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# Integral representation of gamma

Let denote the Euler – Mascheroni constant. Define . Prove that

Solution

Well let ; since increase to on we deduce by the monotone convergence theorem that

Now,

Hence

## 1 Comment

1. The above identity is due to Catalan (1875). The above also satisfies the relation

(1)

which is a typical example of Mahler’s functional equations in the theory of transcedental numbers. Also note that

(2)