See: Math concepts, Symmetry
Develop a notion of analytic symmetry related to solving the equations {$f^{(n)}=f$}.
- Study the differentiation of Taylor series for {$f^{(n)}=f$}.
- How does a superimposed sequence reflect the symmetry of the whole?
Exponential function {$e^x$} is a symmetry from the point of view of differentiation - it is the unit element. And cos and sin are fourth roots. And look for more such roots.
Taylor series for {$e^x$} is based on the symmetric function (inverted) - it is the basis for symmetry in analysis.
Study the symmetry of functions like exponentials, trigonometric, etc by considering the derivative equation {$f^{(n)}=f$} for various integers n. And nonintegers n? Study using Taylor series expansions.
What is special about {$e$}.
- The ratio of arrangements and derangements on n letters goes to e as n goes to infinity. Derangements are counted by using the inclusion-exclusion principle on arrangements.
Symmetry of the whole
- Algorithmic symmetry: Algorithms fold the list of effective algorithms upon itself.
- Analytic symmetry is related to an infinite matrix, which is what we have with the classical Lie algebras. It is also the relation that an infinite sequence (and the function it models) can have with itself. It is thus the symmetry inherent in a recurrence relation, which is the content of a three-cycle, what it is relating, a self to itself.
- Analytic symmetry is governed by the characteristic polynomial which can't be solved in degree 5.
- Euler's manipulations of infinite series (adding up to -1/12 etc.) are related to divisions of everything, of the whole. Consider the Riemann Zeta function as describing the whole. The same mysteries of infinity are involved in renormalization.