在這篇論文中，我們研究在機率分佈函數的史提
司轉換並同時跟機率分佈函數的特徵函數做比較。
在第一節及第二節裡，我們簡介史提司轉換的基
本性質。
在第三節中，我們得到對稱機率分佈函數的史提
司轉換有類似對稱的性質。
在第四節中，我們建立機率分佈函數的史提司轉
換和其密度函數之間的關聯並證明史提司轉換的n 次
導數在上半複數平面是均勻連續的。

In this thesis, we study the Stieltjes transforms of the probability distribution functions
and compare them with the characteristic functions of the probability distribution functions
simultaneously.
In section 1 and section 2, we introduce briefly the Stieltjes transforms.
In section 3, we conclude that the Stieltjes transform is similar to the complexion of
symmetry under the condition of symmetric probability distribution functions.
In section 4, we discuss the relation between Stieltjes transforms of probability
distribution functions and the density of probability distribution functions. We also
show that the nth derivative of Stieltjes transform is uniformly continuous on the upper
complex plane.

1 Introduction
2 Preliminary Properties of Stieltjes Transforms
3 Symmetric Probability Distribution Functions
3.1 Characteristic Functions of Symmetric Probability Distribution Functions
3.2 Stieltjes Transforms of Symmetric Probability Distribution Functions
4 Further Results
4.1 Relation between the Stieltjes Transforms and the Density Functions
4.2 The nth Derivative of Stieltjes transforms

[1] Engene Lukaces, Characteristic Function, Gri n, London, 1970.
[2] Shern-Huh Ou, The Stieltjes Transforms of Probability Distribution Functions,
Master thesis, National Sun Yat-sen University, 1997.
[3] Allen C. Pipkin, A Course on Integral Equations, Springer-Verlag, 1991.
[4] JackW.Silverstein & Sang-Il Choi, Analysis of the Limiting Spectral Distribution of
Large Dimensional Random Matrices, Journal of Multivariate 54, 295-309, 1995.
[5] D. V. Widder, The Stieltjes Transform, Transactions of the American Mathematical
Society, Vol.43, 7-60, 1938.