Topics on Generalized Trigonometric And Hyperbolic Functions
| Autor: | Yu-wen Yang, 楊喻文 |
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| Rok vydání: | 2013 |
| Druh dokumentu: | 學位論文 ; thesis |
| Popis: | 101 This thesis is a continuation of the master thesis of Hui-Yu Chen in 2009. We study the generalized sine functions Sp and generalized cosine functions Cp in more detail. In particular, we evaluate the definite integrals of ∫_0^(πp/2) Sp (x)^α S''p (x)^β dx and ∫_0^(πp/2) C_p (x)^α |C''p (x)|^β dx α> -1 and β>1-p in terms of gamma function. Following the work of Binding et al [2], we also give a proof that the sequence {Sp (nπp x)} form a Riesz basis in L^2(0,1) and a Schauder basis in L^q(0,1) for ( q>6/5 ). On the other hand, we define the generalized hyperbolic sine function Shp and generalized hyperbolic cosine function Chp as x=∫_0^(Shp (x))(1+|t|^p)^(-1/p) dt and |x|=∫_1^Chp (x)(t^p-1)^(-1/p) dt When p=2, they become the hyperbolic sine and cosine functions. We show that the definition is equivalent to the identity and the associated half-linear equation for each function. Furthermore we evaluate the improper integrals ∫_0^∞ Sh_p (x)^α Sh^''p (x)^β dx 和 ∫_0^∞ Ch_p (x)^α Ch''_p (x)^β dx, α>-1,α+β1-p,α+β |
| Databáze: | Networked Digital Library of Theses & Dissertations |
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