About Integral inversion in igor.

I need to find the integral inversion for my data analysis.

For a given function

F(k)=∫p(x)g(k,x)dx, limit(0,∞)

Suppose g(k,x) is a known function. If we know F(k), how can we know p(x)? People have used the method of integral inversion to calculate it mathematically and in Fortran. Can we do in igor to calculate the integral inversion. We can define all those function as wave.??

Hello Rabindra Dulal,

You are dealing with an integral equation (looks like an inhomogeneous Fredholm integral equation of the first kind).  Your function g() is sometimes known as the "kernel".

Depending on the form of the kernel the approach may be different.  If you are lucky, the kernel is a complex exponential and then you are effectively dealing with a Fourier transform.  I assume that is not the case because it would be trivial for you to compute the inverse.

If this is an important enough topic for you I would recommend getting a very good book on the subject "Integral Equations" by F.G. Tricomi (available on Amazon for $8.99) and well worth the price IMO despite the fact that it is does not discuss numerical methods.

My experience with this was with a particular version of the kernel (Hilbert-Schmidt) which allowed me to express the kernel as a sum of functions via the Mercer expansion.  In practical terms you need to know how many terms you need to keep and that, again, depends on your particular form of the kernel function.

Feel free to contact me at support@wavemetrics.com if you want to discuss further details of the problem.



WaveMetrics, Inc.

Thanks for the reply.

If we move to real problem with a function

F(k)=∫p(x)g(k,x)dx, limit(0,∞)

Suppose g(k,x)=2*pi^2*(c)^4*exp(-2*x^2*pi^2*(c)^2)*(1+Besselj(0,2*pi*x*d)). We need to find p(x). How can we do it?

Thanks: Rabindra

Please note that in order to solve for p(x) you would also need an expression for F(k).  Also, your expression for g(k,x) appears to be independent of k and that makes me suggest that you carefully recheck all your parameters.