrr=rr)r)r-r?szasech._eval_rewrite_as_logcKs td|Sr)rqrr)r)r-rCszasech._eval_rewrite_as_acoshcKs:td|dtdd|ttt|ttjSr)rr rfr rr?rr)r)r-rKs,zasech._eval_rewrite_as_asinhcKsttdt|td|tdt| t|tdt|dt|d td|dt|dttd|dSr@)r r rrrr>r)r)r-rAsZ.zasech._eval_rewrite_as_atanhcKs8td|dtdd|tdttt|Sr@)rr r acschr>r)r)r-_eval_rewrite_as_acschszasech._eval_rewrite_as_acsch)r4)Nr)r)r4)rJrKrLrMrcrrxrrr~rr:rgr?rrCrKrArcr)r)r)r-rS&s %     - rSc@seZdZdZdddZeddZeeddZ dd d Z dd dZ dddZ ddZ e ZddZddZddZddZd S)rba ``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/ r4cCs@|dkr2|jd}d|dtdd|dSt||dSrTrFrUr)r)r-rcs  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||  SdSr@)rirrjrGrTrHrkrmrPrrrrrEr is_infinitern)rtrUrIr)r)r-rx 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)Nrr5r4rCr8) rrrTrrzrr?rrr1r)r)r-r~+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||Sr3)r_rr4r rrTr*rrrmrrrrrrrpr rlr5r)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|SrV)r rWr_rr rrbr*rrYrrZr rrrPr:r[rr\rrrmrrrrrrlr]r)r)r-r:RsN     $   *& &  *&     zacsch._eval_nseriescCstSrdrr`r)r)r-rgsz acsch.inversecKs td|td|ddSr@r=rr)r)r-r?szacsch._eval_rewrite_as_logcKs td|Srrerr)r)r-rKszacsch._eval_rewrite_as_asinhcKs:ttdt|tt|dtt|ttjSr)r rrqr rr?rr)r)r-rCs   zacsch._eval_rewrite_as_acoshcKsF|d}|d}t| |ttjt|d |tt|Sr)rr rr?rr)rarUrarg2Zarg2p1r)r)r-rAs zacsch._eval_rewrite_as_atanhcCs |jdjSr)r_rdrr)r)r-rszacsch._eval_is_zero)r4)Nr)r)r4)rJrKrLrMrcrrxrrr~rr:rgr?rrKrCrArr)r)r)r-rbs %  !   0 rbN)I sympy.corerrrZsympy.core.addrsympy.core.functionrrsympy.core.logicrr r sympy.core.numbersr r r sympy.core.symbolr(sympy.functions.combinatorial.factorialsrrr%sympy.functions.combinatorial.numbersrrr$sympy.functions.elementary.complexesrrr&sympy.functions.elementary.exponentialrrr#sympy.functions.elementary.integersr(sympy.functions.elementary.miscellaneousrrrrrr r!r"r#r$r%r&r'sympy.polys.specialpolysr(r3r@rErIr1r\r]r^rrrrrr*rfrqrrrsrSrbr)r)r)r-sZ    4    # "UwV-JG;%/7