四川轻化工大学学报：自然科学版

研究考虑一类欧拉积分公式的计算问题,旨在对其实现简化证明。这类欧拉积分公式是成对出现的,可分别被看作复数的实部和虚部。首先通过应用复数的欧拉公式表示,转化一个含复参变量的广义积分形式,并采用对参变量的求导方法来建立常微分方程,通过求解此微分方程给出了欧拉积分的解析表达式,然后分别取实部和虚部来得出欧拉积分公式。接下来应用所得的欧拉积分公式,利用两无穷限广义积分交换次序,给出了一类广义积分的用实变方法的计算结果,还对相关几类广义积分的计算给出了统一的推导方法,并剖析了几类广义积分之间的相互联系。最后,揭示了Γ函数和欧拉积分公式的重要作用。

It is to consider the calculation problem of a class of Euler's integral formula and the purpose is to realize its simplified proof. This kind of Euler integral formula appears in pairs and can be regarded as the real part and the imaginary part. Firstly,by using the Euler formula of the complex number,a generalized integral form with complex parameter variables is transformed,and the ordinary differential equation is established by using the method of derivation of the parametric variables. The analytic expression of the Euler integral is given by solving the differential equation. Then the Euler integral formula is obtained by taking the real part and the imaginary part respectively. Next,by using the Euler integral formula and the commutative order of two infinite bounded generalized integrals,we give the results of the real variable method for a class of generalized integrals,and give a unified derivation method for the calculation of some related generalized integrals. The interrelation between several kinds of generalized integrals are analyzed. Finally,the importance role of the Γ function and Euler integral formula is revealed.