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By neglecting higher terms of ξ, Eq. (3) or (4) is rewritten as follows: {displaystyle {int}_D^{D+ hleft( xhboxright)}frac{zhbox{{left( x-{x}^{prime}right)}^2+ z{hbox^2}} d{z}^{prime}approx frac{ D hleft({x}^{prime}right)}{{left( x-{x}^{prime}right)}^2+{D}^2} (6).
The result is caused by the neglect of the higher terms of (1−e 2), and these higher (1−e 2) terms give (dot q/q) a positive value but (dot q/q) is much smaller than (|dot a|/a) and (|dot e|/e).
In our experience, employing higher terms of Cauchy's equation gives the fitted curve a freedom to follow the noise in the data and even to develop a minimum, which does not correspond to a theoretically valid dispersion function.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com