PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION

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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: (prited versio); ISSN: (o-lie versio) url: doi: PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION Ampo Dhamacharoe, Portip Kasempi 2,2 Departmet of Mathematics Burapha Uiversity Bagsae, Choburi, 203, THAILAND Abstract: This paper ivestigates the fuctioal th root of the idetity fuctio i the ratioal form f(x) = ax+b. The formula is aalyzed via cx+d liear trasformatio i order to obtai the coditios for a, b, c ad d. Some examples i liear ratioal form or i other forms are provided. The ratioal form of quadratic fuctios is also discussed. AMS Subject Classificatio: 39B72, 39B2, 26C5, 39B72, 40C05 Key Words: fuctioal root, Babbage s equatio, ratioal form, fuctioal equatio, matrix represetatio. Itroductio A fuctioal root is a fuctio which whe applied a give umber of times, equals agive fuctio. Give f(x), the th ordered composite fuctio, f [ (x) ca be computed by applyig the fuctio f, times. The formal defiitio ca be writte as: f [ (x) = f(f [ (x)), =,2,3,..., () where f [0 (x) = x, [6. If the fuctio g(x) is give, the fuctioal th root of Received: August 9, 203 Correspodece author c 204 Academic Publicatios, Ltd. url:

2 0 A. Dhamacharoe, P. Kasempi g is the solutio of the fuctioal equatio f [ (x) = g(x). (2) For example, x 2 +2 is the fuctioal 2 d root of x 4 +4x If g(x) = x, the equatio (2) is called Babbage s equatio (amed after Babbage, [7, 8, 0). I this paper, we cocetrate o the solutios of Babbage s equatio, which are the fuctioal root of the idetity fuctio. (This root is also called the fuctioal th root of uity, or the fuctioal th root of x.) The fuctioal equatio (2), ad other more geeral fuctioal equatios, have bee studied i respect to the properties of the solutios, ad some specific solutios are provided. (see [2, 3, 4, 5, 6, 9, 0, ). For example, the fuctio f(x) = x is the oly real cotiuous solutio of the Babbage equatio whe is odd. The fuctio f(x) = c x, where c is a costat, f(x) = r 2 x 2, 0 x r, ad f(x) = 2x+3 are the fuctioal secod roots. These roots are x 2 also the fuctioal th root of x whe is eve. Some of the fuctioal third roots of uity are: (a) f(x) = x r, where r = + 3i ; 2 (b) f(x) = + 3i x; 2 (c) f(x) = { x 2,if 3k x < 3k x+,otherwise, for k = 0,±,±2,±3,...; (d) f(x) = x 3, where x. x+ (a) ad (b) are the complex valued fuctios; (c) has ifiite discotiuous poits; (d) is iterestig sice it is a real valued fuctio, which is cotiuous but a sigle poit. I accordace with the examples above, the fuctioal th root of x ca be obtaied usig the same patter. (a) f(x) = x r, where r is the th root of uity.

3 ON DERIVATION OF RATIONAL SOLUTIONS... (b) f(x) = rx, where r is the th root of uity. (c) f(x) = { x +,if k x < k x+,otherwise, for k = 0,±,±2,±3,...; (d) f(x) = β +αx γ +δx, where x γ δ. It is ot difficult to verify that all fuctios are solutios. (d) is the ratioal form fuctio where γ ad δ do ot vaish at the same time. This fuctio is a well kow solutio for Babbage s equatio, where the coditio for the ukow costat is: (d) δ = β2 2βγcos 2kπ +γ2 2α(+cos 2kπ ) for some costat iteger k relatively primed to (see [, 9). If f(x) is a fuctioal th root of the idetity fuctio, the the fuctio F, defied by F(x) = g fg(x) is also a fuctioal root, for ay fuctio g ad its iverse g (see [9). This fact is easily see by performig the compositio times. Therefore, oe ca costruct a ifiite umber of such fuctioal roots. The fuctio F(x) ) 3 is the fuctioal third root of this type. =( x /3 3 x /3 + Now we are iterested i the derivatio of the ratioal form solutios as i (d), ad wish to ivestigate this formula to determie whether there is aother solutio of this form. From (d), if δ = 0, the ratioal form becomes a liear fuctio which is reduced to x, or c x if = 2. Assumig that δ 0, the ratioal form ca be reduced to ax+b x+d. Therefore, the problem is to fid a, b ad d so that f(x) = ax+b x+d is the fuctioal th root of x. It is obvious that if m ad are positive itegers ad m, ad if f is a m th fuctioal root of x, the f is also a th fuctioal root of x. Here we cocetrate o the th root, i which is miimal; that is there is o m <, such that f is the m th root.

4 2 A. Dhamacharoe, P. Kasempi 2. Matrix Represetatio ad Derivatio Sice x = x for all real umbers x, so it ca be represeted i the vector form [ x. Thus, the ratioal form f(x) = ax+b ca be writte i the vector x+d form [ [ [ ax+b a b x = x+d d [ a b where is a 2 2 matrix. d The composite fuctio f(f(x)) is a(ax+b)+b(x+d), which ca be represeted i the followig ax+b+d(x+d) form: [ a(ax+b)+b(x+d) ax+b+d(x+d) = [ a b d [ ax+b x+d = [ a b d 2 [ x [ By iductio, the th composite fuctio f [ (x) ca be represeted as [ a b x d For coveiece, i all further aalysis, we shall give a formal defiitio for vector represetatios of ratioal forms. [ u Defiitio. The vector is a represetatio of a umber x if x = v u, where u ad v are real umbers ad v 0. v Defiitio 2. We say that thetwo vectors x ad y areratioal equivalet, writte x y, if both x ad y are represetatios of the same umber. The followigs are useful properties which are easily obtaied from the defiitios: Property. Let x ad y be vectors i R 2, the x y if ad oly if x = ky for some umbers k 0. Property 2. A ratioal equivalet is a equivalet relatio. [ [ x u Property 3. is a represetatio of x. If v 0, the v if ad oly if x = u v. [ x

5 ON DERIVATION OF RATIONAL SOLUTIONS... 3 Property 4. Each vector [ x vector of the form. ad [ u v Defiitio 3. Let s defie the equivalet class: C x = {y R 2 y [ x C = {, with v 0, is ratioal equivalet to a [ x x R},x R}, [ Thus, y C x if ad oly if y is a represetatio of x. We sometimes write x to represet C x. [ a b Defiitio 4. A 2 2 matrix is a ratioal-form trasformatio c d if c 0. Defiitio 5. Two 2 2 matrices A ad B are ratioal equivalet, writte A B, if A = kb for some umber k 0. (Note that the symbol used here, has a differet meaig to that used i elemetary row operatio.) Property 5. Two 2 2 matrices A ad B are ratioal equivalet if ad oly if Ax Bx for all x C. Property 6. The ratioal equivalet for matrices is a equivalet relatio Property 7. Each [ ratioal-form trasformatio is ratioal equivalet to a b a matrix of the form. d Property 8. () The ratioal fuctio f(x) = ax+b is represeted by [ [ x+d a b x. d (2) The th orderedcompositefuctiooff isrepresetedby [ a b d [ x.

6 4 A. Dhamacharoe, P. Kasempi We ow are ready to derive the formula for the root. Cosider the equatio (2) whe g(x) = x : Its represetatio is: [ [ a b x d [ x = k f [ (x) = x., for some o-zero umber k. (3) We are lookig for a, b ad d so that the equatio (3) is satisfied for some o-zero [ umber k, ad for all x R. That is, we seek to fid a, b ad d such [ a b 0 that =k for some umber k. d 0 Let A = [ a b d. The characteristic values of A are: λ = 2 (a+d (a+d) 2 4(ad b)) λ 2 = 2 (a+d+ (a+d) 2 4(ad b)) (4a) (4b) Suppose that λ λ 2, the the matrix A is diagoalizable. There is a ivertible matrix P such that: A = PDP [ λ 0 where D =. The we have 0 λ 2 A = PD P [ λ Sice D = 0 0 λ, demadig that A be a scalar matrix, it implies 2 that λ = λ 2. Therefore, the problem ow becomes fidig a, b ad d so that λ = λ 2. If λ = λ 2, the matrix A is ot diagoalizable, therefore there is o solutio for the case =. For = 2, λ 2 = λ2 2 ad λ λ 2 if ad oly if a+d = 0. Let d = a, the Therefore A = λ 2 = λ 2 2 = (ad b) = ( a 2 b) = a 2 +b. [ a b. a

7 ON DERIVATION OF RATIONAL SOLUTIONS... 5 The fuctioal secod root of the idetity fuctio i the ratioal form is f(x) = ax+b x a (5) forayreal umberaadbsuchthatb a 2. Thisfuctioisalsoafuctioal th root of the idetity fuctio whe is eve. We will ivestigate for the geeral case >. We have that λ λ 2 if ad oly if (a+d) 2 4(ad b) 0. If (a+d) 2 4(ad b) > 0 the λ ad λ 2 are real, λ = λ 2 ad λ λ 2 if ad oly if a +d = 0 ad is eve. This case leads to the doe case = 2, which gives the solutio (5). Now, suppose that (a + d) 2 4(ad b) < 0, the λ ad λ 2 are complex umbers. We write λ = 2 (a+d i 4(ad b) (a+d) 2 ) λ 2 = 2 (a+d+i 4(ad b) (a+d) 2 ) (6a) (6b) Let λ = λ 2 = M, for some real umber M. We write i the complex form: M = M (cos(2kπ +θ)+isi(2kπ +θ)) (7) where M = λ = λ, k = 0,,2,..., ad θ = 0 or π accordig to whether M is positive or egative. From (6a) ad (6b), ote that ad b > 0, we have The λ = λ 2 = 2 (a+d) 2 +4(ad b) (a+d) 2 = ad b (8) M = (ad b) /2 (9) From (7), the de Moivre theorem gives λ ad λ 2 i the form: λ = ( ) ( ) 2kπ +π 2kπ +π ad b(cos +isi ) (0a) or λ = ( ) 2kπ ad b(cos +isi for some umber k = 0,,2,...,. ( 2kπ ) ) (0b)

8 6 A. Dhamacharoe, P. Kasempi Compare (6a) ad (6b) to (0), the we have: 2 (a+d) = ( ) 2kπ +π ad bcos or 2 (a+d) = ( ) 2kπ ad bcos which ca be combied to be sigle equatio: 2 (a+d) = ( ) kπ ad bcos, k = 0,,2,...,2 ( ) kπ If cos = 0, we have a+d = 0. The we have (5). ( ) kπ If cos = ±, the the imagiary part is zero, which is ot the case. I other cases, we have b = ad (a+d)2 ( ) () kπ 4cos 2 ( ) kπ for some iteger k =,2,..., 2 (Sice for k > 2 the value of cos2 is repeated). The umber k should be relatively prime to, sice otherwise the fuctio f will ot be the miimal th root. From the above aalysis, the coclusio will be i accordace with the followig theorem: Theorem. The ratioal fuctioal th root of the idetity fuctio is f(x) = ax+b, where a, b ad d satisfy the equatio (). x+d The result from our aalysis may look a little differet from the solutio give by the formula (d), but with a little work, oe ca show that they actually coicide. The followig is a list of the fuctioal th root of uity for ( 0. ) Note kπ that for > 2, a+d 0; for = 7 ad 9, the exact value of cos is ot kow. 3. Fuctioal Roots of the Form ax2 +bx+c x 2 +dx+e Now we look for the ratioal solutio of the form f(x) = ax2 +bx+c x 2 +dx+e. For example the fuctio f(x) = 2x2 3x+2 x 2 is the fuctioal cube root of x. 2x 3

9 ON DERIVATION OF RATIONAL SOLUTIONS... 7 Table : List of the fuctioal th root of uity f(x) a =,d = ax+b 2 x a, b x+b a2 x, b ax+ad (a+d) 2 x 3 3 x+d x+ ax+ad 2 4 (a+d)2 x x+d x+ ax+ad 2 5 (3± 5)(a+d) 2 x (5±2 5) x+d x+ ax+ad 3 6 (a+d)2 x 3 x+d x+ ax+ad 2 8 (2± 2)(a+d) 2 x (3±2 2) x+d x+ ax+ad 0 0 (5± 5)(a+d) 2 x 5 (5±2 5) x+d x+ This fact ca be easily see by reducig the fractio to be 2x 7 which is x+ i the liear ratioal form, where a = 2 ad d =. We wat to ivestigate this fuctio to see whether the fuctioal roots are of this type. Note that aalyzig this fuctio caot be doe by usig the matrix represetatio, sice the 3 3 matrix does ot match the trasformatio. Before aalyzig the fuctio, we should metio here the importat property of all the fuctioal roots of uity. Firstly, we ivestigate the set S i which the fuctio satisfies the equatio for all x S. Let D be the domai ad R the rage of a root. By defiitio of a composite fuctio, we have R D. Suppose that x D R, the there is o x R, such that x = f(f [ (x)) = f(x). Therefore the set S must be defied so that it is both the domai ad the rage of f. Property 3.. The domai ad rage of a fuctioal root are the same. Property 3.2. Each root is ivertible for all x i its domai. This fact ca be see easily from f(f [ (x)) = x = f [ (f(x)) which implies f (x) = f [ (x). Therefore, ay fuctioal root must be a oe-tooe fuctio. Now we are ready to aalyze the fuctio f(x) = ax2 +bx+c x 2, where the +dx+e fractio caot be reduced ito a liear form. Let S be the set of exteded

10 8 A. Dhamacharoe, P. Kasempi real umbers; that is the set of real umbers with ad. The value of fuctio at or ca be defied accordig to the limit at or. Also f(a - ) ca be defied to be or accordig to the left limit, ad f(a + ) the right limit of f at a. Case : a 0, ad the umerator ax 2 +bx+c ca be factored ito real liear fuctio. The there are two real zeros, which implies that the fuctio is ot oe-to-oe. Thus, f(x) is ot a fuctioal root. If the factors are the same, the the sig of the ratioal fuctio ear that zero are the same. There must be two poits ear the zero such that the ratioal fuctio assumes the same value. Thus, the fuctio is ot oe-to-oe, ad therefore ot a fuctioal root. Case 2 : a 0, ad the umerator ax 2 +bx+c caot be real factored. The the umerator must be positive for all real umbers, or egative for all real umbers. Case 2. : If the deomiator ca be real factored, the there are two poits s ad t, such that the oe-side limit at those poits is or. Sice there are four oe-side limits with oly two values or, the there are two limits which approach the same value or. Sice the fuctio f is cotiuous everywhere, with the exceptio of those two poits, the there exist two distict poits, oe ear s, ad the other ear t, where f assumes the same value. Thus, the fuctio f is ot oe-to-oe. If s = t, the the coclusio will be the same, sice there are two oe-side limits that approach oly oe value. Case 2.2 : If the deomiator caot be real factored, the the fuctio f(x) must be either positive or egative for all real umbers. The the rage of the fuctio is ot the set of exteded real umbers. Thus, f is ot the root of uity. bx+c Case 3 : a = 0, sothe fuctio f is f(x) = x 2. If the deomiator +dx+e ca be real factored, the by the reaso as stipulated i 2., f is ot oe-to-oe. If the deomiator caot be real factored, the the fuctio is bouded sice there is o discotiuity poit, ad lim f(x) = lim f(x) = 0. Therefore the x x rage of the fuctio is ot the set of exteded real umbers. Case 4 : Cosider the fuctio of the form ax2 +bx+c where a 0. x+e Observe that lim f(x) ad lim f(x) approach the differet values or, x x ad also lim ad lim approach the differet values or. x e f(x) x e +f(x) Therefore, i accordace to reasos similar to Case 2., the fuctio is ot oe-to-oe.

11 ON DERIVATION OF RATIONAL SOLUTIONS... 9 From our aalysis, a coclusio ca be draw about the fractios of quadratic fuctios, ad the fractios of polyomials degree greater tha oe, except where they ca be reduced to the fractios of liear fuctios. Theorem 2. There is o fuctioal root of uity i the ratioal form of quadratic fuctios. Corollary. There is o fuctioal root of uity i the ratioal form of polyomials degree greater tha oe. 4. Coclusios There are ifiite solutios for Babbage s equatio. Although the ratioal form solutio of Babbage s equatio is well kow, its derivatio give by Boole [ is somewhat complicated. I this report, we showed the derivatio of the ratioal solutios by mea of special liear trasformatio. The derived formula is equivalet to the old formula, but it is a little differet i its form. Examples of particular solutios for some were also provided. Aalyzig the fuctios of the ratioal form of quadratics shows that there is o solutio i that form, except that ca be reduced ito fractios of liear fuctios. These results ca also be exteded to the fractios of polyomials of degree greater tha oe. Ackowledgmets This work was completed with the support of the Faculty of Sciece Research Fud, Burapha Uiversity. Refereces [ G. Boole, A Treatise o the Calculus of Fiite Differeces, Cambridge, (Prited by C. J. Clay, M.A. at the Uiversity Press) (860), [2 B. Choczewski (Gliwice), O cotiuous solutios of some fuctioal equatios of the -th Order, Aales Poloici Mathematici XI(96), [3 G. M. Ewig ad W. R. Utz, Cotiuous Solutios of the Fuctioal Equatio f (x) = f(x), Caada. J. Math. 5 (953), 0-03.

12 20 A. Dhamacharoe, P. Kasempi [4 R. Isaacs, Iterates of Fractioal Order, Caad. J. Math. 2 (950), [5 M. Kuczma (Krakow), O Some Futioal Equatio Cotaiig Iteratios of the Ukow Fuctio, Aales Poloici Mathematici XI (96), - 5. [6 M. Kuczma (Krakow), O the Futioal Equatio φ (x) = g(x), Aales Poloici Mathematici XI (96), [7 A. Mach, O some Fuctioal Equatios Ivolvig Babbage Equatio, Result i Mathematics 5(-2) (2007), [8 A. D. Polyai ad A. V. Mazhirov, Hadbook of Itegral Equatios: Exact Solutios (Supplemet some fuctioal equatios), Faktorial, Moscow, (998). [9 J. F. Ritt, O Certai Real Solutios of Babbage s Fuctioal Equatio, Aals of Mathematics, Secod Series,7(3) (96), 3-22.(Stable URL: [0 C. G. Small, Fuctioal Equatios ad How to Solve Them, Spriger Sciece Busiess Media, LLC, (2007). [ F. J. McCarthy,Fuctioal th Roots of Uity, The Mathematical Gazette, 64(428), (980).