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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 split the interval of integration to get

.

The change of variables xRule1-x in the first integral gives

, which is the same as the second integral.

Therefore the original integral is .

Unfortunately, Mathematica doesn't currently know this integral directly. However, we can perform a couple steps by hand and compute it:

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We have an integral representation for the elliptic function:

Changing the order of integration, we get





Now we use Mathematica to evaluate the double integrals.

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Again, we have an integral representation for the elliptic function:





Then







Now we use Mathematica to do the double integration.

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We start with the integral representation .

Substituting this integral into the given integral and changing the order of integration gives .

Now we integrate with Mathematica.

Immediate corollaries

It immediately follows that





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We consider the more general integral , where Lambda 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 Lambda 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 Lambda 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 evaluate the series and integral on the right side with Mathematica and then form the original expression.

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Again, the direct approach does not work.

As in the previous representation, we replace by and change the order of summation and integration.



We evaluate the sum and the integral using Mathematica.

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The Mathematica proof is complicated. We use the property of the polygamma function, , to divide the given sum into two sums that we evaluate separately.







The first sum can be rewritten as .

We evaluate it in Mathematica.

Let us name the first sum as Alpha.

Now we consider the second sum, .

We need to introduce the values of of Tan and Cot at .

Now we use Mathematica to calculate the second sum.

We do further simplifications by applying some of transformation rules. The display of the result is suppressed because it is a little complicated; we will simplify it in the next step.

We name the second sum Beta.

We form the original expression using the sums Alpha and Beta.

At the end of the following chain of simplifications we obtain the desired result.

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