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Brook Taylor
16851731 Introduced the Taylor expansion  the main foundation of differential calculus  in words of J. L. Lagrange 
The Taylor Center 

The software, the surrounding theory, and the community
The title of this site represents both a particular
software, the theory surrounding
it, and the community of interested
researchers.
The software is an advanced ODE solver called
the Taylor Center. The solver is
an AllinOne GUI application for
Windows implementing the modern Taylor integration method (Automatic
Differentiation) in a multifunctional user friendly environment with
unique features such as graphing and real time animation of the
trajectories in 2D and 3D stereo, and with the highest number of
significant digits –
the Intel extended
10 byte format with 63bit
mantissa (19 decimal digits).
The theory in the foundation of the method unifies
the
classical concept of elementary
functions, Automatic Differentiation, the Taylor Method for integration
of ODEs, analytic continuation, and the new type of the special points
in holomorphic functions: The so called unremovable
"removable"
singularities.
However there is an open question in this theory, the Conjecture
remaining unsolved since 2004. Due to the nature of its claim, the
Conjecture affects the ways of definition of general elementary
functions and therefore the outline of the entire theory. Thus, as an
organization, the Taylor Center seeks attention of the mathematical
community in order to sooner close the gap in this otherwise
comprehensive theory. This site therefore will provide a forum for
discussing and advancing the solution of the Conjecture, as well as the
updates and upgrades of the software.
Here
is the most brief presentation
(and pdf) of the gap
(slide 4), the Conjecture
(slides
68) and its
context.