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## Print version ISSN 0041-6932

### Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009

Formulas for the Euler-Mascheroni constant

Pablo A. Panzone

Abstract. We give several integral representations for the Euler-Mascheroni constant using a combinatorial identity for . The derivation of this combinatorial identity is done in an elemental way.

2000 Mathematics Subject Classification. 11Mxx
Key words and phrases. Euler-Mascheroni constant.

Introduction. There exist many formulas for Euler-Mascheroni constant , see for example , . Indeed the irrationality of would follow from criteria given in  (see also ).

The purpose of this note is to give integral representations for which seem to be new. As usual we write .

Theorem. If , then  Remark 1. The formulae stated converge more rapidly than the usual definition. For example, notice that for  Indeed this follows form the fact that for one has and if .

Proof. We use the following formula: if and , then (1)   where we set if .

Recall the well-known representation (2)

Notice that in (1), as if and are bounded . We prove this in a moment.

Now i) follows from integrating (2) from 0 to 1 and using (1) with , letting . The first formula of ii) is proved in the same way putting in (1).

Now we prove (1): set ; where , and define if .

Add from to the trivial identity to get  (3) with .

From the equality of the first expression in (3) and the last one, we obtain (1).

We now prove that if , then as (the proof for bounded is similar). Indeed in this range of and one has But , where we have used for , .

This finishes the proof of the theorem. A corollary of formula (1) is the following

Corollary. Let . Set Then Proof. From (1), letting one gets  Now substitute by and by to get for    Now the corollary follows from this last formula inserted in (recall formula (2)) observing that  Remark 2. The corollary stated seems to give clean approximation formulas. Indeed  Numerically we have checked that is always of the form , with a rational number and , .

Notice that (1) or derivates of (1) give formulae for Hurwitz-Riemann's zeta function for .

We mention without proof that formula ii) of theorem 1 is equivalent to  Here is the general hypergeometric function.

References

    A. J. Van der Poorten, A proof that Euler missed... Math.Intelligencer, 1 (Nr.4), 1979, 195-203.        [ Links ]

    P. A. Panzone. Sums for Riemann's Hurwitz function II. Actas del Quinto Congreso A. Monteiro, Bahía Blanca, Universidad Nacional del Sur, 1999, 109-125.        [ Links ]

    J. Sondow. Criteria for irrationality of Euler's Constant. Proc. Amer. Math. Soc. 131, 2003, 3335-3344.        [ Links ]

    J. Sondow. Double integrals for Euler's constant and ln(4/p) and an analog of Hadjicostas's formula. Amer. Math. Monthly, 112, 2005, 61-65        [ Links ]

    J. Sondow and W. Zudilin. Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper. Ramanujan J. 12, 2006, 225-244.        [ Links ]

    X. Gourdon and P. Sebah. A collection of formulae for Euler Constant. 2003, http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.pdf.        [ Links ]

    Eric W. Weisstein. Euler-Mascheroni Constant. From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/Euler-MascheroniConstant.html        [ Links ]

Pablo A. Panzone
Departamento e Instituto de Matemática,
Universidad Nacional del Sur,
Av. Alem 1253,
(8000) Bahía Blanca, Argentina.
ppanzone@uns.edu.ar

Recibido: 18 de octubre de 2008
Aceptado: 3 de Junio de 2009 All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License