Performing a logarithmic substitution, the integral can be rewritten in the algebraic form:
The integral converges for . Here is a graph of on an interval :
Although it isn't known whether the function has a closed-form expression for all values of y, it can be evaluated explicitly for any y which is a rational multiple of ;. We consider several particular cases of in the next section.
This integral belongs to the more common class of integrals
where and are positive integers and is a cyclotomic polynomial. We assume that the integral is convergent. The integral (4) can be envisaged in an alternative form. Performing an integration by parts, we have
where and are positive integers and is a cyclotomic polynomial. In the last section we provide Mathematica code for evaluating integrals (4) and (5). The reader must evaluate that cell in order to calculate all the integrals below. Our results are consistent with a table presented by Baxter, Temperley and Ashley in.
Note that this formula can be rewritten in the following algebraic form:
As far as we know, the exact expression in (6) is new.
Proof. Differentiating the representation (2) relative to y and performing trivial simplifications, we obtain:
The first two integrals are:
Although the third integral could be easily evaluated by Mathematica, a more elegant way is to make use of:
so that the integrand can be rewritten as:
Running the integral in Mathematica, we obtain:
which completes the proof of formula (6). This is connected to the mean cluster density for the bond percolation model on the triangular lattice.
We shall prove that:
which, again, we believe is a new result. Note that this formula can be rewritten in the following algebraic form:
Proof. Differentiating the integral on the right side of representation (2) twice with respect to y, we obtain
and, using differentiation by parameter, formula (5) can be rewritten as:
Then running the integral in Mathematica we get:
Finally, we need to differentiate the above by and find the limit as :
2. S. Finch, Random percolation constants, HTML essay at World Wide Web URL http://www.mathsoft.com/asolve/constant/rndprc/rndprc.html, part of Favorite Mathematical Constants collection (1996).