lemma 100.28 The $d$-dimensional loop integral

open in the book · parts/11-qft-standard-model/02-qed-renormalization.tex:1396 · p. 1915

Rests on

depends_on equation 17.42 eq:ft-gaussian ¶
depends_on equation 17.109 eq:ft-mellin-gamma ¶
depends_on lemma 100.27 Wick rotation ¶
- depends_on equation 100.23 eq:qed-fermion-propagator ¶
- 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 definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.9 Compact set ¶
    - depends_on remark 7.128 What the derivations below take as given ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
      - 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 theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:05-real-analysis@proof-77 ¶
  - proves proof ch:06-complex-analysis@proof-7 ¶
- proves proof ch:02-qed-renormalization@proof-16 ¶
proves proof ch:02-qed-renormalization@proof-17 ¶

Supports

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 ¶

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:ft-gaussian	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1427
`depends_on`	→	eq:ft-mellin-gamma	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1427
`depends_on`	→	Wick rotation	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1427
`depends_on`	←	One-loop effective potential	declared	parts/11-qft-standard-model/08-path-integral-quantization.tex:2208
`depends_on`	←	One-loop electron self-energy	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1803
`depends_on`	←	One-loop vacuum polarization	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1552
`proves`	←	ch:02-qed-renormalization@proof-17	declared	parts/11-qft-standard-model/02-qed-renormalization.tex:1430