Asymptotic Expansions for Product Integration

Symbolic Integration and Asymptotic Expansions

It didn’t do the integrals, but it did apply the sum rule! What’s more, the sum rule was applied twice, even though our rule only indicated one summation. This is a basic feature of Mathematica: it applies simplification rules repeatedly without our having to ask for that explicitly. Let’s tell OurIntegrate about taking constants out of the integral. The syntax at the end of the following rule ...

Asymptotic Expansions for Integrals

Asymptotic Expansions

Asymptotic expansions of functions are useful in statistics in three main ways. Firstly, conventional asymptotic expansions of special functions are useful for approximate computation of integrals arising in statistical calculations. An example given below is the use of Stirling's approximation to the gamma function. Second, asymptotic expansions of density or distribution functions of estimato...

Asymptotic expansions of integrals: the term by term integration method

The classical term by term integration technique used for obtaining asymptotic expansions of integrals requires the integrand to have an uniform asymptotic expansion in the integration variable. A modification of this method is presented in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the relaxation of the uniformity condition provides...

Asymptotic expansions of integrals: The term-by-term integration method

The classical term-by-term integration technique used for obtaining asymptotic expansions of integrals requires the integrand to have an uniform asymptotic expansion in the integration variable. A modi cation of this method is presented in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the relaxation of the uniformity condition provides ...