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 {$f^{(n)}=f$}.

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.

Parsiųstas iš http://www.ms.lt/sodas/Book/AnalyticSymmetry

Puslapis paskutinį kartą pakeistas 2018 lapkričio 11 d., 16:33