Translation by C. Lischka and D. Lischka von. Jörg Arndt, Christoph Haenel, C.
Lischka, D. Lischka. 1. Auflage. Pi - Unleashed – Arndt / Haenel / Lischka / et al.
Unleash your power Lord,. C. D Ebdim. Pour out your grace oh God. Em. D/F# G.
Unleash your purifying fire,. C. Unleash it all,. G. Am G/B. Unleash your spirit ...
We need to be Unleashed from the myth that bigger is better. Let's look at some ...
Read Ephesians 2:10 and 1 Corinthians 12:14-20 ...... He will not force Himself,
or .... body else and they may want to want to script your life to their advant
Dec 12, 2005 - L3 detector at LEP in a data sample collected at centre-of-mass energies ..... 17 INFN Sezione di Firenze and University of Florence, I-50125 ...
Nov 20, 2009 - simple testing ground to study modifications of gravity and their cosmological implications. 1. arXiv:0904.0809v2 [astro-ph.CO] 20 Nov 2009 ...
received an oral dose (10 mg to 60 mg) of PRX-00023. In addition, six subjects received a single 30 mg oral dose of Buspirone (an azapirone 5HT1A agonist).
e TRIUMF, Vancouver, B.C., Canada V6T 2A3 f University of Regina, Regina, Saskatchewan, Canada S4S 0A2 g Physikalisches Institut, Universität Tübingen, ...
Helene F. Rosenberg. Eosinophils unleashed ... phils express PIR-B is in and of
itself not par- ticularly surprising, as PIR-B has already been detected on the ...
These highlights do not include all the information needed to use ONGLYZA safely and ... Bullous Pemphigoid:There have b
24-week data regardless of glycemic rescue. In patients treated with ONGLYZA 2.5 mg, headache (6.5%) was the only advers
Monitor patients for signs and symptoms. (5.2). ⢠Hypoglycemia: In add-on to sulfonylurea, add-on to insulin, and add-
24-week data regardless of glycemic rescue. In patients treated with ONGLYZA 2.5 mg, headache (6.5%) was the only advers
kat, MD, C. Li, PhD, G. G. Sofowora, MD, J. Solus, PhD, H. G. Xie,. MD, PhD ... Ihrie, MS, E. P. Dawson, BS, S. M. Williams, PhD, A. J. Wood, MD,. C. M. Stein, MD ...
Dec 11, 2006 - Recent work in low energy pion physics is reviewed. One of the ...... [40] S. Descotes-Genon and B. Moussallam, arXiv:hep-ph/0607133.
Pi – Unleashed. Trans. Catriona and David Lischka. Berlin: Springer, 2001.
Google Books. 7 Sept. 2009. . ▻ Berlinghoff and
Gouvea ...
Abstract: Active control of the production choke valve is the recommended solution to prevent ... to become unstable after some time, because of inflow.
Feb 6, 2012 - Abstract. In terms of the hypergeometric method, we establish the exten- ... 2√3 π . (2). Following Bailey [4], define the hypergeometric series by. 1+rFs [a0, .... Two examples from Theorem 2 are laid out as follows. .... [12] H.H. Cha
If Asstsmptiorzs 1 a d 2 are satisjied in case (i) then uni- formly for P E y, y E A (P, n), .... IBI < s-krcj. .... P. Steck, Limit theorems for conditiorual distributions, Univ.Missing:
equations and Approximations to $\pi$ â, Ra- manujan offered 17 beautiful series for $1/\pi$ . He then remarks that two of these series, namely,. (1.1).
Feb 2, 2006 - S. D. Foulkes, J. W. Gary, O. Long, B. C. Shen, K. Wang, and L. Zhang .... N. Berger, A. M. Boyarski, R. Claus, J. P. Coleman, M. R. Convery,.
arXiv:hep-ph/9711361v1 18 Nov 1997. KFA-IKP(TH)-1997- ...... e.g. the fit to the
Jul 5, 2007 - recent BES II measurement [18]. For the Ï we switch to a complex pole amplitude, Eq. 8, rather than the spin-0 Breit-Wigner used by E791.
al geodatabases are stored in Microsoft Access, and ArcSDE geodatabases can
be stored in a va- riety of relational database management systems. (DBMS) ...
Jorg Arndt Christoph Haenel. Pi - Unleashed. Translated from the German by
Catriona and David Lischka. With CD-ROM. Springer ...
Jorg Arndt Christoph Haenel
Pi - Unleashed Translated from the German by Catriona and David Lischka
With CD-ROM
Springer
Contents
1. The State of Pi Art
1
2. How Random is TT? 2.1 Probabilities 2.2 Is 7T normal? 2.3 So is TT not normal? 2.4 The 163 phenomenon 2.5 Other statistical results 2.6 The Intuitionists and TT 2.7 Representation of continued fractions
21 21 21 24 25 28 30 32
3. Shortcuts to TT 3.1 Obscurer approaches to TT 3.2 Small is beautiful 3.3 Squeezing TT through a sieve 3.4 TT and chance (Monte Carlo methods) 3.5 Memorabilia 3.6 Bit for bit 3.7 Refinements 3.8 The TT room in Paris
35 35 37 38 39 44 47 49 50
4. Approximations for TT and Continued Fractions 4.1 Rational approximations 4.2 Other approximations 4.3 Youthful approximations 4.4 On continued fractions
51 51 55 63 64
5. Arcus Tangens 5.1 John Machin's arctan formula 5.2 Other arctan formulae
69 69 72
X
6.
Contents
Spigot Algorithms 6.1 The spigot algorithm in detail 6.2 Sequence of operations 6.3 A faster variant 6.4 Spigot algorithm for e
7. Gauss and TT 7.1 The TT AGM formula 7.2 The Gauss AGM algorithm 7.3 Schonhage variant 7.4 History of a formula
77 78 80 82 84 87 87 90 92 94
8. Ramanujan and TT 8.1 Ramanujan's series 8.2 Ramanujan's unusual biography 8.3 Impulses
103 103 105 110
9.
113
The Borweins and TT
10. The B B P Algorithm 10.1 Binary modulo exponentiation 10.2 A C program on the BBP series 10.3 Refinements
12. Miscellaneous 12.1 A TT quiz 12.2 Let numbers speak 12.3 A proof that TT = 2 12.4 The big change 12.5 Almost but not quite 12.6 Why always more?
153 153 154 155 155 156 158
Contents 12.7 TT and hyperspheres 12.8 Viete x Wallis = Osier 12.9 Squaring the circle with holes 12.10An (in)finite funnel
XI 158 160 162 164
13. The History of TT 13.1 Antiquity 13.2 Polygons 13.3 Infinite expressions 13.4 High-performance algorithms 13.5 The hunt for single TT digits Table: History of TT in the pre-computer era Table: History of TT in the computer era Table: History of digit extraction records
165 167 170 185 198 203 205 206 207
14. Historical Notes 14.1 The earliest squaring the circle in history? 14.2 A TT law 14.3 The Bieberbach story
209 209 211 213
15. The Future: TT Calculations on the Internet 15.1 The binsplit algorithm 15.2 The TT project on the Internet
215 215 219
16. 7T Formula Collection
223
17. Tables 17.1 Selected constants to 100 places (base 10) 17.2 Digits 0 to 2,500 of TT (base 10) 17.3 Digits 2,501 to 5,000 of TT (base 10) 17.4 Digits 0 to 2,500 of TT (base 16) 17.5 Digits 2,501 to 5,000 of TT (base 16) 17.6 Continued fraction elements 0 to 1,000 of TT 17.7 Continued fraction elements 1,001 to 2,000 of vr
239 239 240 241 242 243 244 245
A. Documentation for the hfloat Library A.I What hfloat is (good for) A.2 Compiling the library A.3 Functions of the hfloat library A.4 Using hfloats in your own code
247 247 248 248 250
XII
Contents
A.5 Computations with extreme precision A.6 Precision and radix A.7 Compiling & running the 7r-example-code A.8 Structure of hfloat A.9 Organisation of the files A. 10 Distribution policy k, no warranty
