When Mathematics Helps Physics: Calculation of the Integral of Kholodenko and Silagadze

TL;DR

This work provides a physics-free, contour-based derivation of the Landau-Zener-type integral $I_n=\frac{2}{n!}\left(\frac{\pi}{4}\right)^n$ for all $n\ge1$, by recasting the problem into a generating function framework. The authors introduce auxiliary real-valued functions $U,V$ and complex combinations $P,Q$, then derive a chain of relations through a contour integral and a pair of integral transforms, culminating in an imaginary-boundary condition that fixes the generating-function coefficients. The core steps involve an integral equation for a generating function $T(x,t)$, its reduction to a fourth-order ODE, and the use of even and truncated transforms to relate values at infinity to those at the origin, with explicit expressions for $\alpha(t)$ and $\beta(t)$ in terms of $U,V$. A related result shows a symmetric variant $K_n$ yields the same closed form, highlighting a purely mathematical route to a problem connected to the Landau-Zener formula. The approach broadens accessibility to these integrals by avoiding quantum-physics notions and demonstrates how contour methods can unlock exact, rapidly computable formulas for nested cosines integrals.

Abstract

In this paper, we show a physics-free derivation of a Landau-Zener type integral introduced by Kholodenko and Silagadze.

Figures (2)

• Figure 1: Contour $C$
• Figure 2: Numerical solution of ODE for $T(x,t)$