Advanced Calculus fi..

444 Advanced Calculus, Fifth EditionIn general one findsall derivatives being evaluated at (0,O). A series C fn(x, y), in which the fn are givenby (6.65), is known as aTaylor series inx and y, about (0, O), and the function F(x, y)that it represents is termed analytic in the corresponding region. The expansion abouta general point (XI, yl) is obtained by a translation of origin:all derivatives being evaluated at (xl, yl).The general term of the series (6.66) can be interpreted in terms of an nthdiflerential dn F of the function F(x, y):where the (;) are the binomial coefficients.' To indicate the dependence of dnF onXI, yl and the differences x - xl, y - yl, we write:dnF = dnF(xl, yl; x - XI, y - yl).When n = 1 and x - xl = dx, y - yl = dy, one finds:this is the familiar expression for the first differential. The series (6.66) can now bewritten more concisely: