theorem 8.12 Cauchy

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:260 · p. 267

Rests on

depends_on equation 8.5 eq:cpx-cr ¶
depends_on theorem 7.131 Green ¶
- depends_on definition 7.127 Simple regions ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - depends_on definition 7.41 Antiderivative ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof ch:05-real-analysis@proof-77 ¶
proves proof ch:06-complex-analysis@proof-7 ¶

Supports

depends_on corollary 8.15 Deformation of contours ¶
- depends_on lemma 106.1 The short-time kernel ¶
  - depends_on theorem 106.3 Feynman's sum over histories ¶
    - depends_on phenomenon 106.5 The Aharonov–Bohm effect ¶
    - depends_on theorem 106.16 Feynman–Kac ¶
      - depends_on proposition 106.17 Euclidean time projects onto the ground state ¶
      - depends_on proposition 106.21 Reflection positivity is the positivity of the Hilbert space ¶
- depends_on theorem 8.16 Cauchy integral formula ¶
  - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
    - depends_on theorem 8.18 Liouville ¶
      - depends_on corollary A.723 Potential of one infinite row ¶
      - depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - 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 theorem 8.24 Residue theorem ¶ ↺
- depends_on theorem 8.21 Laurent expansion ¶ ↺
- 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 lemma 100.27 Wick rotation ¶
- depends_on lemma 100.28 The $d$-dimensional loop integral ¶
  - depends_on proposition 106.45 One-loop effective potential ¶
  - depends_on theorem 100.35 One-loop electron self-energy ¶
    - depends_on phenomenon 100.66 The Lamb shift ¶
  - depends_on theorem 100.31 One-loop vacuum polarization ¶
    - depends_on phenomenon 100.69 Light scatters off light ¶
depends_on proposition 106.14 The Feynman contour permits the rotation ¶
depends_on theorem 17.78 Causality implies dispersion relations ¶
- depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
- depends_on remark 28.26 Causality is visible in the susceptibility ¶
- depends_on remark 28.62 Neither velocity is a speed limit ¶
depends_on theorem 17.94 Mellin inversion ¶
- depends_on proposition 17.96 Poles give asymptotics ¶ ↺
- depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
  - depends_on corollary 17.101 Coupled channels ¶

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-cr	declared	parts/02-mathematical-methods/06-complex-analysis.tex:267
`depends_on`	→	Green	declared	parts/02-mathematical-methods/06-complex-analysis.tex:267
`depends_on`	←	Deformation of contours	declared	parts/02-mathematical-methods/06-complex-analysis.tex:388
`depends_on`	←	Residue evaluation and causality	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2578
`depends_on`	←	Wick rotation	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1373
`depends_on`	←	The Feynman contour permits the rotation	declared	parts/11-qft-standard-model/08-path-integral-quantization.tex:867
`depends_on`	←	Causality implies dispersion relations	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`depends_on`	←	Mellin inversion	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3258
`proves`	←	ch:06-complex-analysis@proof-7	declared	parts/02-mathematical-methods/06-complex-analysis.tex:270