Abstract
Complex phasors are reinterpreted as real operators in a real vector space of functions. This structure is linked with dilative rotations of the Euclidean plane. Finally we conclude with some methodological ideas related to the teaching of the complex numbers.
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Notes
- 1.
The methods to replace the operation of multiplication by a sum, called prosthaphaeresis, have a long history, even before the discovery of logarithms by Napier. See, for example, the article by Clavius in [5, p. 459].
- 2.
We have learned in the critical analysis by Spivak [6] that we need an analytical definition of circular functions.
- 3.
As customary, the linear combination of two linear operators \(u,v:{\mathscr {E}}\rightarrow {\mathscr {E}}\) is defined as \((\lambda u+\mu v)f:=\lambda (uf)+\mu (vf)\), with \(\lambda , \mu \in {\mathbb R}\).
- 4.
There is certain abuse of notation here; we would distinguish the operator \(j:{\mathscr {E}}\rightarrow {\mathscr {E}}\) from the operator \(j:{\mathscr {E}}_{\omega }\rightarrow {\mathscr {E}}_{\omega }\). This complication, that for each frequency we need a different operator j, makes the phasor technique useless when dealing with circular functions of several frequencies.
References
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Bobillo-Ares, N., Garzón, M.L. (2018). Phasors, Always in the Real World. In: Gil, E., Gil, E., Gil, J., Gil, M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-319-73848-2_75
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DOI: https://doi.org/10.1007/978-3-319-73848-2_75
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