## A simplification of the Laplace method for double integrals. Application to the second Appell function

José L. López and Pedro J. Pagola

### Abstract

The main difficulties in the Laplace method of asymptotic expansions of double integrals result from a change of variables. Generalizing previous work for simple integrals, we propose a variant of the method for double integrals, which avoids this change of variables and simplifies the computations. The calculation of the coefficients of the asymptotic expansion is remarkably simple. Moreover, the asymptotic sequence is as simple as in the standard Laplace's method: inverse powers of the asymptotic variable. A new asymptotic expansion of the second Appell's function $F_2(a,b,b',c,c';x,y)$ for large $b$, $b'$, $c$ and $c'$ is given as an illustration.

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### Key words

asymptotic expansions of integrals, Laplace method for double integrals, second Appell function

### AMS subject classifications

41A60, 41A58, 33C65

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