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July 13th, 2016, 02:26 AM   #1
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These are important integrals in optics but they are never solved

I'm computing light waves in some optical structures. I've found the integrals that nobody has solved before. The solutions, if possible, are important because they can reduce the computational time by several days. If you are looking to work with something that contributes to optics. This is it.

There are two integrals. Both are improper integrals with the integration range -Infinity to +Infinity. Suppose the integration variable is x. Both integrands are

1. Cot(d*Sqrt(k^2-x^2))*[k/Sqrt(k^2-x^2)]*Exp(i*x*c),
where d,k, and c are real number. "i" denotes the Sqrt(-1).

2. Csc(d*Sqrt(k^2-x^2))*[k/Sqrt(k^2-x^2)]*Exp(i*x*c)

Note that Cot(A)=Cos(A)/Sin(A) and Csc(A)=1/Sin(A).
The two integrands are integrated with respect to x from x=-Infinity to x=+infinity.

Anyone can find these integrals?
If you need more detail where these integrals come from, please contact me via p_epsilon@yahoo.com
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July 14th, 2016, 07:30 AM   #2
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This isn't a rare case.

Many special functions were defined in order to have a closed form for some integrals which cannot be expressed with a combination of functions already known.

Suppose that a function be defined by an integral and an original name given to this new function, if this new function is often used in practice and often cited in the literature, this function becomes well known. This function becomes a standard function in the list of the "special functions".

About special functions, a paper for general public : https://fr.scribd.com/doc/14623310/S...ions-speciales

If an integral is so important in optical calculus, why not giving to it an original name, gathering a lot of properties already published, studying and describing more properties and publishing papers about this function.

What has to be done in order to make this function a referenced and useful special
function ?

Quote from https://fr.scribd.com/doc/34977341/S...Dream-Function , page 3 :
'' A special function has to acquire a background of property, descriptions,
formulas and derivations as extended as possible. So, it will be preferable to simply refer to a particular part of the background, instead of searching and redoing a development by
ourselves. Before becoming a referenced special function, its name has to be spread in the
literature in order to become familiar. More importantly, the function should be useful in a
branch of mathematics or physics.''
Thanks from Country Boy
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September 1st, 2016, 12:50 PM   #3
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Math Focus: Theory of analytic functions
Hey JJ,
The paper you shared is in French. Any English versions ?
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