A curious identity in connection with saddle-point method and Stirling's formula

2022/10/20 by Hsien‐Kuei Hwang, Hwang, Hsien-Kuei
Mathematics · #Fractional Differential Equations Solutions #Functional Equations Stability Results #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2210.10989

Abstract

We prove the curious identity in the sense of formal power series: ∫_-∞^∞[y^m] exp(-\fract²2 +∑_j≥3((it)^j)/(j!) y^j-2)d t = ∫_-∞^∞[y^m] exp(-\fract²2+ ∑_j≥3((it)^j)/(j) y^j-2)d t, for m=0,1,…, where [y^m]f(y) denotes the coefficient of y^m in the Taylor expansion of f. The generality of this identity from the perspective of saddle-point method is also examined.

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