Ernest William Hobson

“ Expansion of Functions in Series of Legendre's Coefficients.—If it be assumed that a function given arbitrarily in the interval to , can be represented by a series of Legendre's coefficients and it be assumed that the series converges in general uniformly within the interval, the coefficient a can be determined by using (19) and (20) ; we see that the theorem (19) plays the same part as the property , does in the theory of the expansion of functions in series of circular functions. ”

“ Heine showed (Crelle’s Journal, vol. lxxi., 1870, and in his treatise Kugelfunctionen, vol. i.) that Fourier’s Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the kind. G. Cantor then showed (Crelle’s Journal, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the Math. Ann. vol. v., Cantor extended his investigation to functions having an infinite number of discontinuities. ”

“ The ordinary spherical harmonics of positive integral degree n are those which are rational integral functions of x, y, z. The most general rational integral function of degree n in three letters contains coefficients; if the expression be substituted in (1) , we have on equating the coefficients separately to zero relations to be satisfied; the most general spherical harmonic of the prescribed type therefore contains , or independent constants. There exist, therefore, independent ordinary harmonics of degree n ”

“ For the treatment of potential problems which relate to spaces bounded by special kinds of surfaces, solutions of Laplace's equation are required which are adapted to the particular boundaries, and various classes of such solutions have thus been introduced into analysis. Such functions are usually of a more complicated structure than ordinary spherical harmonics, although they possess analogous properties. As examples we may cite Bessel's functions in connexion with circular cylinders, Lamé's functions in connexion with ellipsoids, and toroidal functions for anchor rings. ”

“ Their objection appears, however, to rest upon a misapprehension as to the meaning of the sum of the series; if x1 be the point of discontinuity, it is possible to make x approach x1, and n become indefinitely great, so that the sum of the series takes any assigned value in a certain interval, whereas we ought to make x = x1 first and afterwards n = ∞, and no other way of going to the double limit is really admissible. ”

“ D’Alembert showed that the ordinate y at any time t of a vibrating cord satisfies a differential equation of the form δ2y/δt2 = a2δ2y/δx2, where x is measured along the undisturbed length of the cord, and that with the ends of the cord of length l fixed, the appropriate solution is y = ƒ (at + x) − ƒ (at − x) , where ƒ is a function such that ƒ (x) = ƒ (x + 2l) ; in another memoir in the same volume he seeks for functions which satisfy this condition. In the year 1748 (Berlin Memoirs, vol. iv.) Euler, in discussing the problem, gave ƒ (x) = α sin πx/l + β sin 2πx/l + . . . ”

“ The question was then considered from a new point of view by Lagrange, in a memoir on the nature and propagation of sound (Miscellanea Taurensia, 1759; Œuvres, vol. i.) , who, while criticizing Euler’s method, considers a finite number of vibrating particles, and then makes the number of them infinite; he did not, however, quite fully carry out the determination of the coefficients in Bernoulli’s Series. These mathematicians were hampered by the narrow conception of a function, in which it is regarded as necessarily continuous ”

“ 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 law, we have
when M is intermediate between the greatest and least values which can have, when x is in the given integral. Now this integral is a continuous function of its upper limit x, and therefore 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. ”

“ The operation
performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis , and is denoted by . The potential function is defined to be the potential due to a singular point of degree zero at the origin; is called the strength of the singular point. Let a singular point of degree zero, and strength , be on an axis , at a distance from the origin, and also suppose that the origin is a singular point of strength ; let be indefinitely increased, and indefinitely diminished, but so that the product is finite and equal to ”

“ If we write , the integral becomes , 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 ƒ (x) , φ (x) are two functions which have finite and determinate values everywhere in the interval except for a finite number of points ”

“ Ann., vol. vi., 1872) and by Hölder (Göttinger Anzeigen, 1894) may be consulted; the most general form of the theorem has been given by Hobson (Proc. London Math. Soc., Series II. vol. vii., 1909) . On the theory of uniform convergence of series, a memoir by W. F. Osgood (Amer. Journal of Math. xix.) may be with advantage consulted. On the theory of series in general, in relation to the functions which they can represent, a memoir by Baire (Annali di matematica, Series III. vol. iii.) is of great importance. ”

“ The degree n may be fractional or imaginary, but we are at present mainly concerned with the case in which n is a positive or negative integer. If x, y, z be replaced by their values , , in polar co-ordinates, a solid spherical harmonic takes the form ”

“ A memoir by Brodén (Math. Annalen, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier’s Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue’s work on Fourier’s Series will be found in his treatise, Leçons sur les séries trigonométriques (1906) ; also in a memoir, “Sur les séries trigonométriques,” Annales sc. de l’école normale supérieure, series ii. vol. xx. (1903) , and in a paper “Sur la convergence des séries de Fourier,” Math. ”

“ This function , thus defined for , is of considerable importance in the potential theory. When , we may in a similar manner obtain two series in ascending powers of , one of which represents , and a certain linear function of the two series represents the analytical continuation of , as defined above. The complete primitive of Legendre's equation is
By the usual rule for obtaining the complete primitive of an ordinary differential equation of the second order when a particular integral is known, it can be shown that (27) is satisfied by
the lower limit being arbitrary. ”

“ If x, y, z are such as to satisfy the two equations , , the equation is also satisfied; the problem of determining the poles is therefore equivalent to the algebraical one of reducing to the product of linear factors by means of the relation , between the variables. Suppose
we see that the plane passes through two of the 2n generating lines of the imaginary cone in which that cone is intersected by the cone . ”

“ Bessel's Functions as Coefficients in an Expansion. — It is clear that eip°<«* = ei* or ew> si » * = g>» satisfy the differential equation (31) , hence if these exponentials be expanded in series of cosines and sines of multiples of 0, the coefficients must be Bessel's functions, which it is easy to see are of the first kind. To expand e'psin *_ put <â– <* = /, we have then to expand eM'-' -1 ) in powers of t. Multiplying together the two absolutely convergent series
we obtain for the coefficient of t m in the product P'" S . ”

“ In case the orthogonal surfaces are concentric spheres, co-axial, circular cones, and planes through the axes of the cones, the parameters are the usual polar co-ordinates r, , , and in this case , thus Laplace's equation becomes
Assume that is a solution, R being a function of r only, of only, of only ”

“ When n is an integer, K„ (z) has n zeros with negative real parts.
29. Spheroidal Harmonics. — For potential problems in which the boundary is an ellipsoid of revolution, the co-ordinates to be used are r, 0, where in the case of a prolate spheroid
= c \V 2 — I sin cos , y = c-Jr* — l sin sin = ”