Mathematical Expressions of Euler's Constant and Its Applications

The series of number: a_n=(1+1[]2+1[]3+…+1[]n)-lnn,converges to Euler's constant γ,and Euler's constant γ has many mathematical forms.We construct three different mathematical forms of Euler's constant γ by two different mathematical forms of Gamma's function,and calculate the improper integral,the sums of some numeral series,the infinite product and so on by Euler's constant γ.