Three essays on Machin’s type formulas
Copyright policy )Three essays on Machin’s type formulas
We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$. These formulas are computationally useful because multiplication or division by a power of two is a very fast operation for most computers. The second one presents a method for finding infinitely many formulas with $N$ terms. In the particular case $N=2$ the method is quite useful. It recovers most known formulas, gives some new ones, and allows to prove in an easy way that there are two terms Machin formulas with Lehmer measure as small as desired. Finally, we correct an oversight from previous result and give all Machin's type formulas with two terms involving arctangents of powers of the golden section.
22 pages, 3 figures
Section (typography), Inverse trigonometric functions, Multiplication (music), Applications of Generalized Functions in Mathematics and Physics, golden section, Mathematical analysis, Theoretical Computer Science, Database, Golden section, FOS: Mathematics, Machin's type formulas, Number Theory (math.NT), Biology, Mathematical Physics, Algebra over a field, Mathematics - Number Theory, Arithmetic, Ecology, 11Y60 (Primary), 11D45, 33B10 (Secondary), Pure mathematics, Cantorian-Fractal Theory of Quantum Physics, Statistical and Nonlinear Physics, Division (mathematics), Measure (data warehouse), Mathematical Concepts, Discrete mathematics, Computer science, computation of \(\pi\), Operating system, History of Mathematics in Education and Research, Computation of π, Physics and Astronomy, Combinatorics, FOS: Biological sciences, Physical Sciences, Machin's-type formulas, Type (biology), Mathematics, Numerical analysis, Lehmer measure
Section (typography), Inverse trigonometric functions, Multiplication (music), Applications of Generalized Functions in Mathematics and Physics, golden section, Mathematical analysis, Theoretical Computer Science, Database, Golden section, FOS: Mathematics, Machin's type formulas, Number Theory (math.NT), Biology, Mathematical Physics, Algebra over a field, Mathematics - Number Theory, Arithmetic, Ecology, 11Y60 (Primary), 11D45, 33B10 (Secondary), Pure mathematics, Cantorian-Fractal Theory of Quantum Physics, Statistical and Nonlinear Physics, Division (mathematics), Measure (data warehouse), Mathematical Concepts, Discrete mathematics, Computer science, computation of \(\pi\), Operating system, History of Mathematics in Education and Research, Computation of π, Physics and Astronomy, Combinatorics, FOS: Biological sciences, Physical Sciences, Machin's-type formulas, Type (biology), Mathematics, Numerical analysis, Lehmer measure
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