July 3rd, 2017, 05:52 PM  #1 
Newbie Joined: Jul 2017 From: US Posts: 12 Thanks: 0  Publishers
Hello, I recently finetuned a mathematical method that creates representations of the integrals of power functions with a tight degree of accuracy around 1% off of the real value at the point of max deviation. I had three questions that you would hopefully be able to answer for me: 1.) Do you believe that this would be significant enough of a finding to get published? 2.) If yes, where should I submit it to? I unfortunately finished it a year after finishing college, but I started working on it in my freshman year. 3.) Where would I get a detailed review of the work? (I am pretty sure many people would rather not read a 50 page paper on a method that may or may not be significant.) Last edited by skipjack; July 4th, 2017 at 08:23 AM. 
July 4th, 2017, 08:06 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,476 Thanks: 495 Math Focus: Yet to find out. 
A quadrature method for approximating power functions?

July 5th, 2017, 04:22 PM  #3 
Newbie Joined: Jul 2017 From: US Posts: 12 Thanks: 0  Nope, what it is, is what I call a mirror function. A mirror function is a function that behaves exactly like or nearly like the function it is based off of. A simple example of this is: ((x5)/2)(sqrt(x^2)/2)+(7/2) which is a mirror of 1. It is pretty much a simple form of Calculus of Variations. https://en.wikipedia.org/wiki/Calculus_of_variations 
July 5th, 2017, 05:56 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,681 Thanks: 437  
July 5th, 2017, 06:07 PM  #5  
Newbie Joined: Jul 2017 From: US Posts: 12 Thanks: 0  Quote:
I am just showing simple functions can be put together to represent anything from something as simple as a constant such as 1 or as complex as the integral of a power function.  
July 16th, 2017, 03:27 PM  #6 
Senior Member Joined: Oct 2009 Posts: 189 Thanks: 74 
Do you mean integrals like $\int x^p dx$? Sorry if I fail to understand, but don't we already know exactly what these integrals are?

July 17th, 2017, 04:56 PM  #7 
Newbie Joined: Jul 2017 From: US Posts: 12 Thanks: 0  Yep, except it is for functions like exp(x^2), x^x, etc, which currently do not have such a representation.
Last edited by Tkipp; July 17th, 2017 at 05:00 PM. 
July 17th, 2017, 05:01 PM  #8 
Senior Member Joined: Oct 2009 Posts: 189 Thanks: 74  OK nice. It has been rigorously proven that these integrals cannot be found analytically in terms of elementary functions. However, we do already know how to approximate these integrals to any degree of accuracy. So if you want to publish this, then you'll need to show somehow that your method is better than the currently known ones. Better could mean that it is more numerically stable, or it is faster, or... If you could show this, then yes, people would be very interested in this result. You don't even have to show it mathematically, doing a simulation study where your technique beats the competition would already be quite good! 
July 17th, 2017, 05:45 PM  #9 
Newbie Joined: Jul 2017 From: US Posts: 12 Thanks: 0  Here is a run I took with exp(x^2). The first is the integral values, the second the derivative of my function divided by exp(x^2) minus 1. As you can see it is fairly accurate, and these graphs capture the max error deviation, which is centered around x=1, 1. As the approaches both positive and negative infinity the error ratio goes to zero. 
July 17th, 2017, 05:49 PM  #10  
Senior Member Joined: Oct 2009 Posts: 189 Thanks: 74  Quote:
So if your method doesn't give the correct answer, what is its benefit? Is it faster? More stable?  