Abstract
This Appendix introduces and defines mathematical tools that are of particular usefulness in the analysis of eddy-current impedance signals. Complex numbers, certain trigonometric functions, important operators, and identities of vector analysis in Cartesian, cylindrical, and spherical coordinate systems are presented. Due to their importance in analysis of eddy-current probe coil fields, Bessel functions are defined and described in some detail.
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Notes
- 1.
The Bessel function of the first kind of order zero is regular, with the power series form
$$ J_0(z)= 1- \frac{x^2}{2^2}+\frac{x^4}{2^24^2}-... =\sum _{m=0}^{\infty }\frac{\left( -\frac{1}{4}z^2\right) ^m}{(m!)^2}. $$ - 2.
It can be shown that
$$ Y_0(z) =\frac{2}{\pi }\left\{ J_0(z)[\log \left( \frac{z}{2}\right) +\gamma ] +\sum _{k=1}^{\infty } (-1)^{k+1}H_k \frac{\left( \frac{1}{4}z^2\right) ^k}{(k!)^2}\right\} $$where \(\gamma \approx 0.577 215 665\) is Euler’s constant and
$$\begin{aligned} H_k=1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{k} \end{aligned}$$is a harmonic number.
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972)
Van Bladel, J.: Electromagnetic Fields. Hemisphere Publishing Corporation, Washington (1985)
Spiegel, M.R.: Schaum’s Outline Series Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis. McGraw-Hill Book Company, New York (1974)
Sadiku, M.N.O.: Elements of Electromagnetics, 4th edn. Oxford University Press, New York (2007)
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Bowler, N. (2019). Appendices. In: Eddy-Current Nondestructive Evaluation. Springer Series in Measurement Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9629-2_10
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DOI: https://doi.org/10.1007/978-1-4939-9629-2_10
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