787] 
FUNCTION. 
537 
Assuming the existence of the expansion of a function fx, in a series of multiple 
cosines, we thus obtain at once the well-known Fourier series, wherein the coefficient 
of cos mx is = çjrir 1 cos mx .fxdx. The question whether the process is applicable is 
J o 
elaborately discussed in Riemann’s memoir (1854), TJeber die Darstellbarkeit einer 
Function durch eine trigonometrische Reihe, No. xii. in the collected works. And again 
we have the Legendrian functions, which present themselves as the coefficients of the 
successive powers of a. in the development of (1 — 2olx+oP)~%, X 0 =l, X x =x, X. 2 =%(x 2 —^), &c.: 
here m, n being any positive integers, j X m X n dx = 0, but 
we have also Laplace’s functions, &c. 
X n 2 dx = 
2 
2w t 1' 
And 
Functions in General. 
17. In what precedes, a review has been given, not by any means an exhaustive 
one, but embracing the most important kinds of known functions; but there are 
questions to be considered in regard to functions in general. 
A function of x + iy has been built up by means of analytical operations performed 
upon x + iy, (x + iy) 2 — x 2 — y 2 + i. 2xy, &c., and the question next referred to has not 
arisen. But observe that, knowing x + iy, we know x and y, and therefore any two 
given functions ^ (x, y), yjr (x, y) of x and y: we therefore also know cf> (x, y) + i^Jr (x, y), 
and the question is, whether such a function of x, y (being known when x + iy is 
known) is to be regarded as a function of x + iy; and if not, what is the condition 
to be satisfied in order that <£ (x, y) + iy]r (x, y) may be a function of x + iy. Cauchy 
at one time considered that the general form was to be regarded as a function of 
x + iy, and he introduced the expression “ fonction monogene,” monogenous function, to 
denote the more restricted form which is the proper function of x + iy. 
Consider for a moment the above general form, say x' + iy' = <fi (x, y) + i)r {x, y), 
where </>, yjr are any real functions of the real variables (x, y); or what is the same 
thing, assume x' = cf) (x, y), y' = (x, y); if these functions have each or either of them 
more than one value, we attend only to one value of each of them. We may then 
as before take x, y to be the coordinates of a point P in a plane II, and x', y' to 
be the coordinates of a point P' in a plane IT. If, for any given values of x, y, the 
increments of (/> (x, y), s\r (x, y) corresponding to the indefinitely small real increments 
h, k of x, y be Ah + Bk, Ch + Dk, A, B, C, D being functions of x, y, then if the new 
coordinates of P are x + h, y + k, the new coordinates of P' will be x + Ah + Bk, 
y' + Ch + Dk; or P, P' will respectively describe the indefinitely small straight paths 
k Ch + Dk 
at the inclinations tan -1 r , tan -1 4; D - ? - to the axes of x, x' respectively; calling 
h 
Ah+Bk 
these angles 6, 6', we have therefore tan 6' - ^ an ■, . Now in order that x'+ iy' 
° A + B tan 6 a 
may be = (f> (x + iy), a function of x + iy, the condition to be satisfied is that the 
increment of x' + iy’ shall be proportional to the increment h -f- ik of x + iy, or say 
that it shall be = (A, + iy) (h + ik), A, y being functions of x, y, but independent of 
C. XI. 68