My Math Forum  

Go Back   My Math Forum > College Math Forum > Complex Analysis

Complex Analysis Complex Analysis Math Forum

Thanks Tree1Thanks
  • 1 Post By JJacquelin
LinkBack Thread Tools Display Modes
July 13th, 2016, 01:26 AM   #1
Joined: Jul 2016
From: Khonkaen, Thailand

Posts: 1
Thanks: 0

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
Piyawath is offline  
July 14th, 2016, 06:30 AM   #2
Senior Member
Joined: Aug 2011

Posts: 334
Thanks: 8

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 :

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 , 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
JJacquelin is offline  
September 1st, 2016, 11:50 AM   #3
Math Team
zaidalyafey's Avatar
Joined: Aug 2012
From: Sana'a , Yemen

Posts: 1,177
Thanks: 44

Math Focus: Theory of analytic functions
Hey JJ,
The paper you shared is in French. Any English versions ?
zaidalyafey is offline  

  My Math Forum > College Math Forum > Complex Analysis

important, integrals, optics, solved

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
A Question about Maclaurin Series Applied in Optics Channeltsui Calculus 0 December 9th, 2012 10:25 AM
Some definite integrals (solved) Rejjy Calculus 20 September 22nd, 2012 11:06 AM
Important axelle Algebra 1 October 28th, 2007 02:07 AM
important sos... sastra81 Abstract Algebra 3 December 26th, 2006 03:04 AM

Copyright © 2018 My Math Forum. All rights reserved.