OA 431 B3 UC-NRLF EXAMPLES ^Dlutioni^ FUNCTIONAL EQUATIONS CHARLES BABBAGE, a.m. f.r.s. l.&e. fx.p.s. AND SECRETARY TO THE ASTRONOMICAL SOCIETV OF LONDON. • '•'•• m m NOTICE. The object of the following Examples of Func- tional Equations^ is to render a subject of considerable interest, more accessible to mathematical students, than it has hitherto been. It is, perhaps, that subject of all others, which most requires the assistance of particular instances, in order fully to comprehend the meaning of its symbols, which are of the most extreme generality ; that assistance is also more particularly required in this branch of science, in consequence of its never yet having found its way into an Elementary Treatise. Oct. 20. 1820. 437522 OF FUNCTIONAL EQUATIONS If a function a is of such a form, that, when it is twice performed on a quantity, the result is the quantity itself, or if a^ (x) = Xy then it is called a periodic function of the second order, if a" (x) = Xy then it is termed a periodic func- tion of the n^^ order, thus when a(x) = a -^ x the second function, or a{ax)=:a{a — x) = a — (a—x)=za — a + x=:x. If a(x) =— L_, then a^X=:a{ax) = I == ^ ~ "^ = ^ " ^ , 1 1 — A? — 1 X I - X and - » ax — I 1 a' a: = a* a a: = = _ a. X = 1 — I — X = Xy the first of these examples is a periodic function of the second, the last is a periodic function of the third order. Prob. 1. To find periodic functions of the second order. Since such functions must satisfy the equation \lf''x = Xy we have or yj/ must be such a function, that it shall be the same as its inverse ; if therefore y :=. \'/ x, we have also .r = \// — 'i/ = \//^, t A (2) or if X and y are connected by some equation, it must be symmetrical relative to x and y, y or "^ x must then be determined from the equation =^ jP { 7, ^ j = 0, for instance, if x -h ^x — a =: 0, \l^ x s= a — x, or if X \l^ X = a"^, \f^ x = -^ , X Another method of determining such functions is as follows : since v/^ J? is of such a form that ^^x = x any sym- metrical function of x and \/^ x remain constant when x is changed into yjy x thus i^ { JT, \l^x \ becomes F { \{^x, \/^'' a } = F { y^ x, x i y if therefore, we can find any particular solution of the equa- tion \lr^ X == Xf containing an arbitrary constant we may sub- stitute such a function for it, but yl^x=a—x is a particular solution therefore \!y X = F{Xy xf/^ x) — X, or x •{■ y\r X =^ F {Xy y\f x)j and by changing the arbitrary function into another of the same form, we find F \ Icy ^ \ = 0, as before. These two methods of determining periodic functions of the second order, are not so convenient as a third process which can be extended to all orders. * Bars placed above quantities under the functional sign, in- dicate that the function is symmetrical relative to those quantities. (3) Assume \!/ x =:

.r = x, this must be equal to x or this equation will be fulfilled if f'*v = v, or if /is a particular solution, and if also <^ "" ' is such an inverse function that <^ -~ ' <^ V s=v. If therefore is arbitrary, and /is a particular solution of y cT = X, then the solution of \|^* j: = a: is \jy X = (p"^ 'y a:. Ex. Let/j: = -, then V^a = 0--'(— ), a: \(px/ if /(Ar)=_-- , x^* = 0-»l-_ r_l; from these may easily be derived the following periodic functions of the second ord&r, ^Iv ~ ^ yy X — • ' X — I X — 1 1 — X \j^ X SS 1 + X v/. a; = \/l - o:^ X + 1 , X v'*' — I >^or=tan-(^'"(''-^>) ^x=log(«-0 Vcos «. cos j:/ o \ / \l^x =1 (a" - Af")'' yj^x =:x - log («^ - 1) X ^j^x= Y ^x =:tdn-' (a- t^nx) (4) Prob. 2. Required periodic functions of the third order, or such as fulfil the equation \lr^ x = x. Assume "^l^x = <}>~^ f

x = X, which will be verified if f(v) is a particular solution of J^ v=.Vy and if ~ * is such an inverse value that "~ *(/> v = i;> hence the solution of the equation is yjy X =z (p~^f(pX, one solution is and hence ^x z=. (h — ^ ( J more particular cases are y^X = y^ X =. a — X \ — 'd X a" , \/a ar^ - a" \I/^X== -— \jyX=: a c — c^ X ' X , ax — d^ . 1 \lrX=: -4^ X =: 1 - X ^"^^ (^zr^y >/^^=-iog(i-.aO (a a?" — a^) ^ yf^x = V _>» y}.X=:log(as'=-a'')-x X I a + b X , , , Prob. 3. To find periodic functions of the /z'^ order, or to solve the equation v/^" x =z x. Assume as before ^x =^ cp-'ftpx then it becomes (5) which is verified if / is a particular solution of/' x = x, an4 if /^(- a) = :c« by putting — x for a^ this becomes and eliminating \/^ ( — x), we have \// j: — a' \/^ ^ = *^ — fl (— xT, hence ^ 1 -a* (2). Given yj^ x — a^r^ =:e* 11 ~ put - for :r, "v// a^l^x == e X and yjy'X = (3). Given (v// :r)^ . v/. 1 - a" 1 — T 1 + X 1 — 2^ put for Xy it becomes ^ 1 4- a: yy^ I + x/ 1 + ^ eliminating x/^ ^ ~~ ^ by means of the former, we find (8) (4). Given xj^ x -^ —^ xj^ ^/O - x") = I + x' putting V(l — ^t'') for x, we have and substituting this value of \/^ ^1 —x"" in the former equa- tion hence and yjy X zi x^, (5). Given ^^ +^ ^^"^> =i 1 +X/.X 1 + .^(-x) put \/^i a^ == • — thus the equation becomes ^lr^x + X yp^i (^— x) = 1, and changing x into — a; we have 4^1 { — X) — ar\/^, (a) = 1, by which eliminating \l^{ — x) from the former, we find ^' 1 + X^' hence yjyyX 1 — X ylr X = I — \l^^x X + x"" (6). Given yj^ x + ^ "^ ^ ^' = c. putting - for x this becomes (9) vf/ _ + (1 -^ X) \I/X ss X r and by eliminating \/r _, we have , 1 (7). Given x/^ i -f- j: v/^ ( 1 - i^^) = 1, putting ]—x for .r, we have v|r(l ~ i) + (1 _ jr)>^(i') = 1, whence, by elimination, 1 - i 1 - r -vl^ r = 1 -^ j^(l -. .r) 1 -.r ^- x' (8). Given /" + .__A(L:l£) = i, put v/., X = -Jl^, then will v/^,(l -ar)= tlLjUf), ^ , and the equation becomes \/^, j: + .r v|/, (1 - x) ss 1, the same as in the last example; let /.? represent the solution there found, then whence -^ >l^ X -- X , l/ X / X — 1 1 — a if we take for fr its value ■ , we have w r = ; . X f B (10) Incase the equation is symmetrical with regard to \/x>r and Vr a ,r, the process of elimination apparently becomes illusory. By a peculiar artifice this difficulty may be over- come, and it happens rather singularly that in all these cases, the solution which is so obtained contains an arbitrary func- tion, and in general the solution is the most extensive which the question admits of. (9). Given \j^ .r = 4^- . X If we put - for .1 , this is changed into ^ ^ = \j^a;, the same X X as the given equation ; it is therefore impossible to eliminate. Let us now suppose yj^ x = a 4^ — \- h, which becomes the given equation when a = l and ^ = 0. By putting _. for x this is changed into ■KJy - = n v/x X 4- h, X and eliminating \// -, we have , __ ah ^- b _ _^_ if 6 = and r/ = 1, this becomes a vanishing fraction whose value is any constant quantity c, and we have \//j: = r, which fulfils the equation. This is a very limited solution, but the following plan will lead us to much more general ones. Take the equation . 1 \^ X =. a ^/ ~ -\- "J (p Xy X which coincides with the given one when ^' = 0and<7= 1 ; also (p X is any arbitrary function of a ; putting - for :( , we have (M) and by elimination, t a (b- + (j) X Let « become 1 + and v become at the same time, then 1] 1-1.1+2.0 + 0^) -2.0 + 0* -2+0 2 and the solution becomes 1 Vx X ~ — , or changing the arbitrary function \lrX=(})X-\- is indefinite. This solution is, in fact, nothing more than an arbitrary symmetrical function of x and - , and may be expressed thus ^^ = xO. 0- Precisely the same course of reasoning will produce tlie solutions of the following equation. (10). yly{x)z= yi^{a - x) V^ X =: xix, a - x). (13) (14). \/^ .X = \^ (a a), \^ X = ;^ (JT, a x), whcrc a* ^ = 1\ (15). The objection which has just been stated occurs in the equation \/^ x -f ^ ( ) = ^, and a similar mode of proceeding will obviate it. The givers equation is a particular case of f + a \lr ( — t — y = c -h v^ Xt with which it coincides, if o = 1 and ij = ; putting for X in tliis, we have and elimination produces If v = and « = 1, this gives \^ .1 = - -h has been changed into another simi- lar one. 16- Given v/^(l -i- *) + \'^ (I — = 1 - i% put 1— i for i, then (13) x]!. X + x/> (2 - a:) = 1 - (i- - 1)» = 2 Jf - .r% this is a particular case of the equation \l^ X + a \l^ {'2, — x) = 2 .1 — .1* + V ipx, with which it agrees li v—0 and fl=:l ; changing x into 2 — r and eliminating xjy {2 — z) from the result, we find x/..r e= \ + { :t-a(2 - i) \ "" i + rt ^ ^ ' 1 - «• • If a = l and v = Of we have 2 t — a^^ (17.) Given -— + 7 = 2, \ + x\l^- X put —- = v/^, r then ^ = v^, - = , ^ X -^ \^x 1 . / 1 I' . .1 ^ -+>/^- 1-1-2 x//i and the equation becomes x/^. .r + x/^, - = 2, a: whose solution may be found by the method just explained to be \/^. 2 = J -f .1 _ ^ ^ , hence , 1 1 + 0.t - ^i X (14) (18). Required the equation of that class of curves which possess the following property, (Part IV. Fig. 1.) a given abscissa A B — a being taken, then the product of any two ordinates at equal distances from B^ shall always be equal to the square of the abscissa a. If y = V^ i represent the equation of the curve, then the condition expressed analy- tically is \ly(a — .?;) . -v// (fl -\- x) = d\ Putting a — X ior i, and then log \'r (x) = \/^i x, we have \l^, 1 -f- "^1 (2 ^ — x) = 2 log a, whose solution is yfy-t X = log a -^ (j) X —

we have (b X — a (t) i - — X ) V -J/ a: = 1 + a 1 - «^ making /?= 1 and v = 0, and changing (p x in 2 r, we have ^,.r = i -^.r + 0(^-^) and therefore In case the coefficient of yf^ a x in the equation \p' X + fx 4r ax =fx, IS of such 2. form that fx .fa .r = 1 , the denominator will vanish, and wo must then have recourse to an artifice similar to that which has already been ex- plained. (20). Ex. 1. Let v|, r + jr"" x/^ i = x''. X put v(^ X + (X^'' -^V(Px)^|^l= r" (16) which coincides with the given equation if vnOi then chang- ing r into -J we have ^ I -\' (x-"" + V(f>i^ ^^X- I -« X and by elimination \'/ X zz .t~"03: :r»« i + x-^" X -\- (f>~ f 2 X hence r" 1 A X This solution differs in form from that which wa$ pre- viously found, but it may be proved to be the same by the following substitution ; since (p is quite an arbitrary function this gives x^''x " x-\ ' /^ ^ \* and restoring the value of \/^, x, we find "^ X = \r^ — 2 X

/', ^= ^ — ^^ ; ^ 0^__ <^^(1 _ ^^2^^ hence 2 v/(l - ^') (19) It would not be difficult to shew the identity of these two apparently different solutions. (23). Given the equation \l^x . V^(l — '^) _ , T" — tt: — ^^ ^> put V^,a- = • .n ; Ti ^l> then x^. ( 1 - :r = X-L-_-/ ; and the equation becomes ^1^^' + ^.(1 —x) = 1, whose general solution is \/^i jc = ^- — ; 0-r+0(l-x)' hence . putting 1 — .r for x, and eliminating v/^ (1 — x), we find [0(1-^) + «^^r (24). Given (^|. xf + (^ -)' = ^L±_f! v^ x . v/. f! •, ^ X ^ X^ X a" divide by v/^ a . \^ — then ^x f_ _ X* + a* ~~aF ^l^ X '^ putting \'/i .r — ^ 3 this becomes (20) ^l^ + ^^ — = — - — > a particular solution of which is ^/^l ^ = x*; hence = .1"^ or v/ X =L ^^ -v// — , « X X and the general solution of this is \^-x^\ (25). Given riL±f + x ^ = 1 -f ^% v/^ X -i- .2? put \^, x = — , then the equation becomes X ^, X + x'' x/., i = 1 + a:% whose solution is >i/^x •=. ^ — , ^-I— - ; X , a? + ^x ix'' + 1)0. r hence ^ = ^^— ; putting - for x and eliminating -v/^ - , we find .^x= ; i — ; — '—rr^^^ I +'"*") X ^ 1/ (21) (26). Given (v/. xf' . {^|r - xT - (^ ^T • (^ - ^)'" = ^ .r ; putting >//, X = (^^ j:)'^ . (Vx — xf, it becomes ^/-.^ - ^^.(- ^)= '^^> whose solution is ^t v = x + x (^> ~ '^)> hence and by the process for eliminating yjy {— x), we shall find I \ x{^» '^T)-^r ] '^J (27). Given y^rx ■{- fx ,^ o.x = /» x, where ax \^ such a function of x that a'' a = cT ; putting « x for i', we have ■y\rs/fx ^{fo.x) derived from the equation /a: ,fax-=:\y we have \/{fa x) . Vx .r + x/(/^) . >/^ « X =/, (.r, a ,r), which is a symmetrical equation, whose general solution is ,, r \ 1 fti'V, ax) .

_iL_| (35), Given ^x ^ a^(^ n/Cj:" - 1)\ _ ^,„^ put — for X, it becomes ^ X ^V .r y Vn/(1 - x')/ x'^ Again, put V_l for i, and we have A, f . — 1 — a-ly X :iz « by eliminating ^ ^\/('^' ~ ^^^and ^^ (^~ ~^) ^^"^ these three equations, we shall find (36). Given +. + + (iJ:|-) + +4^) - the function ;;— is periodic of the third order^ and by 1 ■"" *3 X the process of elimination / \ + X \ ^.x = (37). Given -i~ + ^ + ■- = «• Putting i]/, ;c = — , we have V -^x Tl «- X I — X whose solution is n A\ ^-^ t A^ 'K' . ... rf> ^.^ 1 ^^-l' I — X X hence Y, + whose complete solution is (pT + (pax fx tan~' ^ X = tan~' , hence \^ i- = tan < 1 tan ' > C (/> 2^ + a x r .r > (p X -i- (p aX f i this process is analogous to one employed by M. Laplace, for the integration of a similar equation of differences. * Journal de I'Ecole FolytccnUjiLc , Cuh. l<5. (21) (44). Given ,,,H..(iii)..(-l)..,t^)., 1 +.1- 1 -X being a periodic function of the 4'*" order. ^x= *^ .1-1 V 1 — a: X X -{- 1/ j:.(i-x)'^(. ' T:::ri* x 'x + ii* (45), Given ^.{x, y) + v^ (^^\ i^ = l, Vx 1/ X (♦6). <^(*, y) + ?j)'fMy «^ v, A:^2/), (28) where u^x = a; ft"^ y = y and f{^x^ y) is such a function that /(^, ?/)./(« .r, /3 2/)= 1, then v/^Cr, v) = n^/(«^> /^.V) /x(^, «^, .VW^?/) /(.r, y)/, (^, x) ^ (^^ 2/) = X (^ + ^, 2 y), (53). Given ^/. (.r, y) = (£)' >/. (^, ^) (54). Given x/. (tt - .r) = ^-^, differentiating f yf, (n - x) = ^1^, (2!>) putting TT — .r for x In the given equation ■ ax 1 i« • • d \U (tt — jc) , and eliminating — i— ^ , we have whence by integration y^ X =z b cos J? 4- ^ sin T, and it will be found that c = — ^ ; hence y^r X =. b (cos X — sin X). (55). Given i. (*, y) = d^{x,a-y) ^ d X put a '- y for y^ then differentiate this relative to a, then r/ ^{x, a - y) _ ^^ v/r Tj:, y) which being substituted in the given equation produces whose solution js y^{Xy y) =: e^0?/ + E-^0,^, */) and 0, being two arbitrary functions so constituted as to fulfil the given equation, in order to determine them, put 48—1/ for y and differentiate relative to x, then d \l^ {x, a — jj) , , ^ , , ^ . (30) hence tpi/ =

!> 02> 03 must be determined so as to satisfy the given equation, taking the differential and putting a y for i/, we have U M/ ( X CI lj\ • * ^ =i^ which gives ^'.v)j the next condition is "" yy ^h ^)y the other two conditions are 0,y =^ «''2/» "^^) hence and the general solution of the equation is V^ (X, y) =e^x (y> «i/» "'.y> "'^) + + s-'(a^i/-a^y + cc^—^)x,{yy ccy^ a^y, a^^,) + + ("'i' — .V) K i^i «2/> «* 2/j «'2/) sin ^ + I _i + (a3 1/ - a y) X, (« 1/, «' ?/, a33', 7/) COS X, (60), Given the equation xj. {x, y) = ^l^t^l^lll^ where a is such a function that a" j: = x. This equation may be reduced to the solution of the partial differential equation and the arbitrary functions of y which occur In Its solution, must be determined by the conditions of the equation. (6l). Given the equation d \lr{a — X, y) _ d xjy (.r, b — y) dy dx ^ put a ^ X for Xy also h ^ y for ^, then we have the two equations (33) dyj^jx, y) _ _ (j y^ja •- Xf h - y) dy dx _ d\l^(a — x, b—y) _ dy\^{x, y) dy dx If the first of these be difFerentiated relative to y, and the second relative to x ; then the right side of the first resulting equation vi^Ill be identical with the left side of the second, and we shall have d" x^ (^r, y) ^ d'xl. (x, y) , dy^ dx" the solution of this partial differential equation is ^ (J-, y)=(p{x + y) ^ (^^J + (p, {xy) : to determine the form of (/> and ^j, we have d^(x') = i^^^) + xy '(xy)=--l-^ ' ( J-) The first of these multiplied by d{xy) may be put under the form whose integral is xy(—) 5 \xy/ the solution of which functional equation is ^/ xy V -^ ?/ ^ the solution of the second equation is ^y^ V X \y x^ employing these values of

ix,y)=^xj,.xQ^l) + y x> fd{xy). f— )'x. i^yy — ) \j.'i// V xy/ (63). Given i±j£iJL}l} rz "LtMj^ , dx dy where a^y =^ y and /3* x = x a process nearly similar to that (36) by which the two last equations were solved will lead to the partial differential equation d a y d" \l^ ( JT, t/) __ fLUjE. d' \lr (Xy y) dy d x"" d X dy^ (64). Given y\ra.X — y\ry\rX — y\/''X, It is evident, that whatever be the form of a, this equation can always be satisfied by assuming \|/- x = a ^r, hence the solutions of the following equations, , ax >\rX = -— o -\- C X (65). Given ^ (2a - j:) = x/.» x. Put y^yX = (p—'fipX, then \lr'^X=:(}> — ^f(p and the equation becomes x. (37) This is satisfied by making any symmetrical function of x and 2 a — x\ As an example take fv = — v, also <^ X = X . 2 a — a: —• 2 a X ■— ct", then and y^x =zd>-' fp and a'' x = x, the substitution (p ~ ^f

!/), on both sides If therefore we are acquainted with a particular solution, we find the general one ; let the function ^ 1- be tried, then y 139) y hence A = (pa, and the solution is V^Cr, y) = ^-»(ti0«), a variety of solutions may be found of difFerent forms, such as where a and /5 are any two homogeneous functions of the same degree. (70). If \/^ (j;, 1/) = « a? + byy then y^^(^x^ y) =•• {a + bf-'{ax + 63^), (71). If \}^ {x, y) is any homogeneous function of x, and y of the degree «, then » y) = ! ^ ('*^? Given v/.^a: x/.(^, 1/) Given y^^ix, y) = *~1 7/)}^ X {x/.(l, , or-" • (72). , V) = \/v^ (^, 2/), (73). = ,/^^^ + ^^ 0(1). (40) ^^'^^ (x + y),p(\) 20(f) (74). Given V^'-'(:r, j/) = ]~^f'^l ' ^0(1)-X»0(?) 1/0(1) +*^0(-) (75). xj>~'(i-, 5,) = P^^(a^, ^), provided a and /3 are homogeneous with respect to ^ and ^ ; the first of the « + 1 degree, the second of the n^^, and also at the same time a(l, 1) = /3(1, 1). (76). Given x/.-"^ (^, t^) =^ F y^ {x, ^). Another solution of the same equation is ^^^'y^-'CM'^' where a and /3 are two such functions, that when ^ = ?/, we have also a {X, y) = ^ (x, y), {11), Given ^/.'~M^, ^) = ^/-(^,^), (41) (78). Given ^'•'■{x, y) = \ V'(J-, ,V) ^( (79.) Given ^k'''(.i, y) = ! ^'■' (^, y) I ' ^(..)=^<^±|^| v/2 (80). Given x/x^'=» (.r, y) = - ^{^^y^) = x, 4>y)i putting (x, 3/) = :r, this equation, becomes identical; but making /(r, 3/) = a - ^ -j/, these two equations are verified ; consequently the general solution is r/.(a:, 3^) =(p-'(a - (px - (py). (82). Given v/^^'^ (r, 3/) . v^^'* (i", ?/) = ^'V* (83). Given t vi^^'Cr, »/) = tf v/.''' (>r, ,v), Various methods for the solution of Functional Equations may be found in the following writings : Speculationes Analytico Geometricae, N. Fuss, Mem. de rAcad. Imp. de St. Petersburg, Vol. IV. p. 225. 1811- Memoirs of the Analytical Society, p. 96. 1813. Observations on various points of Analysis, Phil. Trans. J. F. W. Herschel, Essay towards the Calculus of Functions, C. Babbage, 1815. Ditto, Part II. p. 179. 1816. Observations on the analogy which subsists between the Calculus of Functions, and other branches of Analysis, Phil. Trans. 1817. p. 197- C. Babbage. Spence's Essays, 1819. Note by /. F. W. Herschel^ p. 151. Annals of Philosophy, Nov. 1817. Mr. Horner, Journal of the Royal Institution. C. Babbage, ■ (42) Various methods for the solution of Functional Equations may be found in the following writings : Speculationes Analytico Geometricae, N. Fuss. Mem. de I "■^'^>tr-^vvS:^-^ PAMPHLET BINDER Syracuse, N. Y. BDD3DD3aaE RETURN TO the circulation desk ot any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY BIdg. 400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS • 2-month loans may be renewed by calling (510)642-6753 • 1-year loans may be recharged by bringing books to NRLF • Renewals and recharges may be made 4 days prior to due date. DUE AS STAMPED BELOW SEMTONILL JUL 1 7 2001 U. C. BERKELEY 12,000(11/95)