Plus Advent Calendar Door #7: Stokes's phenomenon

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When you look closely at a rainbow, then if you are lucky you will see one or more faint arcs below the main bow that are predominantly green, pink and purple. These are called supernumerary fringes and back in 1838 they attracted the attention of the then Astronomer Royal George Biddell Airy. Little did he know that his attempt to describe those fringes mathematically would found an entire field of mathematics that is still active today.


A rainbow with supernumerary fringes. Photo: Johannes Bahrdt, CC BY-SA 4.0.

The problem which Airy's question eventually uncovered concerns quantities that are so small — exponentially small — that you might be tempted to ignore them. But this you do at your peril. As George Gabriel Stokes showed when he was working on Airy's question, these quantities can grow exponentially large over time and space. This doesn't just happen with supernumerary fringes of rainbows. Understanding these potentially explosive quantities is essential in all sorts of areas beyond maths: from building jet engines to theoretical physics.

This year the Isaac Newton Institute in Cambridge ran a virtual research programme on the topic, bringing together some of the best minds in the field. We were lucky to speak to one of the organisers, Chris Howls, who provided some fascinating insight into Stokes's phenomenon — it's an asymptotic adventure well worth reading about. Find out more here.

Return to the Plus advent calendar 2021.

This article is part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

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