A CURIOUS INTEGRAL

A double integral which came from a cohomology calculation is evaluated explicitly ,sing tile proi)erties of aF2 and 2F hypergeometric functions.


INTRODUCTION.
The 1)roblem of evaluating the integral ,r/2fx/x (1--4cos 2scos2t) /j + 8 cos s cos )3/: ddt has l)een prol)osed by A. Lundell.The computer algebra language Maple tells the user that it can not be evaluated explicitly but evaluates it numerically" to seven decimal places in a couple of seconds.Mathematica, on the other hand, reduces it to the evaluation of a single integral by performing one of the single integrals.
The integral arose as a reduction of a surface integral on a torus which came in relating the cohomology of R 3-(CU L) and R 3-C where C is the circle x2+ y2= a in the xy-plmw and L is the z-axis and where numerical calculations suggested the value rr/4 [2, p.19].The purpose of this note is to prove this conjecture.
}Ve first consider the more general integral [,/2 [,/2 (1 + bcos scos t) I(a,b,c) J0 0 (1 +acos scos2t) cdsdt" (I.I) We find that I(a, b, c) can be expressed as a sum of two 3F2 's with arguement -a.Although there are no explicit general formulas for the analytic continuation of 3F2 's something remark- able happens when c 3/2. hi this case each 3F2 can be expressed as a product of 2F 's of arguement -a which may now be analyticly continued throughout the complex a-plane cut along (-oc,-1].A further simplification occurs when b =-4 with 1(a,-4,3/2) being expressed as a single product of two 2F 's.A final remarkable simplification occurs with a 8 when each of these 1F1 's can be explicitly summed in terms of gamma functions.As an end result we then obtain THEOREM 1.
To prove Theorem 1 we first establish four lemmas.PROOF.This result is well known.An integration by parts yields u, -u,_l,n >_ 1.
We now specialize to the value c 3/2.
PROOF.We use (1.4) for the first F on the right of (2.3) and (1.5) for the second F on the right of (2.3).
Having established (2.4) for [a[ < one may use the.properties of F 's to obtain an analytic continuation of (2.4) throughout the complex a-plane cut along (-c,-1].
We now specialize to the values b -4, c 3/2.