1. Wallis-type infinite product representations of π

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1 Joural of Applied Mathematics ad Computatioal Mechaics 04, 3(), NOTE ON SOME INFINITE PRODUCTS FOR Peter Kahlig, Jausz Matkowski Sciece Pool Viea, Sect. Hydrometeorology Viea, Austria Faculty of Mathematics, Computer Sciece ad Ecoometrics Uiversity of Zieloa Góra, Zieloa Góra, Polad Abstract. After a brief review of (slowly covergig) Wallis-type ifiite products for, (faster covergig), Dido-type ifiite products for are treated. The otio of alteratig products facilitates error checkig. Keywords: ifiite product, Wallis-type product, Dido fuctioal equatio, Dido sequece, algebraic umber, trascedetal umber, costructible umber Mathematics Subject Classificatio: Primary 40A0; Secodary 39B Dedicated to Ludwig Reich o the occasio of his 75 th birthday Itroductio I Sectio we review Wallis-type ifiite product represetatios of. I Sectio we touch o the Dido fuctioal equatio, which is used i Sectio 3 to costruct coveiet Dido-type ifiite product represetatios of. Computatioal aspects are treated i Sectio 4 to certify that (cotrary to some opiios amog physicists) ifiite products may be useful eve i umerical work. The otio of alteratig products (Sectio 3) facilitates error checkig i Sectio 4.. Wallis-type ifiite product represetatios of Wallis famous ifiite product (origially obtaied by a iterpolatio process, cf. [, ]) reads W =, where W =

2 44 P. Kahlig, J. Matkowski Takig reciprocals, this is equivalet to k (k + ) 4 k ( k + ) = = (k + ) (k + ) k= k= Λ. () = ( (k ) ) = ( 3 ) ( 5 ) ( 7 ) k = = Let us call () Wallis first product. Multiplyig by (ad groupig carefully to maitai a distict formatio law) leads to aother versio, ow geerally called Wallis product (cf. [3, 4]) or, more properly, Wallis secod product, ( k) ( k) = = () k= (k )(k + ) k= ( k) Λ = ( ( k) ) = ( ) ( 4 ) ( 6 ) = k = Remark. Of course, istead of (), / could be expressed by doublig (), = = = ( (k + ) ) (3) k = = ( 3 ) ( 5 ) ( 7 )Λ = (with a leadig scale factor of before the mai product with formatio law). This is differet from the (vaishig) diverget product ( k) = = ( + ( ) ) ( + ) k (4) k= k k= = ( + ) (+ 4 ) (+ 6 ) Λ = 0, ad differet from the (idefiitely growig) diverget product ( k) = = ( ( ) ) ( ) k (5) k= k k= = ( ) ( 4 ) ( 6 ) Λ =. 9 5

3 Note o some ifiite products for 45. The Dido fuctioal equatio Suppose that a cotiuous fuctio f :[, ) [0, ) satisfies the Dido fuctioal equatio + + (6) x f ( x) = f ( x) f ( x), x, related to the aciet isoperimetric problem of Dido (cf. [5]). I [6] it is show that if the costat is the asymptote of f at ifiity, that is if limx f ( x) =, the f ( x) = cot, x [, ). x x (7) Restrictig x to iteger values, we have the Dido sequece f ( ) = cot( ) for =,3,4,..., ad we may iterpret f () as the ier radius r (or area A ) of a regular polygo of order with fixed perimeter P, scaled by half-perimeter P / This yields explicitly f ( ) = r /( P/) = A /( P/). (8) P P P r = f ( ) = cot, =,3, 4,... (polygos), with r = (circle); P P P A = f ( ) = cot, =,3,4,... (polygos), with A = (circle) Alteratively, if the outer radius R of a regular polygo of order is to be used, the Dido sequece becomes f ( ) = R cos( ) / ( P / ), leadig to the expressio implyig P P R cos = f ( ) = cot,

4 46 P. Kahlig, J. Matkowski P P R =, =,3,4,... (polygos), with R = (circle). si( ) The iequality f ( ) for =,3,4,... resembles the well-kow isoperimetric iequality (cf. [7]), P P r or A, =,3,4,..., 4 (9) where equality holds for (circle). 3. Dido-type ifiite product represetatios of It is coveiet (cf. [7]) to use the followig: Defiitio. A algebraic umber is called costructible if it is a aggregate of fiitely may ratioals ad/or square roots. Remark. It is well kow (cf. [7, 8]) that a regular -go is costructible by ruler ad compass if ad oly if its Dido value f( ) ( fixed) is a costructible algebraic umber. (Otherwise the -go is ot costructible; its Dido value might cotai, for istace, a cube root.) Remark 3. Utilizig well-kow product represetatios of cos ad si to form cot = cos/ si, we obtai ((k ) / ) f ( ) = cot =, =,3, 4,... k. (0) k = ( ) Obviously, this expressio is a alteratig product ; explicitly, it may be writte ( ) j+. () j= f ( ) = ( ( j / ) ), =,3, 4,... Remark 4. I aalogy to Leibiz s well-kow criterio for (coditioally coverget) alteratig series [amely, the remaider of a alteratig series has the sig of the first eglected term, ad is closer to 0 tha the first eglected term], we may formulate a criterio for alteratig products: the remaider of a

5 Note o some ifiite products for 47 alteratig product is > or < just like the first eglected factor, ad is closer to tha the first eglected factor. Remark 5. Usig (0), we ca compile values of the Dido sequece f () for regular polygos of order =,3,,0. Costructible -gos are marked by *. Order Dido value f ( ) = cot( ) um. value of f( ) related ifiite product for f( ) * 0 0 3* 0.95 (3 / ) (9 / ) (5 / ) * * (5/ ) (5/ ) (5/ ) * f (7) (7) = (7 / ) (/ ) (35 / ) f 7 4 8* ( + ) f (9) (9 / ) (7 / ) (45 / ) f (9) * f () () = (/ ) (33/ ) (55/ ) f 33 * ( + 3) f (3) 0.3 (3) = (3/ ) (39 / ) (65 / ) f f (4) f (4) 4 8 4

6 48 P. Kahlig, J. Matkowski 5* 6* 7* A A (5 / ) (45 / ) (75 / ), B 0.34 B , where A = (+ 5)( ) ad B =+ (+ + ) 5+ D 7 7 D D (7/ ) (5/ ) (85/ ), 7 7 D * where D = f (8) f (8) f (9) (9) = (9 / ) (57 / ) (95/ ) f E E , where E = For we obtai the trascedetal umber (Dido value for circle) f ( ) = lim cot = Remark 6. The items of the list i Remark 5 admit several iterpretatios. For istace, by (0) we obtai a multitude of ifiite products for, ((k ) / ) =, = 3,4,5,..., f ( ) ( ) () k = k where / f ( ) is a scale factor, it is for geeral < < a geeral algebraic umber > [but reasoably simple for *, i.e. for f( ) a costructible algebraic umber (cf. Remark ), which implies i this case that also / f ( ) (the scale factor) is a costructible algebraic umber]. For < <, the product represets a positive trascedetal umber <. For, the scale factor approaches while the product approaches. I the extreme case =, the scale factor grows idefiitely, / f () =, while the product degeerates to 0. Thus, Dido-type represetatios of the trascedetal umber cosist i geeral (amely for < < ) of two factors: a algebraic scale factor (of the same magitude as )

7 Note o some ifiite products for 49 ad a trascedetal ifiite product (of magitude ). I priciple, also Wallistype products like () respectively () may be iterpreted as such a decompositio: = 4 Π... [used e.g. i (3) below] respectively = Π...; Wallis-type products lack a coveiet error estimatio (i cotrast to Dido-type products, see ext sectio). 4. Computatioal aspects Approximatios (of order N N) to Wallis s first product () may be writte with pi () = N ( ) = 4 ( (k+ ) ), pi N (3) k=. To 3 ad 6 sigificat digits we get pi (300) = ad pi (400000) = , exhibitig rather slow covergece, ad promptig statemets like the followig [3]: These ifiite products have a variety of uses i aalytical mathematics. However, because of rather slow covergece, they are ot suitable for precise umerical work. Yet we will show presetly that Dido-like ifiite products may be umerically useful. For etry 4 of the list i Remark 5, i.e. takig = 4 i (), we have approximatios (of order N N) (4k ) Pi( N) = 4, N (4) k = (4 k) with Pi( ) =. To 3, 6 ad 9 sigificat digits we obtai here Pi (0) = , Pi(300) = =... ad Pi (0000) = , showig a acceptable rate of covergece. Moreover, we ca calculate the expected error: accordig to Remark 4, we just have to look at the first eglected factor ν i (4) [i compariso to ()], amely ν ( N ) = (4( N + ) ) ; the remaider R is closer to tha ν, i.e. R( N ) < ν( N), ad we obtai for the absolute error E (whe retaiig N factors oly) the expressio E N R N N N+ (5) ( ) = 4 ( ) < 4 ν( ) = 4(4( ) ), where the leadig 4 is the scale factor from (4).

8 50 P. Kahlig, J. Matkowski Thus, by (5), expected errors for (4) are E E E (0) <.3 0, (300) <.8 0 ad (0000) =.5 0. Verificatio: empirical errors of (4) are Pi Pi Pi (0) =.0 0, (300) =. 0 ad (0000) =.0 0. Ackowledgemets The authors are idebted to Adrzej Majczak for writig this paper i the Word editig system. Refereces [] Wallis J., Arithmetica Ifiitorum, Oxford 656. [] Chabert J.-L. (ed.), A History of Algorithms: From the Pebble to the Microchip, Spriger-Verlag, Berli, Heidelberg, New York 999. [3] Arfke G.B., Weber H.J., Mathematical Methods for Physicists, Academic Press, Sa Diego, New York 995. [4] Berggre L., Borwei J., Borwei P. (eds.), Pi: A Source Book, Spriger-Verlag, New York 004. [5] Kahlig P., Matkowski J., O the Dido fuctioal equatio, A. Math. Siles. 999, 3, [6] Kahlig P., Matkowski J., Sharkovsky A.N., Dido s fuctioal equatio revisited, Roczik Naukowo-Dydaktyczy Akademii Pedagogiczej w Krakowie, Prace Matematycze 000, 7, [7] Courat R., Robbis H., Stewart I., What is Mathematics? Oxford Uiversity Press, Oxford 996. [8] va der Waerde B.L., Algebra I, Spriger-Verlag, Berli, Heidelberg, New York 966.