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Airy Functions Demystified — I

The Mathematical Physics of an Innocent Looking Differential Equation — Basics and Numerical Solutions

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Abstract

Airy functions have been long important in various areas of mathematics, physics, and astronomy. Even though the Airy differential equation looks extremely simple, its solutions are generally ‘notorious’ in nature. In this expository article, we have given a thorough description of the Airy functions with real arguments. Majority of the books (with the exception of specialized books on the subject) do not pay attention in detail to these functions. Taking this as a motivating factor, we have written this article for postgraduates and students beginning research to help them understand the nitty-gritty of these beautiful functions. The solutions of the Airy differential equation called the Airy and Bairy functions are explained in detail. Finally, we have given some ideas to numerically obtain these functions.

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Suggested Reading

  1. George Biddell Airy, On the intensity of light in the neighborhood of a caustic, Transactions of the Cambridge Philosophical Society, Vol.6, pp.37–402, 1838.

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  2. Milton Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications Inc., 1965.

  3. James Ward Brown and Ruel Churchill, Complex Variables and Applications, Mc Graw-Hill, 1943.

  4. J L M Quiroz Gonzalez and D Thompson, Getting started with Numerov’s method, Computers in Physics, Vol.11, pp.514–515, 1997.

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  5. H M Antia, Numerical Methods for Scientists and Engineers, Hindustan Book Agency, 2012.

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Authors and Affiliations

  1. Department of Physics, Visvesvaraya National Institute of Technology, Nagpur, 440 010, India

    M. S. Ramkarthik

Authors
  1. M. S. Ramkarthik
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  2. Elizabeth Louis Pereira
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Corresponding author

Correspondence to M. S. Ramkarthik.

Additional information

M S Ramkarthik (left) is an Assistant Professor at the Department of Physics, Visvesvaraya National Institute of Technology, Nagpur. He is working in the field of quantum information theory, quantum computing, entanglement and mathematical physics.

Elizabeth Louis Pereira (right) is a second-year M.Sc. student working under the guidance of Dr M S Ramkarthik at the Department of Physics, Visvesvaraya National Institute of Technology, Nagpur and her interests are in theoretical and mathematical physics.

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Ramkarthik, M.S., Pereira, E.L. Airy Functions Demystified — I. Reson 26, 629–647 (2021). https://doi.org/10.1007/s12045-021-1166-4

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  • DOI: https://doi.org/10.1007/s12045-021-1166-4

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