By Steven Dale Cutkosky

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**Additional resources for An Introduction to Galois Theory [Lecture notes]**

**Example text**

Then, the subset of eigenvectors of B with nonzero eigenvalues form a basis for the image X of <6 under B. Consequently, the components of α lying in X can be eliminated by a proper choice of Γ . Then, the component X c of <6 , which does not lie in X and represents a subspace complementary to X, is spanned by the eigenvectors of B corresponding to zero or near-zero eigenvalues. Consequently, the resonance and near-resonance terms are in the subspace of <6 spanned by the eigenvectors of B corresponding to zero or near-zero eigenvalues.

However, the algebra involved in attacking the complex-valued equation is much less than that involved in attacking the two realvalued equations. 78) where T0 D t and T1 D t. 82) where A is an arbitrary function of T1 to this order; it is determined by eliminating the secular terms from ζ1 . 59). 83) P because D1 A D A. 51). 88) where cc stands for the complex conjugate of the preceding terms. 89) N i T0 i D1 Ae i T0 C 3α 5 C α 7 C i(α 6 C 3α 8 ) A2 Ae C cc C NST . 59). 83). Again, the algebra involved in attacking the Cartesian real form of the problem is more than that involved in attacking the complex-valued form of the problem.

96), there are no resonance terms. If we are in doubt, we seek a function h 1 that can be used to eliminate all of the perturbation terms. If we are successful in ﬁnding a smooth function h 1 that eliminates all of the perturbation terms, then g 1 D 0. , singular and near-singular terms) in h 1 . 69). Because the obtained Γm are regular, there are no resonance terms and g 1 D 0. 100) and choose g 2 to eliminate all of the resonance and near-resonance terms. Again, because η / e i ω t , only the term proportional to η 2 ηN is a resonance term.