proposition 8.23 Residue at a simple pole

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:583 · p. 271

Rests on

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 ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - 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 definition 6.17 Simply connected space ¶
      - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 6.15 Path-connected space ¶
    - depends_on theorem 8.12 Cauchy ¶
      - depends_on equation 8.5 eq:cpx-cr ¶
      - depends_on theorem 7.131 Green ¶
      - proves proof ch:06-complex-analysis@proof-7 ¶
    - proves proof ch:06-complex-analysis@proof-9 ¶
  - 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 ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶ ↺
      - proves proof ch:06-complex-analysis@proof-4 ¶
    - 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 ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - proves proof ch:06-complex-analysis@proof-14 ¶
  - proves proof ch:06-complex-analysis@proof-15 ¶
depends_on theorem 8.20 Taylor expansion ¶ ↺
proves proof ch:06-complex-analysis@proof-16 ¶

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type	direction	node	provenance	where
`depends_on`	→	Isolated singularities; residue	declared	parts/02-mathematical-methods/06-complex-analysis.tex:589
`depends_on`	→	Taylor expansion	declared	parts/02-mathematical-methods/06-complex-analysis.tex:589
`proves`	←	ch:06-complex-analysis@proof-16	declared	parts/02-mathematical-methods/06-complex-analysis.tex:592