example 8.25 A real integral by residues

Rests on

• depends_on equation 8.16 eq:cpx-simple-pole ¶
• depends_on lemma 8.9 ML estimate ¶
  • depends_on equation 8.6 eq:cpx-contour-integral ¶
  • proves proof ch:06-complex-analysis@proof-4 ¶
• depends_on theorem 8.24 Residue theorem ¶
  • depends_on corollary 8.15 Deformation of contours ¶
    • depends_on definition 6.17 Simply connected space ¶
      • depends_on definition 6.6 Continuous map ¶
        • depends_on definition 3.53 Preimage ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
      • depends_on definition 6.15 Path-connected space ¶
        • depends_on definition 6.6 Continuous map ¶ ↺
    • depends_on theorem 8.12 Cauchy ¶
      • depends_on equation 8.5 eq:cpx-cr ¶
      • depends_on theorem 7.131 Green ¶
        • depends_on definition 7.127 Simple regions ¶
        • depends_on remark 7.128 What the derivations below take as given ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • proves proof ch:05-real-analysis@proof-77 ¶
      • proves proof ch:06-complex-analysis@proof-7 ¶
    • proves proof ch:06-complex-analysis@proof-9 ¶
  • depends_on definition 8.22 Isolated singularities; residue ¶
    • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      • depends_on equation 8.6 eq:cpx-contour-integral ¶ ↺
      • depends_on proposition 8.4 Euler's formula ¶
        • depends_on equation 8.2 eq:cpx-exp-series ¶
        • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • proves proof ch:06-complex-analysis@proof-2 ¶
      • proves proof ch:06-complex-analysis@proof-6 ¶
    • depends_on theorem 8.21 Laurent expansion ¶
      • depends_on corollary 8.15 Deformation of contours ¶ ↺
      • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
      • depends_on theorem 8.16 Cauchy integral formula ¶
        • depends_on corollary 8.15 Deformation of contours ¶ ↺
        • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
        • depends_on lemma 8.9 ML estimate ¶ ↺
        • proves proof ch:06-complex-analysis@proof-10 ¶
      • depends_on theorem 8.20 Taylor expansion ¶
        • depends_on equation 8.12 eq:cpx-cif ¶
        • depends_on equation 8.13 eq:cpx-derivative-formula ¶
        • depends_on lemma 8.9 ML estimate ¶ ↺
        • depends_on proposition 7.46 Geometric series ¶
        • proves proof ch:06-complex-analysis@proof-14 ¶
      • proves proof ch:06-complex-analysis@proof-15 ¶
  • 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 ¶

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