theorem 8.20 Taylor expansion

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:498 · p. 270

Rests on

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 equation 8.6 eq:cpx-contour-integral ¶
- proves proof ch:06-complex-analysis@proof-4 ¶
depends_on proposition 7.46 Geometric series ¶
- depends_on corollary A.46 Monotone convergence ¶
  - depends_on theorem A.40 Least-upper-bound property ¶
    - depends_on definition A.36 Cut ¶
    - depends_on definition A.38 Order on $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - proves proof app:A-long-proofs@proof-29 ¶
  - proves proof app:A-long-proofs@proof-33 ¶
- depends_on definition 7.45 Series ¶
  - depends_on definition 7.2 Absolute value ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
- proves proof ch:05-real-analysis@proof-27 ¶
proves proof ch:06-complex-analysis@proof-14 ¶

Supports

depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
- depends_on proposition A.609 The exponent is a quadratic polynomial ¶
  - depends_on proposition A.611 The constraint on $\Gamma$ ¶
    - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
depends_on proposition 8.23 Residue at a simple pole ¶
depends_on theorem A.598 Hudson ¶
depends_on theorem 8.21 Laurent expansion ¶
- depends_on definition 8.22 Isolated singularities; residue ¶
  - depends_on proposition 8.23 Residue at a simple pole ¶ ↺
  - depends_on proposition 17.96 Poles give asymptotics ¶
  - depends_on theorem 8.24 Residue theorem ¶
    - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on corollary 17.76 Heaviside's expansion theorem ¶
      - depends_on proposition 17.77 Poles and stability ¶
    - depends_on example 8.25 A real integral by residues ¶
    - depends_on proposition 17.96 Poles give asymptotics ¶ ↺
    - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
    - depends_on theorem 31.68 Kutta–Joukowski ¶
      - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
      - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
      - depends_on theorem A.737 The lift-curve slope of the flat plate ¶
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶
- depends_on theorem 8.24 Residue theorem ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:cpx-cif	declared	parts/02-mathematical-methods/06-complex-analysis.tex:505
`depends_on`	→	eq:cpx-derivative-formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:505
`depends_on`	→	ML estimate	declared	parts/02-mathematical-methods/06-complex-analysis.tex:505
`depends_on`	→	Geometric series	declared	parts/02-mathematical-methods/06-complex-analysis.tex:505
`depends_on`	←	An entire function with a quadratic bound on its real part	declared	appendices/A-long-proofs.tex:29104
`depends_on`	←	Residue at a simple pole	declared	parts/02-mathematical-methods/06-complex-analysis.tex:589
`depends_on`	←	Hudson	declared	appendices/A-long-proofs.tex:28684
`depends_on`	←	Laurent expansion	declared	parts/02-mathematical-methods/06-complex-analysis.tex:539
`proves`	←	ch:06-complex-analysis@proof-14	declared	parts/02-mathematical-methods/06-complex-analysis.tex:508