菲涅尔积分的几种计算方法

Calculation Methods of Fresnel Integral

考虑菲涅尔积分的多种计算方法的来源问题,介绍了通过引入收敛因子转化为二重广义积分计算的方法,并指出这种方法发现的思想来源。对菲涅尔积分和广义菲涅尔积分给出了利用广义积分交换次序定理的计算方法,没有通过引入收敛因子就解决了问题,方法自然且具有一般性。对一类欧拉积分公式,给出了对参变量求导的简便计算方法,指出了一类欧拉积分公式对广义菲涅尔积分计算的应用,发现菲涅尔积分、广义菲涅尔积分、狄利克雷积分都可以是一类欧拉积分公式的特例,沟通了这些积分之间的关系。

One of the calculation methods of Fresnel integral is achieved by means of the conversion to double improper integral and the origin of this method is introduced. The calculation method of Fresuel integral and generalized Fresnel integral via the theorem of integral exchange order is discussed without the aid of the convergence factor, which generalizes the method. The simplified calculation method of the Euler integral formula via the derivation of the parameter is demonstrated, which explains the application of Euler integral formula to generalized Fresnel integral calculation and establishes the relation- ship among Fresnel integral, generalized Fresnel integral and Dirichlet integral which are all found to be the special cases of Euler integral formula.