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I am looking for an asymptotic expansion of the oscillatory integral of the form $$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$ as $\lambda_i\to \infty$ independently, where $f\in C^\infty_c(\mathbb{R}^n)$ and $\phi_i$ are smooth.

When all $\lambda_i$ are equal an asymptotic expansion can be given by usual stationary phase analysis. I am unable to find any literature which discusses about different $\lambda_i$.

Thanks for any reference.

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The case of two parameters was worked out by D. Sather in Stationary phase for two parameters with an application to Bessel functions.

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