Beam Models and Quantum Systems with Linear Potentials Based on Airy Functions
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Abstract
This article provides a rigorous study of Airy functions, emphasizing their construction
by power series, their relation with Bessel functions, and their canonical integral representation. We
highlight their asymptotic properties and structural role in mathematical physics. To illustrate their
applicability, we develop three analytical models: the Euler–Bernoulli beam under self-weight, the
quantum bouncer with a linear gravitational potential, and the particle in a uniform electric field.
In each case, the quantization conditions and physical scales naturally emerge from the zeros of
the Airy function. These results confirm the central role of Airy functions in bridging differential
equations, special functions, and applied physics.
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References
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