proposition 9.43 The unmodulated line, and the feet of the tongues

Rests on

• depends_on corollary 9.41 Hill's equation ¶
  • depends_on corollary 9.27 Linear stability ¶
    • depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.14 Linear independence ¶
      • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on definition 5.56 Endomorphism ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶
      • depends_on theorem 9.23 Solution of a constant-coefficient system ¶
        • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
        • depends_on proposition 9.22 The exponential and its derivative ¶
        • proves proof ch:07-odes-sturm-liouville@proof-12 ¶
      • proves proof ch:07-odes-sturm-liouville@proof-13 ¶
    • depends_on theorem 9.26 Structure of the solutions ¶
      • depends_on theorem 8.24 Residue theorem ¶
        • depends_on corollary 8.15 Deformation of contours ¶
        • depends_on definition 8.22 Isolated singularities; residue ¶
        • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
        • depends_on theorem 8.21 Laurent expansion ¶
        • proves proof ch:06-complex-analysis@proof-17 ¶
      • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
        • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
        • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • proves proof ch:03-linear-algebra-representations@proof-26 ¶
      • depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
      • proves proof ch:07-odes-sturm-liouville@proof-14 ¶
    • proves proof ch:07-odes-sturm-liouville@proof-15 ¶
  • depends_on proposition 9.38 Liouville's formula ¶
    • depends_on definition 9.37 Fundamental matrix ¶
      • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶ ↺
      • depends_on theorem 9.13 Dimension of the solution space ¶
        • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶ ↺
        • depends_on definition 9.2 Linear equation; homogeneity ¶
        • proves proof ch:07-odes-sturm-liouville@proof-6 ¶
    • depends_on proposition 9.15 Abel's identity ¶
      • depends_on definition 9.2 Linear equation; homogeneity ¶ ↺
      • depends_on definition 9.14 Wronskian ¶
      • proves proof ch:07-odes-sturm-liouville@proof-7 ¶
    • proves proof ch:07-odes-sturm-liouville@proof-20 ¶
  • depends_on theorem 9.39 Floquet ¶
    • depends_on corollary 9.27 Linear stability ¶ ↺
    • depends_on definition 9.37 Fundamental matrix ¶ ↺
    • depends_on proposition 9.38 Liouville's formula ¶ ↺
    • proves proof ch:07-odes-sturm-liouville@proof-21 ¶
  • proves proof ch:07-odes-sturm-liouville@proof-22 ¶
• depends_on equation 9.42 eq:slt-mathieu ¶
• depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
• proves proof ch:07-odes-sturm-liouville@proof-23 ¶

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