Hi,
Thanks all for the replies.
Context:
I have performed an experiment in which I have measured the complex part of the KK relations, and wish to use it to calculate the real components.
Sample of dataset:
Wavelength, µm n k 0.5 2.693 0.0000121 0.52 2.65 0000043 0.54 2.62 0.0000015 0.56 2.596 8.00E-07 0.58 2.578 6.00E-07 0.6 2.56 5.00E-07 0.62 2.548 4.00E-07 0.64 2.537 4.00E-07 0.66 2.528 4.00E-07 0.68 2.521 3.00E-07
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I've read some stuff which makes me believe that dw' is supposed to mean "take the integral, and change the sign", kind of like adding the signum function on to the answer of the integral, but I don't know if it applies in this case?
In my specific case, I wanted to work out the X_1 component to give me an answer for the real component, and then validate using the second one, but I can't see how to apply it to the dataset in front of me. In the above table, X_1 is the n column and X_2 is the k column. This is from data where the 2 components are known, so I can tell if I have the right answer. Imagine at this point that the n column does not exist.
(apologies, yes it is principal, not principle - put it down to frazzled brain
)
Regarding the Cauchy Principal Value - I know I need to calculate the residue around some point Z_0, but I don't have a function so to speak - I have just a dataset. I could do a Laurent series expansion around one point on the dataset (wavelength), setting the point under consideration as Z_0, and all the other points as Z and then using that to work out the CPV? Is that a valid approach? I don't know what the singularity would be in the dataset given that I'm supposed to be able to apply this across the wavelengths and get an answer.
The w I take to represent the k point from my dataset corresponding to the wavelength under consideration (is this valid?), so would w' be taken as the tangent to that point, worked out by looking at the 2 points either side, fitting a quadratic to it, and taking the tangent of this?
I know this is a bit heavy going, so I appreciate the effort for any assistance you can give.
(I'm using MATLAB to perform my analysis, so I'm not over worried about the tedium in working out)
Thanks