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Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous. A modern derivation can be found by examining ∫ 0 π sin n x d x {\displaystyle \int _{0}^{\pi }\sin ^{n}x\,dx} for even and odd values of n {\displaystyle n} , and noting that for large n {\displaystyle n} , increasing n ...
John Wallis (26 December 1650 – 14 March 1717), [7] MP for Wallingford 1690–1695, married Elizabeth Harris (d. 1693) on 1 February 1682, with issue: one son and two daughters Elizabeth Wallis (1658–1703 [ 8 ] ), married William Benson (1649–1691) of Towcester, died with no issue
In hyperbolic geometry (where Wallis's postulate is false) similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by George David Birkhoff (see Birkhoff's axioms ) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening ...
In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps are conformal maps of the upper half-plane H to triangles on the Riemann sphere, bounded by circular arcs. This mapping is a generalization of the Schwarz–Christoffel mapping to triangles with circular arcs.
In physics, a chain of coupled baker's maps can be used to model deterministic diffusion. As with many deterministic dynamical systems, the baker's map is studied by its action on the space of functions defined on the unit square. The baker's map defines an operator on the space of functions, known as the transfer operator of the map.
The sequence () is decreasing and has positive terms. In fact, for all : >, because it is an integral of a non-negative continuous function which is not identically zero; + = + = () () >, again because the last integral is of a non-negative continuous function.
Frontispiece of Operum Mathematicorum Pars Prima (1657) by John Wallis, the first volume of Opera Mathematica including a chapter entitled Mathesis Universalis.. Mathesis universalis (from Greek: μάθησις, mathesis "science or learning", and Latin: universalis "universal") is a hypothetical universal science modelled on mathematics envisaged by Descartes and Leibniz, among a number of ...
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.