Ode to Bessel Functions

Ok here is what I believe is the first ever poem to be published on Bessel Functions. Those who studied partial differential equations will know that if a function is expressed in cylindrical coordinates the separation of variables technique will usually end up with one of the equations resulting in Bessel's equations, named after Bessel who first came across it when investigating planetary ,motion. If the equation is solved by assuming a series solution by the Froebenius method a set of orthogonal functions is obtained called Bessel Functions. These are essentially similar to cosines and sines in rectangular coordinates and any function in cylindrical coordinates can be expressed in terms of them. 

Since their discovery (or invention) they have been used for all sorts of purposes to model the acoustic waves in a cylindrical tube, the electromagnetic waves in a cylinder, heat conduction in a cylinder and many other things so it is important to become familiar with them, Unfortunately the Open University Applied  maths and physics courses only mention them in passing and their properties are hardly developed. A good little book although out of print but easy to find on Amazon which discusses Bessel functions and their properties is by Sneddon 

https://www.amazon.co.uk/Special-functions-mathematical-physics-chemistry/dp/B00316KJ0Q

 Perhaps the most amazing application of Bessel Functions was their use by Crick to deduce the diffraction pattern of the Double Helix. He realised that if DNA was a helix then it's diffraction pattern could be expressed in terms of Bessel Functions. Resulting in a 'Saltire' like pattern as shown in the photograph below. What is even more amazing was the fact that if there were two helices intertwined with each other then the 4 peak would be minimised and again this can be seen in the photograph below. So there we have it almost definitive proof that DNA the molecule of life is a double helix and ultimately dating back to Bessel's Functions

I have to add a sorry note to this marvelous tale, the woman who did all the hatd work in obtaining the diffraction photographs was a French woman called Rosalind Franklin, Being a women she was treated really appallingly and Watson in his book makes some quite sexist remarks about her. Alsu unfortunately she died at a fairly young age too late to get any credit when the Nobel prize was awarded. So just as in the case of Jocelyn Bell, sexist males got all the credit for a woman's work. There was a very good Horizon programme about this made in the mid 1980's I don't know if it is still available if you can get a copy it is definitely worth trying to track down. 

Anyway here is the tribute to Mr Bessel and his amazing set of functions 

                                                        Bessel

                                            What an amazing function,

                                            We can use it without compunction.

                                            Any field that’s in a tube,

                                            You have put it in the groove.

 

                                            Aided by your Function,

                                            Franklin, Crick and Watson,

                                            After a long, long trial,

                                            Deduced life’s double spiral.

 

Rosalind Franklin's photograph of DNA, note the missing peak this indicates that there are two helices interwined with each other.