2—Infinite Series 392.5 Power series, two variables
The idea of a power series can be extended to more than one variable. One way to develop it is to use exactly
the same sort of brute-force approach that I used for the one-variable case. Assume that there is some sort of
infinite series and successively evaluate its terms.
f(x,y) =A+Bx+Cy+Dx^2 +Exy+Fy^2 +Gx^3 +Hx^2 y+Ixy^2 +Jy^3 ···Include all the possible linear, quadratic, cubic, and higher order combinations. Just as with the single variable,
evaluate it at the origin, the point(0,0).
f(0,0) =A+ 0 + 0 +···Now differentiate, but this time you have to do it twice, once with respect to x whileyis held constant and once
with respect to y whilexis held constant.
∂f
∂x(x,y) =B+ 2Dx+Ey+··· then∂f
∂x(0,0) =B
∂f
∂y(x,y) =C+Ex+ 2Fy+··· then∂f
∂y(0,0) =C
Three more partial derivatives of these two equations gives the next terms.
∂^2 f
∂x^2(x,y) = 2D+ 6Gx+ 2Hy···∂^2 f
∂x∂y(x,y) =E+ 2Hx+ 2Iy···∂^2 f
∂y^2(x,y) = 2F+ 2Ix+ 6Jy···Evaluate these at the origin and you have the values ofD,E, andF. Keep going and you have all the coefficients.
This is awfully cumbersome, but mostly because the crude notation that I’ve used. You can make it look
less messy simply by choosing a more compact notation. If you do it neatly it’s no harder to write the series as
an expansion about any point, not just the origin.
f(x,y) =∑∞
m,n=0Amn(x−a)m(y−b)n (11)