# Kramers-Kronig relations on a finite data set

1. Aug 26, 2013

### Niles

Hi

Say I have a finite data set (frequency, absorption) and I would like to find the corresponding dispersion. For this I could use the Kramers-Kronig (KK) relation on the absorption data. What I would do is to make a qubic spline and then perform the KK-transformation.

However, the absorption data naturally doesn't run from ±∞, but what I would do is simply to use the extremes of my frequency-data instead - this will naturally introduce some numerical error. What do professional people do in this case, do they quantify the error? Or is there not a way to extract the dispersion from the absorption data?

2. Aug 26, 2013

### Zarqon

Hmm, if you have a data set, you probaby want to calculate the Kramers-Kroning integrals numerically. I wouldn't recommend first building a spline because they are terribly inaccurate outside of the range where you have data points, and using that could lead to very uncontrolled errors.

3. Aug 26, 2013

### DrDu

I have no idea what professional packages do, but have some general information on how the dielectric constant should behave asymptotically. Namely ε-1 should fall off like 1/ω at very high frequencies and should go to a constant in the limit ω→0. There are also lots of sum rules which provide further information on the relevant constants as far as you cannot infer them from your data.

4. Aug 26, 2013

### Niles

I only integrate (numerically!) from the first and last frequency data point, so I never go outside the range.

Thanks. They behave as anticipated, but I'm worried about the precision.

5. Aug 27, 2013

### DrDu

I meant that you could integrate over the corresponding asymptotic expressions in the range where you don't have data.