Thirty-Three Representations of Catalan's Constant

Victor Adamchik

Catalan's constant is a numerical constant (called Catalan in Mathematica) that appears in many combinatorial and analytic settings. This notebook presents thirty-three integral and series representations of Catalan's constant, some of them new. For each representation, a proof is given, accessible by pressing the proof button. Often the proof consists of simply evaluating the representation directly in Mathematica. In other cases, explicit symbolic manipulation in Mathematica is needed to develop a proof.

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We use an integral representation for the elliptic function:

Then

Now we use Mathematica to do the double integration.

Proof

Immediate corollaries

It immediately follows that

Proof

We consider the more general integral , where is an arbitrary parameter. We can evaluate this integral with Mathematica.

To find the original integral we need to find the limit of this expression as tends to 0. This in turn means that we need to find the asymptotic expansion of . For that we convert the hypergeometric function into a series and look at the general term.

We expand this into a series with respect to around zero.

Now we sum the series with this as the general term.

Finally, in the result of the integration at the beginning, we replace by this asymptotic expansion.

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The direct approach does not work.

We observe that .
Substitute this into the series and change the order of summation and integration.

We evauate the series and integral on the right side with Mathematica and then form the original expression.

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