| 用参数展开法计算一类反常积分 |
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| 引用本文： | 王碧桂,邱为钢.用参数展开法计算一类反常积分[J].数学学习,2011,14(1):85-86. |
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| 作者姓名： | 王碧桂 邱为钢 |
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| 作者单位： | 湖州师范学院理学院,浙江,湖州,313000 |
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| 基金项目： | 浙江省高等学校创新团队项目，湖州师范学院省级精品课程项目，湖州师范学院高等教育研究项目 |
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| 摘 要： | 对含参数反常积分I(t,s)=∫+∞0 x-1(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.
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| 关 键 词： | 反常积分 对数函数 参数展开法 |
Evaluating a Class of Improper Integrals by Parameter-Expansion Method |
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| WANG Bi-gui,QIU Wei-gang.Evaluating a Class of Improper Integrals by Parameter-Expansion Method[J].Studies In College Mathematics,2011,14(1):85-86. |
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| Authors: | WANG Bi-gui QIU Wei-gang |
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| Institution: | (School of science, Huzhou Teacher＇s College, Huzhou 813000, PRC) |
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| Abstract: | 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. |
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| Keywords: | improper integral logarithmic function parameter-expansion method |
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