<Dl> <Dd> P A = 2 π h c 2 (k T h) 4 ∫ 0 ∞ u 3 e u − 1 d u . (\ displaystyle (\ frac (P) (A)) = (\ frac (2 \ pi h) (c ^ (2))) \ left ((\ frac (kT) (h)) \ right) ^ (4) \ int _ (0) ^ (\ infty) (\ frac (u ^ (3)) (e ^ (u) - 1)) \, du .) </Dd> </Dl> <Dd> P A = 2 π h c 2 (k T h) 4 ∫ 0 ∞ u 3 e u − 1 d u . (\ displaystyle (\ frac (P) (A)) = (\ frac (2 \ pi h) (c ^ (2))) \ left ((\ frac (kT) (h)) \ right) ^ (4) \ int _ (0) ^ (\ infty) (\ frac (u ^ (3)) (e ^ (u) - 1)) \, du .) </Dd> <P> The integral on the right is standard and goes by many names: it is a particular case of a Bose - Einstein integral, or the Riemann zeta function, ζ (4) (\ displaystyle \ zeta (4)), or the Polylogarithm . The value of the integral is π 4 15 (\ displaystyle (\ frac (\ pi ^ (4)) (15))), giving the result that, for a perfect blackbody surface: </P> <Dl> <Dd> j ⋆ = σ T 4, σ = 2 π 5 k 4 15 c 2 h 3 = π 2 k 4 60 ħ 3 c 2 . (\ displaystyle j ^ (\ star) = \ sigma T ^ (4) ~, ~ ~ \ sigma = (\ frac (2 \ pi ^ (5) k ^ (4)) (15c ^ (2) h ^ (3))) = (\ frac (\ pi ^ (2) k ^ (4)) (60 \ hbar ^ (3) c ^ (2))).) </Dd> </Dl>

A spherical blackbody of radius r radiates power p and its rate of cooling is r then