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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. (English) Zbl 0693.41033
Summary: The integral $$ F\sb{\lambda}(z,\alpha)=\int\sp{\infty}\sb{0}t\sp{\lambda -1}e\sp{-zt-\alpha /t}f(t)dt $$ is considered for large values of the real parameter z; $\alpha$ and $\lambda$ are uniformity parameters in [0,$\infty)$. The asymptotic expansion is given in terms of the modified Bessel function $K\sb{\lambda}(2\sqrt{\alpha z})$. The asymptotic nature of the expansion is discussed and error bounds are constructed for the remainders in the expansions. An example is given for confluent hypergeometric or Whittaker functions. In this example the integrals are transformed to standard forms and the mappings are investigated.

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E15Asymptotic representations in the complex domain
33B20Incomplete beta and gamma functions
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