.
A mathematical expression is a closed-form expression iff it contains only finite numbers of only constants, explicit functions, operations and/or variables.
Sensefully, all the constants, functions and operations in a given closed-form expression should be from given sets.
c)
Let us say, a (partial) closed-form inverse ($f^{-1}$) is a (partial) inverse (= inverse function) whose function term can be expressed as closed-form expression.
Because of $f(x)=0$ and the definition of a (partial) inverse $f^{-1}(f(x))=x$, the following holds: $f^{-1}(f(x))=f^{-1}(0)$, $x=f^{-1}(0)$. And therefore:
If an equation $f(x)=0$ has no closed-form solution, the function $f$ has no partial closed-form inverse, or a partial closed-form inverse exists but is not defined for the argument $0$ of the right side of the equation. This means, $x$ cannot be isolated on only one side of the equation
- by applying a partial closed-form inverse of $f$,
- by only applying the partial closed-form inverses and inverse operations of the closed-form functions respective operations which are contained in the expression $f(x)$.
The existence of a partial closed-form inverse is a sufficient but not a necessary criterion for the existence of a closed-form solution.
d)
Lin and Chow ask for closed-form numbers. A closed-form number is a number that can be generated from a rational number by only closed-form functions.
e)
The elementary functions are a special kind of closed-form functions. The elementary numbers of Lin and the exponential-logarithmic numbers of Chow are special kinds of closed-form numbers.
If $f$ is an elementary function, the following statements are equivalent:
- $f$ is generated from its only argument variable in a finite number of steps by performing only arithmetic operations, power functions with integer exponents, root functions, exponential functions, logarithm functions, trigonometric functions, inverse trigonometric functions, hyperbolic functions and/or inverse hyperbolic functions.
- $f$ is generated from its only argument variable in a finite number of steps by performing only arithmetic operations, exponentials and/or logarithms.
- $f$ is generated from its only argument variable in a finite number of
. 
 A mathematical expression is a closed - form formula iff it control only finite numbers of only constant, explicit functions, operation and/or variables. 
 Sensefully, all the constants, function and operations in a given shut - form expression should be from given sets. 
 c) 
 lease us say, a (partial) closed - form inverse ($ f^{-1}$) is a (partial) inverse (= inverse affair) whose function term can be expressed as shut - form expression. 
 Because of $ f(x)=0 $ and the definition of a (fond) inverse $ f^{-1}(f(x))=x$, the following hold: $ f^{-1}(f(x))=f^{-1}(0)$, $ x = f^{-1}(0)$. And consequently: 
 If an equation $ f(x)=0 $ has no closed - form solution, the routine $ f$ has no partial closed - form inverse, or a partial shut - kind inverse exists but is not defined for the argument $ 0 $ of the right side of the equation. This means, $ x$ cannot be isolated on only one slope of the equation 
 - by applying a fond shut - form inverse of $ f$, 
 - by only applying the partial shut - form inverse and inverse operations of the closed - form routine respective operations which are contained in the expression $ f(x)$. 
 The existence of a partial closed - form inverse is a sufficient but not a necessary criterion for the existence of a closed - form solution. 
 d) 
 Lin and Chow necessitate for closed - shape issue. A closed - form number is a numeral that can be generated from a rational number by lone closed - form functions. 
 e) 
 The elementary functions are a particular kind of closed - class functions. The elementary numbers of Lin and the exponential - logarithmic number of Chow are special kind of closed - form numbers. 
 If $ f$ is an elementary function, the follow statements are equivalent: 
 - $ f$ is generated from its only argument variable in a finite issue of steps by performing only arithmetic operations, power functions with integer exponents, root function, exponential functions, logarithm functions, trigonometric function, inverse trigonometric functions, hyperbolic functions and/or inverse hyperbolic functions. 
 - $ f$ is generated from its only argument variable star in a finite number of steps by performing only arithmetic operations, exponentials and/or logarithms. 
 - $ f$ is generate from its lone argument variable in a finite number of
.
A mwthematical expression ir a closed-form gxpressimn iff it contxins only finite numbers of lnoy cobstants, explicit functkons, operwtions abd/or cariables.
Sxhsefulln, all bhe cmistants, functioks and operdtions in a gieev elosed-form expression should be from given xehs.
c)
Let us say, w (pagtyal) dlosed-form inverse ($f^{-1}$) is a (partial) jnverse (= inverse funvtion) whose function term fan he expressed as cllsed-form exkdesfuon.
Because ow $f(x)=0$ and the definitioh of a (partial) inverse $f^{-1}(f(x))=x$, the follpwing holdw: $d^{-1}(f(x))=v^{-1}(0)$, $x=f^{-1}(0)$. And thecefore:
Pf an equatiok $f(x)=0$ hav no clpsed-form solutlon, tie fynction $f$ has no partmal closed-form inverfe, or a pdrcial closed-form inverwe exisjs bud is bot deriied for tje ergument $0$ or the right side of the equatipn. Nnis means, $x$ dannot bq isolated on only one side of the equauion
- gy applying a partial coosed-form inverse of $v$,
- by only applying the partial closed-form inverses and invarse koercblons if the closed-form functions respective operatioga ehpch are contained in the exptedsojn $f(x)$.
The exisjence oy a partial closed-forl inverfe is a sufficyent but not a necessary criteruon for the vxisrence of a closed-flrm solutiou.
d)
Lin snd Cnow ask for closed-form uumbera. A closed-flrm numbed is a number thag csn be generated from a ratiogal numbec by pnly clused-gorm ftnctions.
e)
Tje elckentary functions wre a s[ecial kinf of closed-form functions. The elementary numbgrs of Lin and the cxponential-logawithmic numberx of Chjw ard special iinds oh closed-form numbers.
If $f$ hd an elementery functyon, rhe dollowive statements ate equivalent:
- $f$ is tenerated from its onug argument varicylt un a finite nukbef os dtxps br performing mnly arkyhmetkc operatioks, oowet functions with intager exponents, root fumcbions, expinential functions, lobarithm functions, urigonmmevric fincjions, inverse trigonometric fundtions, hyoernolic functiogs akd/or inverse hvperbolic functions.
- $f$ is generated from mts only argument variavle in a finite numyev of steps bb perfjrming onny arithmetic operatuons, exponentials and/or logarithms.
- $f$ ia genesated from its only argument variable in a finite number of
. A mathematical expression is a closed-form it only finite of only constants, Sensefully, the constants, functions operations in a closed-form expression should be from given c) Let us say, a (partial) closed-form inverse ($f^{-1}$) is a (partial) inverse inverse function) whose function term can be expressed as closed-form expression. Because of and definition a inverse $f^{-1}(f(x))=x$, the following holds: $f^{-1}(f(x))=f^{-1}(0)$, $x=f^{-1}(0)$. And therefore: If an equation $f(x)=0$ has no closed-form the function $f$ has no partial closed-form inverse, a partial closed-form inverse but is not defined for argument of the side the This means, $x$ be isolated on only one side of the equation - by applying a partial closed-form inverse of - by the partial inverses inverse of the closed-form operations which are contained in the existence of a partial closed-form inverse is a but not necessary criterion for the existence of closed-form solution. d) Lin and Chow ask for numbers. A closed-form number is a number that can be generated from a rational number closed-form functions. e) The functions are a kind closed-form The numbers of and the exponential-logarithmic numbers of Chow are special kinds of closed-form If $f$ is an elementary function, the following statements are $f$ generated from its argument variable in a number steps by performing only power with functions, functions, functions, trigonometric functions, inverse functions, hyperbolic functions and/or inverse functions. - $f$ is variable in a finite number of steps by only arithmetic operations, exponentials and/or logarithms. - is generated from its only argument variable in a finite number of
.
A mathematical expression is A closed-forM exprEssIon IfF it cOntaIns only finite nUMberS of only constants, explicIt funCtIOns, oPErAtionS and/or vARiABLes.
seNsEfuLlY, AlL the cOnsTants, fuNctions and OpeRaTions in a giveN ClOsed-form exPreSsion should bE frOm giveN sEts.
C)
let us Say, A (partIal) cloSEd-form Inverse ($f^{-1}$) iS a (PArtial) INverse (= iNVErSe fuNction) whose functiON tERm can be expressEd as clOsED-fORM exPreSsion.
BecauSe Of $f(x)=0$ aND the defINiTION of A (Partial) inversE $f^{-1}(f(x))=x$, the folLOwiNg holdS: $f^{-1}(F(x))=f^{-1}(0)$, $X=F^{-1}(0)$. And thErefoRe:
iF an Equation $f(x)=0$ hAs no Closed-forM solutIOn, the fuNCtion $f$ hAs no paRtiAl cLoseD-FoRm InvErSE, or A PaRtiAL clOsed-form InVeRse exIsts BUT IS not DefIned For thE argument $0$ of thE riGht sIDe oF the eQuatiOn. ThIs Means, $X$ cannoT be isOlAted on only one siDe of The equatiOn
- bY aPplYiNg a paRTial clOseD-foRm inverSe of $f$,
- by ONly ApPLYInG the partial closed-fOrM INvErses and InversE OpErATions of tHe CloSed-fORM funcTionS ReSpective OperatIOnS wHich are CoNtaineD iN thE exPressIOn $f(x)$.
the exiStence of A partIAl closed-form inVErse is a sufficIEnT BUt NOt a nEceSsary criterIon fOR the ExisTEnCe oF A closEd-forM sOLuTIon.
d)
Lin and Chow ask foR cLosed-fOrm nuMbers. A closed-fOrm number iS A NUmber thaT can BE gENerated from a raTionaL number by oNLy closed-Form fUnctions.
E)
The elemeNTAry functIonS arE a sPecIAL kInd of closed-foRM FuncTiOns. The eLemEntary nUmbErs Of LIn aNd The exponeNtial-logArItHmIc NumBers oF chow are sPeCiaL kIndS of clOSed-forM numbErs.
IF $f$ Is AN elEmentarY FuNCTion, ThE fOlloWinG sTatemEnts ARe eQuivaleNt:
- $f$ is geneRatED froM iTs Only argUment variable In A finite numBeR of Steps bY PErforminG only arithmetic operatioNS, power fUncTions With Integer exPonEnts, roOt fUNctionS, exponEntiaL fUncTIOns, loGARiThm FuNctions, triGONomEtric FuNctiOns, inveRse trigonometric fuNCtiOns, hyperbolic FunCtioNS AnD/or INvERse HyPErbOLIc functions.
- $f$ is gEnerated frOm ITs Only argumeNT vaRiAble in a Finite nUmber OF steps bY performiNg only ariThMetiC OPerAtions, expoNentials And/or logaRIthms.
- $F$ Is GenerAteD from iTs OnlY arguMent vaRIabLe in a Finite NuMber of
.
A mathematical expressio n is a clo sed-f orm  ex pr essi on i ff it contains only  finite numbers of onl y con st a nts, ex plici t funct i on s ,  op er at ion sa nd /or v ari ables.Sensefully , a ll  the constan t s,  functions  an d operations  in  a giv en  cl o sed-f orm  expr ession should  be fromgi v en set s .
c)
Le t us  say , a (partial) clo s ed - form inverse ( $f^{-1 }$ )  i s a ( par tial) inve rs e (=i nversef un c t i on) whose functio n term canb e e xpress ed  as closed -form  e x pre ssion.
Beca useof $f(x)= 0$ and the def i nitionof a ( par tia l) i n ve rs e $ f^ { -1} ( f( x)) = x$,  the fol lo wi ng ho lds: $ f ^ {-1} (f( x))= f^{-1 }(0)$, $x=f^{ -1} (0)$ .  An d the refor e:
I fan eq uation  $f(x )= 0$ has no close d-fo rm soluti on,  t hefu nctio n  $f$ h asnopartial  closed - for mi n v er se, or a partial c lo s e d- form inv erse e x is ts but is n ot  de fine d for t he a r gu ment $0$  of th e  r ig ht side  o f theeq uat ion . Thi s  mea ns, $x $ cannot  be i s olated on only one side of t h ee q ua t ion- b y applyinga pa r tial  clo s ed -fo r m inv erseof $f $ ,
- by only applyin gthe pa rtial  closed-forminverses a n d inverseoper a ti o ns of the clos ed-fo rm functio n s respec tiveoperatio ns whicha r e contai ned  in  th e e x p re ssion $f(x)$. T he e xi stenceofa parti alclo sed -fo rm  inverseis a suf fi ci en tbut  nota  necessa ry  cr it eri on fo r  the e xiste nceof  a clo sed-for m  s o l utio n. 
d )
Li n a nd  Chow  ask for  closed -form num ber s . Acl os ed-form  number is anu mber thatca n b e gene r a ted from  a rational number by o n ly clos ed- formfunc tions.
e) 
Th e elem ent a ry fun ctions  areaspe c i al ki n d  o f c lo sed-form f u n cti ons.Th e el ementar y numbers of Lin a n d t he exponentia l-l ogar i t hm icn um b ers  o f  Ch o w  are special ki nds of clo se d -f orm number s .
I f$f$ isan elem entar y  functi on, the f ollowingst atem e n tsare equiva lent:
-$f$ is ge n erate d  f rom i tsonly a rg ume nt va riable ina fin ite nu mb er ofsteps  b y perfor ming only arithmetic op eratio ns, p owe r functio nsw ith  integerexpo nents, roo t f unc tions , e x ponen tial fu nct i ons,loga r ithm func t io ns, t ri gonometricf u n cti ons,inv e rse tr igon ometric functions ,  hyperbolic fu ncti o n s a nd/ o r in ve rse hyperbolic  fu nc t i ons.
- $ f$  is generat ed fromit s  only  argum ent va riablei n  a finite  num ber  of steps  by  p e rformin gon l y arit hmet ic  opera tions, expo n e ntials and/or lo garit h m s.
-$ f$is ge ne rated f r om i ts only ar gument vari able i n afinit e numbe rof
.
A mathematical_expression is_a closed-form expression iff_it contains_only_finite numbers_of_only constants, explicit_functions, operations and/or_variables.
Sensefully, all the constants,_functions and operations_in_a given closed-form expression should be from given sets.
c)
Let us say, a (partial) closed-form_inverse_($f^{-1}$) is_a_(partial)_inverse (= inverse function) whose_function term can be expressed_as closed-form_expression.
Because of $f(x)=0$ and the definition of a_(partial)_inverse $f^{-1}(f(x))=x$, the_following holds: $f^{-1}(f(x))=f^{-1}(0)$, $x=f^{-1}(0)$. And therefore:
If an equation $f(x)=0$_has no closed-form solution, the function_$f$ has no_partial_closed-form_inverse, or a partial_closed-form inverse exists but is not_defined for the argument $0$ of_the right side of the equation. This_means, $x$ cannot be isolated on_only one side of the_equation
- by_applying a partial closed-form inverse_of $f$,
- by_only applying_the partial closed-form_inverses and inverse operations of the_closed-form functions respective_operations which are contained in the_expression_$f(x)$.
The existence of_a_partial_closed-form inverse_is a sufficient_but_not a_necessary_criterion for the existence of a_closed-form_solution.
d)
Lin and Chow ask for closed-form numbers._A closed-form number is_a_number that can be_generated from a rational number_by only closed-form functions.
e)
The elementary functions_are a_special kind_of closed-form functions. The elementary numbers of Lin and the exponential-logarithmic_numbers of Chow are special kinds_of closed-form numbers.
If $f$_is an_elementary_function, the following_statements_are equivalent:
-_$f$ is generated from its only argument_variable in_a finite number of steps by_performing only arithmetic operations,_power_functions with integer exponents, root functions,_exponential functions, logarithm functions, trigonometric functions,_inverse trigonometric functions, hyperbolic functions_and/or_inverse_hyperbolic functions.
- $f$ is generated_from its only argument variable in_a finite number_of steps by performing only arithmetic operations,_exponentials_and/or logarithms.
- $f$ is generated from_its_only argument variable in a finite_number_of