Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
He attended lectures on the numerical solution of equations and on definite integrals by M.
He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial differential equations, and Jacobi's on analytical mechanics and higher algebra.
He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved.
The integrals are then properly functions of the direction in which the light is to be estimated.
If when the aperture is given, the wave-length (proportional to k1) varies, the composition of the integrals is unaltered, provided E and n are taken universely proportional to X.
We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y.
If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) by merely limiting the integration to the transparent parts of the aperture.
The intensity I 2, the quantity with which we are principally concerned, may thus (be expressed I 2 = 3 fcos27rv 2 .dv} 2 2 t 2 These integrals, taken from v =o, are (known as Fresnel's integrals; we will denote them by C and S, so that C = fo cos 27rv 2 .dv, S = fjsinv 2 .dv.
Cauchy, without the use of Gilbert's integrals, by direct integration by parts.
(19), 1 abA) ' ' we may write 12= (cos 27rv 2 .dv) 2 + (f sin zirv 2 .dv) 2 (20), or, according to our previous notation, 12 = (2 - C 2 +(z - Sv)2= G2 +H2 Now in the integrals represented by G and H every element diminishes as V increases from zero.
In the general motion again of the liquid filling a case, when a = b, 52 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d =- 2+ 2 J, = 2 x22111, d = 2 2`2 (+/'15-Om) (1 yy y n`t dt a +c dt a +c dt a +c) dc2, a2-1-c2 d122 a2 c2 dt ="2) +a2= G2y 71' dt = 121 1 - a 2 -c 2SJ, (19) of which three integrals are e +777 r z y 2= L -?2J2, (20) (a2 + c2) 2 2 121+14 =M+ 2c2(a2-c2)1 ' (21) 121+522hN = + x24 2,2 and then (dt / 2 = (a2 + c 2) 2(° v 2 - 12171) 2 4C4 2 2 - (+ c2)2?(E+77) (?
To determine the motion of a jet which issues from a vessel with plane walls, the vector I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.
A torsion of the ellipsoidal surface will give rise to a velocity function of the form 4)--- where SZ can be expressed by the elliptic integrals in a similar manner, since dX/P3.
When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical equations in (3) express the fact that x, x, xs.
Introducing Euler's angles 0, c15, x1= F sin 0 sin 0, x 2 =F sin 0 cos 0, xl+x 2 i =iF sin 0e_, x 3 = F cos 0; sin o t=P sin 4+Q cos 0, dT F sin 2 0d l - dy l + dy 2x = (qx1+ryi)xl +(qx2+ry2)x2 = q (x1 2 +x2 2) +r (xiyi +x2y2) = qF 2 sin 2 0-Fr (FG - x 3 y 3), (16) _Ft (FG _x 323 Frdx3 (17) F x3 X3 elliptic integrals of the third kind.
In elliptic integrals, the amplitude is the limit of integration when the integral is expressed in the form f 4) 1% I - N 2 sin e 4) d4.
The above expressions for the capacity of an ellipsoid of three unequal axes are in general elliptic integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk.
Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g.
His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.
The third volume (1816) contains the very elaborate and now well-known tables of the elliptic integrals which were calculated by Legendre himself, with an account of the mode of their construction.
Legendre had pursued the subject which would now be called elliptic integrals alone from 1786 to 1827, the results of his labours having been almost entirely neglected by his contemporaries, but his work had scarcely appeared in 1827 when the discoveries which were independently made by the two young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely.
The remainder of the first volume relates to the Eulerian integrals and to quadratures.
The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.
The latter portion of the second volume of the Traite des fonctions elliptiques (1826) is also devoted to the Eulerian integrals, the table being reproduced.
In 1788 Legendre published a memoir on double integrals, and in 1809 one on definite integrals.
Bezout's method of elimination, the other on the number of integrals of an equation of finite differences.
In pure mathematics, his most important works were his series of memoirs on definite integrals, and his discussion of Fourier's series, which paved the way for the classical researches of L.
Riemann's results are contained in the memoirs on " Abelian Integrals," &c. (Crelle, t.
Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.