A rational approximation of the arctangent function and a new approach in computing pi

Size: px

Start display at page:

Download "A rational approximation of the arctangent function and a new approach in computing pi"

Allison Davidson
5 years ago
Views:

1 arxiv:63.33v math.gm] 4 Mar 6 A rational approimation of the arctangent function and a new approach in computing pi S. M. Abrarov and B. M. Quine March 4, 6 Abstract We have shown recently that integration of the error function erf ( represented in form of a sum of the Gaussian functions provides an asymptotic epansion series for the constant pi. In this work we derive a rational approimation of the arctangent function arctan ( that can be readily generalized it to its counterpart sgn ( π/ + arctan (, where sgn ( is the signum function. The application of the epansion series for these two functions leads to a new asymptotic formula for π. Keywords: arctangent function, error function, Gaussian function, rational approimation, constant pi Derivation Consider the following integral ] e y t erf (t dt = ( y π arctan, ( y Dept. Earth and Space Science and Engineering, York University, Toronto, Canada, M3J P3. Dept. Physics and Astronomy, York University, Toronto, Canada, M3J P3.

2 where we imply that all variables t, and y are real. Assuming that y = the integral ( can be rewritten as arctan ( = π e t erf (t dt. ( The error function can be represented in form of a sum of the Gaussian functions (see Appendi A erf ( = lim π e (l /. (3 Consequently, substituting this limit into the equation ( leads to arctan ( = π lim e t t π e (l / t } {{ } erf(t Each integral term in this equation is analytically integrable. Consequently, we obtain a new equation for the arctangent function Since it follows that arctan ( = 4 lim π = 6 lim π = 4 arctan ( dt. (l + 4. (4 (l + 4. (5 It should be noted that the limit (5 has been reported already in our recent work ].

3 Ε Fig.. The difference ε over the range at = (blue, = (red, = 3 (green, = 4 (brown and = 5 (black. Truncation of the limit (4 yields a rational approimation of the arctangent function arctan ( 4 (l + 4. (6 Figure shows the difference between the original arctangent function arctan ( and its rational approimation (6 ε = arctan ( 4 (l + 4 over the range at =, =, = 3, = 4 and = 5 shown by blue, red, green, brown and black curves, respectively. As we can see from this figure, the difference ε is dependent upon. In particular, it increases with increasing argument by absolute value. Thus, we can conclude that the rational approimation (6 of the arctangent function is more accurate when its argument is smaller. Consequently, in order to obtain a higher accuracy we have to look for an equation in the form π = N a n arctan (b n, b n <<, n= 3

4 where a n and b n are the coefficients, with arguments of the arctangent function as small as possible by absolute value b n. For eample, applying the equation (6 we may epect that at some fied the approimation π = 4 arctan ( = 6 (l + 4 is less accurate than the approimation based on the Machin s formula 3, 4] ( ( ] π = 4 4 arctan arctan ( 4 (/5 (l (/5 + 4 /39 (l (/ Furthermore, with same equation (6 for arctan ( we can improve accuracy by using another formula for pi 4] ( ( ( ] π = 4 arctan + 8 arctan 5 arctan ( (/8 6 (l (/ (/57 (l (/ (/39 (l (/ due to smaller arguments b n of the arctangent function. Application. Counterpart function Once the rational approimation (6 for the arctangent function is found, from the identity ( arctan + arctan ( = π sgn (, where, > sgn ( =, =, < 4.

5 is the signum function 5], it follows that 4 (l + 4 π sgn ( + arctan (. (7 Figure shows the epansion series (7 computed at = (blue curve. The arctangent function is also shown for comparison (red curve. As we can see from this figure, on the left-half plane the epansion series (7 is greater than the original arctangent function by π/, while on the right-half plane it is smaller than the original arctangent function by π/. Approimation for Π sgn arctan Fig.. The epansion series (7 computed at = (blue curve resembling the function sgn ( π/ + arctan (. The original arctangent function (red curve is also shown for comparison. The approimation (7 can be replaced with eact relation by tending the integer to infinity and taking the limit as 4 lim (l + 4 = π sgn ( + arctan (. (8 Since this limit represents a simple generalization of the equation (4, the function sgn ( π/ + arctan ( can be regarded as a counterpart to the arctangent function arctan (. 5

6 . Asymptotic formula for pi Using the limits (4 and (8 for the arctangent function arctan ( and its counterpart function sgn ( π/ + arctan (, we can readily obtain an asymptotic epansion series for pi. et us rewrite the equation (8 as follows arctan ( = π sgn ( 4 lim The difference of the equations (9 and (4 yields = π (4 sgn ( lim (l + 4 }{{} eq. (9 or or 4 lim ] (l (l + 4 π = 8 lim (l + 4. (9 ( 4 lim (l + 4 }{{} eq. (4 = π sgn ( ] (l (l + 4 ( since sgn ( = / 5]. Obviously, the equation ( can be interpreted as π = ( ] arctan + arctan (. Remarkably, although the argument is still present in the limit ( this asymptotic epansion series remains, nevertheless, independent of. This signifies that according to equation ( the constant π can be computed at any real value of the argument R. The limit ( can be truncated by an arbitrarily large value >> as given by π 8 (l (l ]. (

7 We performed sample computations by using Wolfram Mathematica 9 in enhanced precision mode in order to visualize the number of coinciding digits with actual value of the constant pi The sample computations show that accuracy of the approimation limit ( depends upon the two values and (the dependence on the argument in the equation ( is due to truncation now. For eample, at = and =, we get }{{} , 5 coinciding digits while at same = but smaller = 9, the result is }{{} coinciding digits Comparing these approimated values with the actual value for the constant pi one can see that at = and = 9 the quantity of coinciding digits are 5 and 33, respectively. It should be noted, however, that the argument cannot be taken arbitrarily small since its optimized value depends upon the chosen integer. 3 Conclusion We obtain an efficient rational approimation for the arctangent function arctan ( that can be generalized to its counterpart function sgn ( π/ + arctan (. The application of the epansion series of the arctangent function and its counterpart results in a new formula for π. The computational test we performed shows that the new asymptotic epansion series for pi may be rapid in convergence. Acknowledgments This work is supported by National Research Council Canada, Thoth Technology Inc. and York University. The authors would like to thank Prof. H. Rosengren and Prof.. Tournier for review and useful information. 7

8 Appendi A Consider an integral of the error function (see integral on page 4 in 6] erf ( = π e u sin ( u du u. This integral can be readily epressed through the sinc function {sinc ( = sin ( /, sinc ( = = } by making change of the variable v = u leading to or erf ( = π = 4 π e v sin (v vdv v e v sin (v dv v erf ( = 4 π = π e v sinc (v dv. e v sin (v dv v The factor in the argument of the sinc function can be ecluded by making change of the variable t = v again. This provides or erf ( = 4 π erf ( = π e t /4 sinc (t dt e t /4 sinc (t dt. (A. As it has been shown in our recent publication, the sinc function can be epressed as given by 7] sinc ( = lim 8 ( l /. (A.

9 From the following integral sinc ( = (u du = (t dt (A.3 it is not difficult to see that the ine epansion (A. of the sinc function is just a result of integration of equation (A.3 performed by using the midpoint rule over each infinitesimal interval t = /. There are many ine epansions of the sinc function can be found from equation (A.3 by taking integral with help of efficient integration methods 8]. For eample, another ine epansions of the sinc function can be found by using the trapezoidal rule + ( sinc ( = lim + and the Simpson s rule sinc ( = lim + ( ( l / ( ] l (A.4 + ( ] l. (A.5 It is interesting to note that the limit (A.5 can also be derived trivially as a weighted sum of equations (A. and (A.4 in a proportion /3 to /3 as follows sinc ( = 3 lim + 3 lim ( l / + ( + ( ] l. Any of these or similar ine epansions of the sinc function can be used in integration to obtain epansion series for the error function erf ( and, consequently, for the constant pi as well. However, as a simplest case we consider an application of equation (A. only. Thus, substituting the ine epansion (A. of the sinc function into the integral (A. yields erf ( = π lim ep ( t /4 ( l / t } {{ } sinc(t dt. 9

10 Each terms in this equation is analytically integrable. Therefore, its integration leads to the epansion series (3 of the error function. The more detailed description of the epansion series (3 of the error function is given in our work ]. References ] H.A. Fayed and A.F. Atiya, An evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral, Math. Comp., 83 ( S / ] S.M. Abrarov and B.M. Quine, A new asymptotic epansion series for the constant pi, arxiv: ] J.M. Borwein, P.B. Borwein and D.H. Bailey, Ramanujan, modular equations, and approimations to pi or how to compute one billion digits of pi, Amer. Math. Monthly, 96 (3 ( stable/356 4] J.M. Borwein and S.T. Chapman, I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly, Amer. Math. Monthly, (3 ( math.monthly ] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, nd ed., Chapman & Hall/CRC 3. 6] E.W. Ng and M. Geller, A table of integrals of the error functions, J. Research Natl. Bureau Stand. 73B ( ( ] S.M. Abrarov and B.M. Quine, A rational approimation for efficient computation of the Voigt function in quantitative spectroscopy, J. Math. Research, 7 ( ( Preprint version: arxiv:54.3 8] J.H. Mathews and K.D. Fink, Numerical methods using Matlab, 4 th ed., Prentice Hall 999.

Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi

arxiv:1604.03752v1 [math.gm] 10 Apr 2016 Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi S. M. Abrarov and B. M. Quine April 10, 2016 Abstract