特定類有理合成函數之分數微分與積分問題

Some Differintegral Formulas of Composite and Rational Functions

陳志威(Chih-We Chen)

指導教授 ： 林賜德

中原大學/理學院/數學研究所/碩士(2008年)

https://doi.org/10.6840/cycu200900127

摘要

本篇論文主要針對有理合成函數，利用分數微積分之基本定義及常用性質，導證出不論是有理數或是複數階微分都可以成立的有理合成函數之微分積分法則，並以Fox-Wright函數來表示其最後結果。第一章首先介紹由日本西本教授定義的分數微積分定義以及本篇論文引用到的引理與性質，以及超幾何函數與Fox-Wright函數之定義。第二章推導有理合成函數
[(z−α)μ−ζ_]k 與(αz +β)ρ/[(λz +μ)σ −ζ_]k 之分數微分積分法則，並以Fox-Wright函數來表示其最後結果，並且將此二函數推廣到類似函數如1/[(z−α)μ−ζ_]k 或是整數階微分。第三章是探討使用分數微積分之方法與普通微積分之方法二者之間的關係，將定理一與定理二中的有理合成函數，分別對n階微分來討論，其中的n ∈ N0，並且實際將n = 0; 1，分別使用分數微積分之方法與普通微積分之方法，推導出二者的結果，並且比較二者之結果是否相同；若相同，表示用分數微積分之基本定義與性質導證出的有理合成函數之微分積分法，也能夠適用在正整數n階微分，其中包含n = 0。

關鍵字

有理合成函數 ； 超幾何函數 ； 分數微積分 ； 橢圓積分 ； Fox-Wright函數 ； Gamma函數

並列摘要

This paper mainly aims at holds some composite and rational function to prove some differintegral formulas of composite and rational functions by the basic definition and the commonly used property of fractional calcalus , no matter the rational number or complex number and expresses its final result by the Fox-Wright function. In first chapter,the first of introduced is the definition, the principle and the property of fractional calculus which are defined and proved by Professor Katsuyaki Nishimoto of Japan, and
the definition of hypergeometry function and Fox-Wright function. In second chapter, we prove the differintegral formulas of composite and rational functions about [(z−α)μ−ζ_]k and (αz +β)ρ/[(λz +μ)σ −ζ_]k, and expresses its final result by the Fox-Wright function. Besides, it generalizes the two functions to the similar function like 1/[(z−α)μ−ζ_]k or integer step differential. Third chapter discusses about the relation of the method of fraction calculus and the method of the basic calculus, and some composite and rational functions of theorem one and theorem two separately discusses the n step differential and n ∈ N0. Besides we separately uses the method of faction calculus and the method of the basic calculus to prove their result by n=0,1, and compare they are the same or not; if they are the same, it express that the differintegral formulas of composite and rational functions also can be suitable in the positive integer n step differential containing n=0.

並列關鍵字

composite and rational functions ； fractional calculus ； hypergeometry function ； Fox-Wright function

參考文獻

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