Problem #PRU-61455

Problems Calculus Functions of several variables

Problem

Definition. Let the function f(x,y) be valid at all points of a plane with integer coordinates. We call a function f(x,y) harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: f(x,y)=1/4(f(x+1,y)+f(x1,y)+f(x,y+1)+f(x,y1)). Let f(x,y) and g(x,y) be harmonic functions. Prove that for any a and b the function af(x,y)+bg(x,y) is also harmonic.