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FOURIER, JEAN BAPTISTE JOSEPH (1768-1...
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See also:FOURIER, See also:JEAN See also:BAPTISTE See also:JOSEPH (1768-1830) , See also:French mathematician, was See also:born at See also:Auxerre on the 21st of See also: He took an important See also:part in the preparation of the famous Description de l'Egypte and wrote the See also:historical introduction He held his prefecture for fourteen years; and it was during this See also:period that he carried on his elaborate and fruitful investigations on the See also:conduction of See also:heat. On the return of See also:Napoleon from See also:Elba, in 1815, Fourier published a royalist See also:proclamation, and left See also:Grenoble as Napoleon entered it. He was then deprived of his prefecture, and, although immediately named prefect of the See also:Rhone, was soon after again deprived. He now settled at Paris, was elected to the See also:Academic See also:des Sciences in 1816, but in consequence of the opposition of See also: See also:Freeman (See also:Cambridge, 1872), and a See also:German by Weinstein (See also:Berlin, 1884). His mathematical researches were also concerned with the theory of equations, but the question as to his priority on several points has been keenly discussed. After his See also:death Navier completed and published Fourier's unfinished See also:work, Analyse des equations indeterminees (1831), which contains much original See also:matter. In addition to the See also:works above mentioned, Fourier wrote many See also:memoirs on scientific subjects, and 'loges of distinguished men of science. His works have been collected and edited by Gaston Darboux with the See also:title Euvres de Fourier (Paris, 1889-189o). For a See also:list of Fourier's publications see the See also:Catalogue of Scientific Papers of the Royal Society of See also:London. Reference may also be made to See also:Arago, " Joseph Fourier," in the Smithsonian See also:Report (1871). FOURIER'S SERIES, in mathematics, those series which proceed according to sines and cosines of multiples of a variable, the various multiples being in the ratio of the natural See also:numbers; they are used for the See also:representation of a See also:function of the variable for values of the variable which See also:lie between prescribed finite limits. Although the importance of such series, especially in the theory of vibrations, had been recognized by D. See also:Bernoulli, See also:Lagrange and other mathematicians, and had led to some discussion of their properties, J. B. J. Fourier (see above) was the first clearly to recognize the arbitrary See also:character of the functions which the series can represent, and to make any serious See also:attempt to prove the validity of such representation; the series areconsequently usually associated with the name of Fourier. More See also:general cases of trigonometrical series, in which the multiples are given as the roots of certain transcendental equations, were also considered by Fourier. Before proceeding to the See also:consideration of the See also:special class of series to be discussed, it is necessary to define with some precision what is to be understood by the representation of an arbitrary function by an See also:infinite series. Suppose a function of a variable x to be arbitrarily given for values of x between two fixed values a and h; this means that, corresponding to every value of x such that a~x —b, a definite arithmetical value of the function is assigned by means of some prescribed set of rules. A function so defined may be denoted by f(x); the rules by which the values of the function are determined may be embodied in a single explicit See also:analytical See also:formula, or in several such formulae applicable to different portions of the See also:interval, but it would be an undue restriction of the nature of an arbitrarily given function to assume a priori that it is necessarily given in this manner, the possibility of the representation of such a function by means of a single analytical expression being the very point which -we have to discuss. The variable x may be represented by a point at the extremity of an interval measured along a straight See also:line from a fixed origin; thus we may speak of the point c as synonymous with the value x=c of the variable, and of f(c) as the value of the function assigned to the point c. For any number of points between a and b the function may he discontinuous, i.e. it may at such points undergo abrupt changes of value; it will here be assumed that the number of such points is finite. The only discontinuities here considered will be those known as See also:ordinary discontinuities. Such a discontinuity exists at the point c if f(c+e), f(c—e) have distinct but definite limiting values as c is indefinitely diminished; these limiting values are known as the limits on the right and on the left respectively of the function at c, and may be denoted by f(c+o), f(c—o). The discontinuity consists therefore of a sudden See also:change of value of the function from f(c—o) to f(c+o), as x increases through the value c. If there is such a discontinuity at the point x=o, we may denote the limits on the right and on the left respectively by f(+o), f(—o). Suppose we have an infinite series a1 (x) +u2(x)+... +u,,,(x)+.. . in which each See also:tennis a function of x, of known analytical See also:form; let any value x = c(a =c = b) be substituted in the terms of the series, and suppose the sum of n terms of the arithmetical series so obtained approaches a definite limit as n is indefinitely increased; this limit is known as the sum of the series. If for every value of c such that a gc b the sum exists and agrees with the value of f(c), the series '±'u„ (x) is said to represent the function(fx) between the values a, b of the variable. If this is the See also:case for all points within the given interval with the exception of a finite number, at any one of which either the series has no sum, or has a sum which does not agree with the value of the function, the series is said to represent " in general " the function for the given interval. If the sum of n terms of the series be denoted by Se(c), the See also:condition that S,,(c) converges to the value f(c) is that, corresponding to any finite See also:positive number 5 as small as we please, a value in of n can be found such that if n~nh f(c)—S, (c)I<5. Functions have also been considered which for an infinite number of points within the given interval have no definite value, and series have also been discussed which at an infinite number of points in the interval cease either to have a sum, or to have one which agrees with the value of the function; the narrower conception above will however be retained in the treatment of the subject in this See also:article, reference to the wider class of cases being made only in connexion with the history of the theory of Fourier's Series. See also:Uniform Convergence of Series.—If the series u1(x)+u2(x)+...+ u2(x)+...converge for every value of x in a given interval a to b, and its sum be denoted by S(x), then if, corresponding to a finite positive number 5, as small as we please, a finite number n, can be found such that the arithmetical value of S(x) —S,,(x), where n n1 is less than 5, for every value of x in the given interval, the series is said to converge uniformly in that interval. It may however happen that as x approaches a particular value the number of terms of the series which must be taken so that S(x) —Se(x) may be <5, in-creases indefinitely; the convergence of the series is then infinitely slow in the neighbourhood of such a point, and the series is not uniformly convergent throughout the given interval, although it See also:con-verges at each point of the interval. If the number of such points in the neighbourhood of which the series ceases to converge uniformly be finite, they may be excluded by taking intervals of finite magnitude as small as we please containing such points, and considering the convergence of the series in the given interval with such sub-intervals excluded; the convergence of the series is now uniform throughout the See also:remainder of the interval. The series is said to be in general uniformly convergent within the given interval a to b if it can be made uniformly convergent by the exclusion of a finite number of portions of the interval, each such portion being arbitrarily small. It is known that the sum of an infinite series of continuous terms can be discontinuous only at points in the neighbourhood of which the convergence of the series is not uniform, but non-uniformity of convergence of the series does not necessarily imply discontinuity in the sum. Form of Fourier's Series.—If it be assumed that a function f(x) arbitrarily given for values of x such that o x <l is capable of being represented in general by an infinite series of the form Ai See also:sin l +Az sin -r-+ ...+Az sinnix+ and if it be further assumed that the series is in general uniformly convergent throughout the interval o to 1, the form of the coefficients A can be determined. Multiply each See also:term of the series by sin nix, and integrate the product between the limits o and 1, then in virtue of the See also:property f o sin n l x sin ! xdx = 0, or Z1, See also:accord- has a definite meaning for every value of n. Before we proceed to examine the See also:justification for the assumptions made, it is desirable to examine the result obtained, and to deduce other series from it. In See also:order to obtain a series of the form Bs+BI See also:cos d +B2 cos2ix+ ... +Bs cos nl x+ .. . for the representation of f(x) in the interval o to 1, let us apply the series (I) to represent the function f(x) sin ; we thus find l T. sin nxx fJ o f (x) sin ix sin - dx, l sin -- i f(x) - cos (n—1 )max—cos (n-{ ~ )'rx dx. On rearrangement of the terms this becomes 1 ax l 2 l narx l sin i f f(x)dx+T sin d ax cos nl,rxf f(x) cos dx. hence f(x) is represented for the interval o to 1 by the series of cosines 1 f f (x)dx+l2, cos- x f f (x) cos -dx . (2) o We have thus seen, that with the assumptions made, the arbitrary function f(x) may be represented, for the given interval, either by a series of sines, as in (I), or by a series of cosines, as in (2). Some important See also:differences between the two series must, however, be noticed. In the first See also:place, the series of sines has a vanishing sum when x=o or x=l; it therefore does not represent the function at the point x=o, unless f(o) =o, or at the point x=l, unless Al) =o, whereas the series (2) of cosines may represent the function at both these points. Again, let us consider what is represented by (I) and (2) for values of x which do not lie between o and 1. As f(x) is given only for values of x between o and 1, the series at points beyond these limits have no necessary connexion with f(x) unless we suppose that f(x) is also given for such general values of x in such a way that the series continue to represent that function. If in (I) we change x into -x, leaving the coefficients unaltered, the series changes sign, and if x be changed into x--21, the series is unaltered; we infer that the series (I) represents an See also:odd function of x and is periodic of period 21; thus (I) will represent f(x) in general for values of x between too , only if f(x) is odd and has a period 21. If in (2) we change x into -x, the series is unaltered, and it is also unaltered by changing x into x+21; from this we see that the series (2) repre.-sents f (x) for values of x between too , only if f (x) is an even function, and is periodic of period 21. In general a function f(x) arbitrarily given for all values of x between too is neither periodic nor odd, nor even, and is therefore not represented by either (I) or (2) except for the interval o to 1. From (I) and (2) we can deduce a series containing both sines and cosines, which will represent a function f(x) arbitrarily given in the interval -1 to 1, for that interval. We can See also:express by (I) the function 2{f(x)-f(-x)} which is an odd function, and thus this function is represented for the interval -1 to +1 by 7 E sin nix f o2{ f(x)-f(-x)} sin-ixdx; we can also express 2{f(x)-}-f(-x)}, which is an even function, by means of (2), thus for the interval -1 to +1 this function is represented by Z f o2lf(x)+f(—x)}dx+ L 2 °0 E cos nia Jot{.f(x)+f(—x)}cos !if x. It must be observed that f(-x) is absolutely See also:independent of f(x), the former being not necessarily deducible from the Iatter by putting -x for x in a formula; both f(x) and f(-x) are functions given arbitrarily and independently for the interval o to 1. On adding the expressions together we obtain a series of sines and cosines which represents f(x) for the interval -1 to(1. The integrals f of(-x) cos nixdx' J of(-x) sin n'-Z2-de are See also:equivalent to folf(x) cos nixdx, +f f(x) sin -T-dx, thus the series is 1 I°p n,rx 1 mrx max 1 max 2l f if (x)dx+l cos l f 1f (x) cos l dx+ 7! sin f f(x)sin i dx, which may be written 21 f lf(x')dx'+l2if(x') cosna(xi-x)dx' . . . (3) The series (3), which represents a function f(x) arbitrarily given for the interval -1 to 1, is what is known as Fourier's Series; the expressions (I) and (2) being regarded as the particular forms which (3) takes in the two cases, in which f(-x) = -f(x), or f(-x) =f(x) respectively. The expression (3) does not represent f(x) at points beyond the interval -1 to 1, unless f(x) has a period 21. For a value of x within the interval, at which f(x) is discontinuous, the sum of the series may cease to represent f(x), but, as will be seen-hereafter, has the value If f(x+o)+f(x-o)}, the mean of the Iimits at the points on the right and the left. The series represents the function at x=o, unless the function is there discontinuous, in which case the series is 2{ f(+o)+f(-o)}; the series does not necessarily represent the function at the points 1 and -1, unless f(l) =f(-l). Its sum at either of these points is Zl f(l)-ff(-l)}. Examples of Fourier's Series.—(a) Let f(x) be given from o to 1, by Ax) =c, when olx<21, and by f(x)-c from 21 to 1; it is required to find a sine series, and also a cosine series, which shall represent the function in the interval. We have J o f (x) sin nixdx=c f olsin nixdx-c f 31 sin nixdx cl =na (cos nzr-2 cos sna+1). This vanishes if n is odd, and if n=4m, but if n=4m+2 it is equal to 4cl/na; the series is therefore 4c. 1 2irx 1 6ax 1 10ax a ~2 sin l + sin l +5 sin d + . . > For unrestricted values of x, this series represents the ordinates of the series of straight lines in fig. I, except that it vanishes at the points o, 21, 1, g1 .. . -21 -1 0 ?1 We find similarly that the same function is represented by the series 4c ( ax 1 3ix+1 cos 5zrx- a cos l' cos -1--+5 -7- ) during the interval o to 1; for general values of x the series repre- sents the See also:ordinate of the broken line in fig. 2, except that it vanishes at the points 21, l . . -21 =1 b 7 FIG. 2. (b) Let f(x)=x from o to 21, and f(x)=l-x, from 21 to 1; then 1 See also:flax Ii1 max f f(x) sin l dx= See also:fox sin l dx+ f ~l(l-x) sin nidx z z / 2t la cos 2 +n—212„-2 sin 2 na (cos -cos nal 12 12 na l2 na 212 na + na cos na-`na cos 2 +n,,See also:r2 sin 2 =-2a2 sin 2 See also:ing as n' is not, or is, equal to n, we have 2lA,, = f of (x) sin-1 xdx, and thus the series is of the form Z 2 sin-lx f f(x) sin-,xdx . . . (I) This method of determining the coefficients in the series would not be valid without the See also:assumption that the series is in general uniformly convergent, for in accordance with a known theorem the sum of the integrals of the See also:separate terms of the series is otherwise not necessarily equal to the integral of the sum. This assumption being made, it is further assumed that f (x) is such that f of(x)sinnl xdx or hence the sine series is ~l / nx 1 3xx 1 5rx 1 az `sin -3zsin 1 +z sin-~-- ... J For general values of x, the series represents the ordinates of the See also:row of broken lines in fig. 3. The cosine series, which represents the same function for the interval o to 1, may be found to be 1, 21 r 2rx 1 6rx 1 10arx 4_- `cos l +3zcos l +5zcos l + ... J This series represents for general values of x the ordinate of the set of broken lines in fig. 4. Dirichlet's Integral.—The method indicated by Fourier, but first carried out rigorously by Dirichlet, of proving that, with certain restrictions as to the nature of the function f(x), that function is in general represented by the series (3), consists in finding the sum of n+l terms of that series, and then investigating the limiting value of the sum, when n is increased indefinitely. It thus appears that the series is convergent, and that the value towards which its sum converges is if f(x+o)+f(x-o)}, which is in general equal to f(x). It will be convenient throughout to take -s to a as the given interval; any interval -1 to l may be reduced to this by changing x into lx/a, and thus there is no loss of generality. We find by an elementary See also:process that a+cos (x-x') + cos 2(x-x')+ ... + cos n(x-x') sin2tt2 I (x'-x) =2 sin 1(x' -x) Hence, with the new notation, the sum of the first n+I terms of (3) is x 2n2 I (x'- x) ~f~ f( ,)sin 2 sin %(x'-x) dx'. If we suppose f(x) to be continued beyond the interval —ir to a, in such a way that f(x)=f(x+2a), we may replace the limits in this integral by x+a, x—a respectively; if we then put x'-x=2z, and let f(x') =F(z), the expression becomes f _~ F(z)ssin zzdz, where m=2n+I; this expression may be written in the form 1 f'; sin zd 1 sin mz 7^-J o F(z) sin mz z+ f o*F( z) sin z dz .... (4) We require therefore to find the limiting value, when m is indefinitely increased, off o F(z)ss n zzdz ; the form of the second integral being essentially the same. This integral, or rather the slightly more general onej oF(z)ssin zzdz, when 0< is known as Dirichlet's integral. If we write X(z) =F(z) the integral sIn z becomes f hX(z)si Zmz dz, which is the form in which the integral is frequently considered. The Second Mean-Value Theorem.—The limiting value of Dirichlet's integral may be conveniently investigated by means of a theorem in the integral calculus known as the second mean-value theorem. Let a, b be two fixed finite numbers such that a<b, and suppose f(x), 4(x) are two functions which have finite and determinate values everywhere in the interval except for a finite number of points; suppose further that the functions f(x), 0(x) are integrable threwhout the interval, and that as x increases from a to b the function f(x) is monotone, i.e. either never diminishes or never increases; the theorem is that f a f(x)¢(x)dx=f(a+0) f ac&(x)dx+f(b-0) f bo(x)dx when is some point between a and b, and f(a), f(b) may be written for f(a+o), f(b-o) unless a or b is a point of discontinuity of the function f(x). To prove this theorem, we observe that, since the product of two integrable functions is an integrable function, f bf(x)(p(x)dx exists, and may be regarded as the limit of the sum of a series f(xo)c/)(xo) (xi–xo) +f(xi)4'(xi) (x2–x,)+ . . . +f(xn_~)~(x -~) (x, x„–,) where xo=a, x„=b and x1, x2 . x„_1 are n—I intermediate points. We can express ¢(xr) (x,+l-xr) in the form Y,1.1-Yr, by K=r putting Yr= ¢(xK_1) (xx-xx-1), Yo=o. K=1 See also:Writing Xr for f(xr), the series becomes Xo(Yi-Yo) +X1(Y2-Y1) +. . . +X„_1(Y.-Y,.-1) or Yl(Xo-X1)+Yz(X1-X2)+...+Y„(X,,-1-X„)+Y„X,,. Now, by supposition, all the numbers Y1, Y2 . . . Y are finite, and all the numbers X,_1-Xr are of the same sign, hence by a known algebraical theorem the series is equal to M (Xo-X„) +Y„X„ where M is a number intermediate between the greatest and the least of the numbers Yl, Y2, . . . Y,,. This remains true however many partial intervals are taken, and therefore, when their number is increased indefinitely, and their breadths are diminished indefinitely according to any See also:law, we have fa bf(x)O(x)dx=[f(a)f(b)}M+f(b) f a¢(x)dx when M is intermediate between the greatest and least values which f acp(x)dx can have, when x is in the given integral. Now this integral is a continuous function of its upper limit x, and there- fore there is a value of x in the interval, for which it takes any particular value between the greatest and least values that it has. There is therefore a value between a and b, such that m = f ~4(x)dx, a hence faf(x)¢(x)dx=If(a)-f(b)} fa4(x)dx+f(b) fac(x)dx =f(a) fQo(/x)dx+f(b) f d,(x)dx. If the interval contains any finite numbers of points of discontinuity of f(x) or ck(x), the method of See also:proof still holds See also:good, provided these points are avoided in making the subdivisions; in particular if either of the ends be a point of discontinuity off (x), we write f(a +o) or f(b-o), for f(a) or f(b), it being assumed that these limits exist. Functions, with Limited Variation.—The condition that f(x), in the mean-value theorem, either never increases or never diminishes as x increases from a to b, places a restriction upon the applications of the theorem. We can, however, show that a function f(x) which is finite and continuous between a and b, except for a finite number of ordinary discontinuities, and which only changes from increasing to diminishing or See also:vice versa, a finite number of times, as x increases from a to b, may be expressed as the difference of two functions fi(x), fz(x), neither of which ever diminishes as x passes from a to b, and that these functions are finite and continuous, except that one or both of them are discontinuous at the points where the given function is discontinuous. Let a, 0 be two consecutive points at which f(x) is discontinuous, consider any point xi, such that axi 0, and suppose that at the points MI, M2 . Mr between a and xl, f(x) is a maximum, and at ml, m2 . . mr it is a minimum; we will suppose, for example, that the ascending order of values is a, Ml, m1, M2, mz . Mr, mr, xi; it will make no essential difference in the See also:argument if m1 comes before M1, or if Mr immediately precedes xl, Mr_i being then the last minimum. Let ',(xi)=[f(Ml)f(a+o)l+[f(M2)f(mi)]+ ... +If(Mr)-f.(mr_1)]+[f (xi)f(mr)] ; now let xi increase until it reaches the value Mr+1 at which f(x) is again a maximum, then let %,&(xl) =If(Ml)f(a+o)]+[f(Mi)f(ml)]+ . +[f(Mr)f(mr–1)l +[f(Mr+l) f(mr)l ; and suppose as x increases beyond the value Mr+l, ¢(xi) remains See also:constant until the next minimum mr+1 is reached, when it again becomes variable; we see that ¢(xi) is essentially positive and never diminishes as x increases. Let x(xi) =[f(Mi)f(mi)]+[f(M2)f(mi)]+ . . +[f(Mr)-f(mr)l, then let xi increase until it is beyond the next maximum Mr+l, and then let x(xi)=If(Mi)f(mi)]+[f(M2)f(mi)]+ .. . +[f (Mr)f (mr)l +[f (Mr+l)f (xl)] thus x(xi) never diminishes, and is alternately constant and variable. We see that '(xt)-x(xi) is continuous as xi increases from a to /3, and that ¢(xi)-x(xi) =f(xi) f(a+o), and when x1 reaches /3, we have +~(R)–x(xl) =f(0-o)-f(a+o). Hence it is seen that between a and f(x)=[,&(x)+f(a+o)]-x(x), where >G(x)+f(a+o), x(x) are continuous and never diminish as x increases; the same reasoning 756 applies to every continuous portion of f(x), for which the functions 0(x), x(x) are formed in the same manner; we now take fl(x) = 1(x)+ f(a+o) +C, See also:f2 (x) = x(x) +C, where C is constant between consecutive discontinuities, but may have different values in the next interval between discontinuities; the C can be so chosen that neither 1(x) nor f2(x) diminishes as x increases through a value for which f(x) is discontinuous. We thus see that f(x) =f1(x) f2(x), where fi(x), f2(x) never diminish as x increases from a to b, and are discontinuous only where f(x) is so. The function f(x) is a particular case of a class of functions defined and discussed by See also:Jordan, under the name " functions with limited variation " (functions a variation bornee); in general such functions have not necessarily only a finite number of See also:maxima and minima. Proof of the Convergence of Fourier's Series.—It will now be assumed that a function f(x) arbitrarily given between the values ~r and +7r, has the following properties: (a) The function is everywhere numerically less than some fixed positive number, and continuous except for a finite number of values of the variable, for which it may be ordinarily discontinuous. (b) The function only changes from increasing to diminishing or vice versa, a finite number of times within the interval; this is usually expressed by saying that the number of maxima and minima is finite. These limitations on the nature of the function are known as Dirichlet's conditions; it follows from them that the function is integrable throughout the interval. On these assumptions, we can investigate the limiting value of Dirichlet's integral; it will be necessary to consider only the case of a function F(z) which does not diminish as z increases from o to 4w, since it has been shown that in the general case the difference of two such functions may be taken. The following lemmas will be required: i. Since J o sin zz dz =J o 11+2 cos 2z+2 cos 4z+... +2 cos 2nz}dz = 2 ; this result holds however large the odd integer m may be. 2. Ifo<a« J a ja sin sin mzdz=si 1 ,J a sin mz dz+ i.n- sin mz dz z n o. s ,B where a < y < /3, hence IP sin mz, 2 ( 1 1 . \ 4 z <— ` -1 a sin z m sin a+sm <m sin a' a precisely similar proof shows that I f a sl z mzdz l< 4 Ma' hence the integrals f a sisnri zdz sinzmz dz, converge to the limit zero, as m is indefinitely increased. 3. If a>O, f a si dO cannot exceed For by the mean-value theorem If k s' Bde l< Q +., a hence I Lh=ee f h sin a; a in particular if a 5 x, f sl B BdB 2 2 Again da f: siBdO_/ -slam a>0, therefore sl B BdO increases as a diminishes, when B < a < 2r ; but Ern a o f a si BdB= , hence sin 0,10 I <z' a where a <,r, and <-2 where a = 7r. It follows that x (NsinBdB zr, provided 04 a<$. a B I _ To find the limit of foF(z) sin mzdz, we observe that it may be sin z +{F(µ+o)-F(o)} fµs in zzdz+{F(l,r-o)-F(o)}[e ss n zzdz when i;' lies between/µ and 2ir. When m is indefinitely increased, the two last integrals have the limit zero in virtue of lemma (2). To evaluate the first integral on the right-See also:hand See also:side, let G(z) {F(z)-F(o)1 sin z, and observe that G(z) increases as z increases from o to µ, hence if we apply the mean-value theorem f('o G( )sinzmzdz =I G(µ) (µsinzmzdz J f~ See also:GOA)J m; sle BdO <lrG(µ), where o < E <µ, since G(z) has the limit zero when z = o. If a be an arbitrarily chosen positive number, a fixed value of µ may be so chosen that arG (µ) <1E, and thus that f o G(z) sin zmzdz < -y' e. When µ has been so fixed, in may now be so chosen that f o,rF(z) sin zzdz-2F(o) I <o. sin m It has now been shown that when m is indefinitely increased f f o F(z)ss. zzdz - 2 F(0) has the limit zero. 1 Returning to the form (4), we now see that the limiting value of 1- f' F (z)sin mz dz+ l C' F (_ z) sin mz dz is z { F(+O) +F (-0) 1; a o sin z o sin z hence the sum of n+I terms of the series 20 ltf (x)dx +l f 1f(xi) cos n'r(i-xl)dx converges to the value z { f(x+o) +f (x-o) }, or to ff(x) at a point where f(x) is continuous, provided f(x) satisfies Dirichlet's conditions for the interval from -1 to 1. Proof that Fourier's Series is in General Uniformly Convergent.—To prove that Fourier's Series converges uniformly to its sum for all values of x, provided that the immediate neighbourhoods of the points of discontinuity of f(x) are excluded, we have If ;F(z) sin mzdz-2 F(o) I <7rG(µ)+m sinµ{F(µ+o)-F(o)} +m s n to {F(4,r-o)-F(o)} < SI .f(x+2,L) f(x) +m s n µ {.f(x+2u)-f(x)} +ms ntl{.f(x+r)—f(x)}. Using this inequality and the corresponding one for F(-z), we have IS2„+1(x) f(x)I <µ cosec µ}If(x+2µ)-f(x)1 +If(x-2µ)-f(x)I] +A I m cosec where A is some fixed number independent of m. In any interval (a, b) in which f(x) is continuous, a value µl of µ can be chosen such that, for every value of x in (a, b), lf(x+2µ) f(x)l, I f(x-2µ)-f(x) I are less than an arbitrarily prescribed positive number e, provided µ=µ1. Also a value µ2 of µ can be so chosen that eµ2 cosec <4n, where 'n is an arbitrarily assigned positive number. Take for µ the lesser of the numbers µl, µ2, then I Si - f (x)I <n+AI m cosecµ for every value of x in (a, b). It follows that, since n and in are independent of x, I S2„+1 f (x) I <2e, provided n is greater than some fixed value n1 dependent only on e. Therefore S2„+1 converges to f(x) uniformly in the interval (a, b). Case of a Function with Infinities.—The See also:limitation that f(x) must be numerically less than a fixed positive number throughout the interval may, under a certain restriction, be removed. Suppose F(z) is indefinitely great in the neighbourhood of the point z=c, and is such that the limits of the two integrals f ±F F(z)dz are both zero, as e is indefinitely diminished, then f o F(z) Ssri zzdz denotes the limit when e=o, e1=o of JL -F(z) si n zzdz+ f zEl F(z)sin. n. zm zdz, both these limits existing; the Jo c+ sin first of these integrals has zorF(+o) for its limiting value when m is in-definitely increased, and the second has zero for its limit. The theorem therefore holds if F(z) has an infinity up to which it is absolutely integrable;' this will, for example, be the case if F(z) near the point C is of the form x(z) (z-c)`µ+>1(z), where x(c), ,'(c) are finite, and o < o <1. It is thus seen that See also:fix) may have a finite number of infinities within the given interval, provided the function is integrable through any one of these points; the function is in that case still representable by Fourier's Series. The Ultimate Values of the Coefficients in Fourier's Series.—If f(x) is everywhere finite within the given int¢'eval -2r to +vr, it can be shown that an, b,,, the coefficients of cos nx, sin nx in the series which represent the function, are such that na,,, nb,,, however. written in the form F(0)fa sinn mzdz+ f o{F(z)-F(0)}ss n mzdz m +P µ {F(z)-F(0)1 ssin zzdz where µ is a fixed, number as small as we please; hence if we use lemma (I), and apply the second mean-value theorem, o (z)s. zzdz- 2F(0) = f o {F(z)-F(0)}sin z sin mzaz f~ sin n sin nE ffi(x) cos nxdx =fi( +0) n- +fi(r-0) n with a similar expression, with f2(x) for f,(x), being between s-and -is; the result then follows at once, and is obtained similarly foothe other coefficient. a +nirf of'(x)cos? dx where a represent the points where f(x) is discontinuous. Hence if f(x) is represented by the series Ea„ sin nix, and f'(x) by the series Eb„ cos T ,we have the relation bn= `See also:tea",-i [f(+0) =f (l-0) + E cos na{ f(a+0) f(a-0)}] hence only when the function is everywhere continuous, and f(+o), f(l-o) are both zero, is the series which represents f'(x) obtained at once by differentiating that which represents f(x). The form of the coefficient as discloses the discontinuities of the function and of its See also:differential coefficients, for on continuing the integration by parts we find a„=n [f(+0) ref(1-0) + cosnla{ 91-0)}] +n z[f'(+0) ref'(l-0) + sin-Ti ~f'($+0) f'(0-0)}] +&c. where p are the points at which f'(x) is discontinuous. Additional information and CommentsThere are no comments yet for this article.
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