<Dl> <Dd> H (X) = − ∫ − ∞ ∞ f (x) log ⁡ f (x) d x = 1 2 (1 + log ⁡ (2 σ 2 π)) (\ displaystyle H (X) = - \ int _ (- \ infty) ^ (\ infty) f (x) \ log f (x) \, (\ rm (d)) x = (\ tfrac (1) (2)) (1 + \ log (2 \ sigma ^ (2) \ pi))) </Dd> </Dl> <Dd> H (X) = − ∫ − ∞ ∞ f (x) log ⁡ f (x) d x = 1 2 (1 + log ⁡ (2 σ 2 π)) (\ displaystyle H (X) = - \ int _ (- \ infty) ^ (\ infty) f (x) \ log f (x) \, (\ rm (d)) x = (\ tfrac (1) (2)) (1 + \ log (2 \ sigma ^ (2) \ pi))) </Dd> <P> where f (x) log ⁡ f (x) (\ displaystyle f (x) \ log f (x)) is understood to be zero whenever f (x) = 0 (\ displaystyle f (x) = 0). This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified variance, by using variational calculus . A function with two Lagrange multipliers is defined: </P> <Dl> <Dd> L = ∫ − ∞ ∞ f (x) ln ⁡ (f (x)) d x − λ 0 (1 − ∫ − ∞ ∞ f (x) d x) − λ (σ 2 − ∫ − ∞ ∞ f (x) (x − μ) 2 d x) (\ displaystyle L = \ int _ (- \ infty) ^ (\ infty) f (x) \ ln (f (x)) \, (\ rm (d)) x - \ lambda _ (0) \ left (1 - \ int _ (- \ infty) ^ (\ infty) f (x) \, (\ rm (d)) x \ right) - \ lambda \ left (\ sigma ^ (2) - \ int _ (- \ infty) ^ (\ infty) f (x) (x - \ mu) ^ (2) \, (\ rm (d)) x \ right)) </Dd> </Dl>

Would the graph represent a normal density function