INFINITE PRODUTS Oe defies a ifiite product as- F F F... F x [ F ] Takig the atural logarithm of each side oe has- l[ F x] l F l F l F l F... So that the iitial ifiite product will coverge oly if the sum of the log terms coverges. Oe of the earliest ifiite products is the Wallis product of 6. It reads ad is easily established by otig that- / si x 6 6 6 6...... 7... / dx, si x 6 6 dx ad realizig that as goes to ifiity the two itegrals should be equal. This yields the Wallis result-.7796... Lookig at the log covergece test, oe fids that the log series for the Wallis product sums to l/ ad thus the Wallis product coverges but, ufortuately, does so very slowly. The ext importat historical cotributios to ifiite products were those of the Swiss mathematicia Leoard Euler 77-78. I particular he looked at the fuctio Fxsix/ x ad made use of the fact that this fuctio has simple zeroes at all iteger values betwee mius ad plus ifiity. Accordigly he postulated that oe should be able to expad thigs as follows-
si x x A x ad recogized that A must equal / sice the fuctio must vaish at x,,,, etc. As a result oe has the coverget ifiite product- si z z z z z z... O settig z/, we have the ifiite product-.666977... O settig z to /6, oe fids- 6 6 76 7 9... 899 ad by equatig the coefficiets of the x terms i the equality, oe has his famous ifiite series result-... 6 ζ Followig similar argumets to those used by Euler, oe ca also show that the zeroth order Bessel fuctio which has roots at λ, λ, λ, yields the ifiite product- J x x x λ ad that cosx, which has its zeros at /, satisfies-
cos x x Aother ifiite product, agai goig back to Euler, is that for the Riema zeta fuctio. It ca be expressed i terms of prime umbers p as- ζ z z p A good discussio of how this result is derived ca be foud i the book Prime Obsessio by Joh Derbyshire ad likely will play a role i the fial proof of the Riema cojecture. Let us ext demostrate how oe may covert a cotiued fractio ito a ifiite product. osider the square root of two. Oe of its cotiuous fractio represetatio is- whose covergets are give by- [;,,,,,...].., We thus have, 7, 7, with 8 8 8 The sequece of umbers, -,8, appearig i the deomiator of this ifiite product expasio is geerated by- N so that the ext umbers will be -9, 87, -67, etc.
As aother example of costructig a ifiite product, we look at exp.788889 Startig with its computer obtaied simple cotiued fractio exp[;,,,,,,,6,,,8,,, ] we geerate the covergets to fid- 8 9,,,,, 7 This time oe ca fid o simple recurrece relatio to geerate the higher s because the cotiued fractio patter i the square bracket above has varyig iteger values. Nevertheless, oe ca cotiue fidig higher values of by brute force calculatio so that the resultat fiite product ca approximate exp to ay desired order of accuracy. We will stop here with. The result is the approximatio- exp B 9 87 77 68 It would be ice if someoe out there would fid the geeral form of B i this last expressio so that a exact ifiite product for exp could be give. It will probably ivolve the use of the semi-periodic form of the coefficiets i [ ] which go as [ -,,,,,, ]. Alteratively oe could search for a more symmetric form of a partial fractio for exp which leds itself to geeral evaluatio of B. Recall from above that the Wallis formula gives a exact value for although its simple partial fractio array has the o-periodic form [; 7,,, 9,,,,,,,,,,,,,,,, ]. To sum certai ifiite products ivolvig quotiets of fuctios of it is sometimes coveiet to first expad the umerator ad deomiator term i the quotiet ad the proceed to cacel various terms i the resultat expasio. A good example of such a procedure ivolves the evaluatio of the ifiite product- S Expadig this product out produces- 7. 7 6 7 S. After cacellig terms we are left with-
.... S Aother example is-... 6 6 For most cases the ifiite product of a quotiet of fuctios of will be either zero or ifiity. Thus- ad