A Curious Identity Arising From Stirling's Formula and Saddle-Point Method on Two Different Contours

  • Hsien-Kuei Hwang


We prove the curious identity in the sense of formal power series:
y^{j-2}\right)\mathrm{d} t
= \int_{-\infty}^{\infty}[y^m]
y^{j-2}\right)\mathrm{d} t,
for $m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$, which arises from applying the saddle-point method to derive Stirling's formula. The generality of the same approach (saddle-point method over two different contours) is also examined, together with some applications to asymptotic enumeration.

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