Generalized trigonometric functions in complex domain

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F15%3A43925683" target="_blank" >RIV/49777513:23520/15:43925683 - isvavai.cz</a>
Result on the web
<a href="http://mb.math.cas.cz/mb140-2/10.html" target="_blank" >http://mb.math.cas.cz/mb140-2/10.html</a>
DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Generalized trigonometric functions in complex domain
Original language description
We study extension of $p$-trigonometric functions $sin_p$ and $cos_p$ to complex domain. For $p=4, 6, 8, dots$, the function $sin_p$ satisfies the initial value problem which is equivalent to $$-(u')^{p-2}u"-u^{p-1} =0, quad u(0)=0, quad u'(0)=1 leqno(*)$$ in $mathbb{R}$. In our recent paper, Girg, Kotrla (2014), we showed that $sin_p(x)$ is a real analytic function for $p=4, 6, 8, dots$ on $(-pi_p/2, pi_p/2)$, where $pi_p/2 = int_0^1(1-s^p)^{-1/p}$. This allows us to extend $sin_p$ to complex domain by its Maclaurin series convergent on the disc ${zinmathbb{C}colon|z|<pi_p/2}$. The question is whether this extensions $sin_p(z)$ satisfies (*) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of $sin_p$ to complex domain for $p=3,5,7,dots$ Moreover, we show that the structure of the complex valued initial value
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
S - Specificky vyzkum na vysokych skolach

Publication year
2015
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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