Using the n i ≫ ≫ 1 approximation and using Stirling's approximation for the factorials ( x ! ≈ ≈ x x e − − x 2 π π x ) gives f ( n i ) = ∑ ∑ i ( n i + g i ) ln ⁡ ⁡ ( n i + g i ) − − n i ln ⁡ ⁡ ( n i ) + α α ( N − − ∑ ∑ n i ) + β β ( E − − ∑ ∑ n i ε ε i ) + K , where K is the sum of a number of terms which are not functions of the n i . Taking the derivative with respect to n i , and setting the result to zero and solving for n i , yields the Bose–Einstein population numbers: n i = g i e α α + β β ε ε i − − 1 .