 If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by formula_72, is maintained at the temperature gradient formula_73 degrees Celsius, for formula_9 in formula_75, then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by
Here, sinh is the hyperbolic sine function.This solution of the heat equation is obtained by multiplying each term of   by formula_77.While our example function formula_1 seems to have a needlessly complicated Fourier series, the heat distribution formula_79 is nontrivial.