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is a solution of Bessel s di erential equation x2 y 00 xy 0 x2 p2 y 0 and is thus called a Bessel function of order p. See Problems 11.46, 11.110 through 11.113. Similarly, the hypergeometric function F a; b; c; x 1 a B a a 1 b b 1 2 x x 1 c 1 2 c c 1 17
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is a solution of Gauss di erential equation x 1 x y 00 fc a b 1 xgy 0 aby 0. These functions have many important properties. 1 X an zn , where z x iy In nite series of complex terms, in particular power series of the form
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and an may be complex, can be handled in a manner similar to real series. Such power series converge for jzj < R, i.e., interior to a circle of convergence x2 y2 R2 , where R is the radius of convergence (if the series converges only for z 0, we say that the radius of convergence R is zero; if it converges for all z, we say that the radius of convergence is in nite). On the boundary of this circle, i.e., jzj R, the series may or may not converge, depending on the particular z. Note that for y 0 the circle of convergence reduces to the interval of convergence for real power series. Greater insight into the behavior of power series is obtained by use of the theory of functions of a complex variable (see 16). 1 X In nite series of functions of two (or more) variables, such as un x; y can be treated in a
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manner analogous to series in one variable. In particular, we can discuss power series in x and y having the form a00 a10 x a01 y a20 x2 a11 xy a02 y2 18
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using double subscripts for the constants. As for one variable, we can expand suitable functions of x and y in such power series. In particular, the Taylor theroem may be extended as follows.
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TAYLOR S THEOREM (FOR TWO VARIABLES) Let f be a function of two variables x and y. If all partial derivatives of order n are continuous in a closed region and if all the n 1 partial derivatives exist in the open region, then
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    @ @ 1 @ @ 2 h k f x0 h; y0 k f x0 ; y0 h k f x0 ; y0 f x0 ; y0 @x @y 2! @x @y  n 1 @ @ h k f x0 ; y0 Rn n! @x @y where   1 @ @ n 1 h k Rn f x0 h; y0 k ; n 1 ! @x @y and where the meaning of the operator notation is as follows:   @ @ h k f hfx kfy ; @x @y   @ @ 2 h k h2 fxx 2hkfxy k2 fyy @x @y 0<<1
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 @ @ n k by the binomial theorem. @x @y Note: In alternate notation h x x x0 , k y y y0 . If Rn ! 0 as n ! 1 then an unending continuation of terms produces the Taylor series for f x; y . Multivariable Taylor series have a similar pattern. and we formally expand h 4. Double Series. Consider the array of numbers 0 u11 u12 B u21 u22 B B u31 u32 @ . . . . . . (or functions) 1 u13 . . . u23 . . . C C u33 . . . C A . . .
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