這篇論文是延續陳惠瑜在2009年的碩士論文。在此，我們更為仔細地研究廣義正弦函數Sp 和廣義餘弦函數 Cp 。特別在 α> -1和 β>1-p 的條件下， 我們計算出兩不定積分 ∫_0^(πp/2) Sp (x)^α S'p (x)^β dx 和 ∫_0^(πp/2) C_p (x)^α |C'p (x)|^β dx 能以伽瑪函數的形式呈現。接著Binding [2]等人的工作，我們對函數列 {Sp (nπp x)} 在 L^2 (0,1) 空間組成 Riesz 基底和在 L^q (0,1) 空間 ( q>6/5 ) 組成 Schauder 基底給出一個簡單的證明。
另一方面，我們定義了廣義雙曲正弦函數 Sh_p 和廣義雙曲餘弦函數 Ch_p 如下:
x=∫_0^(Shp (x))(1+|t|^p)^(-1/p) dt 和 |x|=∫_1^Chp (x)(t^p-1)^(-1/p) dt
當p=2時，即為雙曲正弦函數和雙曲餘弦函數。我們列出了和上述定義等價的恆等式及相關的半線性方程式。再者，我們推導出兩個瑕積分，∫_0^∞ Sh_p (x)^α Sh^'p (x)^β dx 和 ∫_0^∞ Ch_p (x)^α Ch'_p (x)^β dx 分別在 α>-1,α+β<0 和 β>1-p,α+β<0的條件下，亦能以伽瑪函數的形式呈現。

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+β<0 and β>1-p,α+β<0, again in terms of gamma functions.

1.Introduction........................................................................1
2.Integration of generalized trigonometric functions.....9
3.Generalized hyperbolic functions................................19
3.1 Generalized hyperbolic sine functions................19
3.2 Generalized hyperbolic cosine functions............24
3.3 Integration of generalized hyperbolic functions.34
3.4 Other generalized hyperbolic functions...............37
4. Basis properties of generalized sine functions......42
Appendices.........................................................................48
1 The gamma function.....................................................48
2 Table of properties of Sp(x), Cp(x), Shp(x) and Chp(x)...................................................................................53
3 The graph of πp, Shp(x) and Chp(x) ..........................55

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