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Andrius Kulikauskas

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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

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Book.AnalyticSymmetry istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2018 lapkričio 11 d., 16:33 atliko AndriusKulikauskas -
Pridėtos 16-17 eilutės:

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.
2018 lapkričio 11 d., 16:30 atliko AndriusKulikauskas -
Pridėtos 14-15 eilutės:

Taylor series for e^x is based on the symmetric function (inverted) - it is the basis for symmetry in analysis.
2018 lapkričio 11 d., 16:27 atliko AndriusKulikauskas -
Pridėtos 7-10 eilutės:
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* Study the differentiation of Taylor series for {$f^{(n)}=f$}.
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Pridėta 13 eilutė:
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.
2018 lapkričio 11 d., 16:12 atliko AndriusKulikauskas -
2018 lapkričio 11 d., 15:59 atliko AndriusKulikauskas -
Pakeista 11 eilutė iš:
* The ratio of permutations and [[https://en.wikipedia.org/wiki/Derangement | derangements]] on ''n'' letters goes to ''e'' as ''n'' goes to infinity. Derangements are counted by using the inclusion-exclusion principle on arrangements.
į:
* The ratio of arrangements and [[https://en.wikipedia.org/wiki/Derangement | derangements]] on ''n'' letters goes to ''e'' as ''n'' goes to infinity. Derangements are counted by using the inclusion-exclusion principle on arrangements.
2018 lapkričio 11 d., 15:59 atliko AndriusKulikauskas -
Pakeista 11 eilutė iš:
* The ratio of permutations and [[https://en.wikipedia.org/wiki/Derangement | derangements]] on ''n'' letters goes to ''e'' as ''n'' goes to infinity.
į:
* The ratio of permutations and [[https://en.wikipedia.org/wiki/Derangement | derangements]] on ''n'' letters goes to ''e'' as ''n'' goes to infinity. Derangements are counted by using the inclusion-exclusion principle on arrangements.
2018 lapkričio 11 d., 15:58 atliko AndriusKulikauskas -
Pridėtos 8-11 eilutės:


What is special about ''e''.
* The ratio of permutations and [[https://en.wikipedia.org/wiki/Derangement | derangements]] on ''n'' letters goes to ''e'' as ''n'' goes to infinity.
2018 lapkričio 11 d., 15:57 atliko AndriusKulikauskas -
Pridėtos 1-7 eilutės:
>>bgcolor=#E9F5FC<<
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See: [[Math concepts]], [[Symmetry]]

'''Develop a notion of analytic symmetry related to solving the equations {$f^{(n)}=f$}.'''
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AnalyticSymmetry


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