用参数展开法计算一类反常积分

Evaluating a Class of Improper Integrals by Parameter-Expansion Method

对含参数反常积分I(t,s)=∫0+∞x-t(1+x)-sdx,由贝塔函数的积分表示得到I(t,s)的伽马函数表示,再由伽马函数的级数展开,得到I(t,s)的参数级数展开.I(t,s)可在积分符号内按参数展开,参数系数是含对数函数的反常积分.对比同类参数的系数,可得一系列含对数函数反常积分的值.

By its relation with Beta functions, the improper integral I（t,s）=∫0^＋∞（1＋z）^-x dx can be expressed in terms of Gamma functions. Using the power series expansion for Gamma function, the power series for I（t,s） in terms of parameters t and s can be obtained. On the other hand, the integrand can be expanded into power series of parameters t and s also. Comparing the coefficients of the two power series, many improper integrals of logarithmic functions and their values are obtained.