lemma 8.9 ML estimate

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:194 · p. 267

Rests on

depends_on equation 8.6 eq:cpx-contour-integral ¶
proves proof ch:06-complex-analysis@proof-4 ¶

Supports

depends_on example 8.25 A real integral by residues ¶
depends_on lemma 17.70 Jordan's lemma on the Bromwich contour ¶
- 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 106.2 Gaussian integrals ¶
- 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 106.27 The free generating functional and the Feynman propagator ¶
- depends_on theorem 106.54 The gauge-fixed action ¶
  - depends_on proposition 106.55 Ghosts are required by unitarity ¶
depends_on proposition 17.96 Poles give asymptotics ¶
depends_on proposition 106.14 The Feynman contour permits the rotation ¶
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 definition A.724 The staggered street ¶
      - depends_on proposition A.725 The street translates rigidly ¶
    - 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 theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on lemma 15.12 Artin decomposition ¶
      - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
- 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 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 theorem 9.26 Structure of the solutions ¶
  - depends_on theorem 8.24 Residue theorem ¶ ↺
depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶ ↺
depends_on theorem 8.13 Goursat ¶
depends_on theorem 8.20 Taylor expansion ¶
- depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶ ↺
- 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 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 ¶

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-contour-integral	declared	parts/02-mathematical-methods/06-complex-analysis.tex:198
`depends_on`	←	A real integral by residues	declared	parts/02-mathematical-methods/06-complex-analysis.tex:647
`depends_on`	←	Jordan's lemma on the Bromwich contour	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2548
`depends_on`	←	Gaussian integrals	declared	parts/11-qft-standard-model/08-path-integral-quantization.tex:245
`depends_on`	←	Poles give asymptotics	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:3320
`depends_on`	←	The Feynman contour permits the rotation	declared	parts/11-qft-standard-model/08-path-integral-quantization.tex:867
`depends_on`	←	Cauchy integral formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:409
`depends_on`	←	Derivatives of all orders; Cauchy estimates	declared	parts/02-mathematical-methods/06-complex-analysis.tex:445
`depends_on`	←	Goursat	declared	parts/02-mathematical-methods/06-complex-analysis.tex:301
`depends_on`	←	Taylor expansion	declared	parts/02-mathematical-methods/06-complex-analysis.tex:505
`depends_on`	←	Causality implies dispersion relations	declared	parts/02-mathematical-methods/15-fourier-integral-transforms.tex:2772
`proves`	←	ch:06-complex-analysis@proof-4	declared	parts/02-mathematical-methods/06-complex-analysis.tex:201