proposition 7.77 Special values, periodicity, and the kernel

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2059 · p. 232

Rests on

depends_on definition 7.74 $\pi$ ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on lemma 7.73 $\cos 2 < 0$ ¶
  - depends_on equation 7.53 eq:ana-cos-sin-series ¶
  - depends_on proposition 7.52 Alternating series test ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - 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 ¶
    - proves proof ch:05-real-analysis@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-48 ¶
- depends_on theorem 7.23 Intermediate value theorem ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - proves proof ch:05-real-analysis@proof-8 ¶
- depends_on theorem 7.51 Termwise differentiation ¶
  - 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 ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶ ↺
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - 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 ¶
    - depends_on theorem 7.8 Cauchy criterion ¶ ↺
    - proves proof ch:05-real-analysis@proof-29 ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - depends_on theorem 7.50 Power series; radius of convergence ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶ ↺
    - proves proof ch:05-real-analysis@proof-31 ¶
  - proves proof ch:05-real-analysis@proof-32 ¶
depends_on equation 7.58 eq:ana-half-angle ¶
depends_on lemma 7.65 The integer part ¶
- depends_on axiom 3.33 Peano axioms ¶
- depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
  - depends_on definition A.36 Cut ¶
  - depends_on theorem A.40 Least-upper-bound property ¶ ↺
  - proves proof app:A-long-proofs@proof-32 ¶
- depends_on lemma 7.64 Well-ordering of $\N$ ¶
  - depends_on axiom 3.33 Peano axioms ¶ ↺
  - depends_on axiom 3.34 Principle of induction ¶
  - proves proof ch:05-real-analysis@proof-41 ¶
- proves proof ch:05-real-analysis@proof-42 ¶
depends_on lemma 7.75 The first quadrant ¶
- depends_on corollary 7.36 cor:ana-mvt-consequences ¶
  - depends_on theorem 7.35 Mean value theorem ¶
    - depends_on proposition 7.29 Linearity ¶
      - depends_on definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - proves proof ch:05-real-analysis@proof-20 ¶
- depends_on definition 7.74 $\pi$ ¶ ↺
- depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
  - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
  - depends_on equation 5.3 eq:lin-binomial ¶
  - depends_on proposition 7.49 Cauchy product ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-30 ¶
  - depends_on theorem 7.51 Termwise differentiation ¶ ↺
  - proves proof ch:05-real-analysis@proof-46 ¶
- depends_on theorem 7.23 Intermediate value theorem ¶ ↺
- depends_on theorem 7.51 Termwise differentiation ¶ ↺
- proves proof ch:05-real-analysis@proof-49 ¶
depends_on lemma 7.72 Addition theorems ¶
- depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
- depends_on equation 5.3 eq:lin-binomial ¶ ↺
- depends_on proposition 7.49 Cauchy product ¶ ↺
- depends_on proposition 7.48 Ratio test ¶ ↺
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-47 ¶
proves proof ch:05-real-analysis@proof-51 ¶

Supports

depends_on lemma 7.78 Chord, arc, tangent ¶
- depends_on proposition 7.79 The polygon recursion ¶
  - depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
  - depends_on proposition 7.83 Viète's product ¶
    - depends_on remark 7.84 One construction, two faces ¶
  - depends_on remark 7.82 What Archimedes proved ¶
  - depends_on remark 7.80 Why these are the polygon perimeters ¶
  - depends_on remark 7.84 One construction, two faces ¶ ↺
depends_on proposition 7.85 Irrationality of $\pi$ ¶
- depends_on remark 7.94 Three numbers, two kinds ¶
- depends_on remark 7.87 Priority, and what irrationality does not give ¶
depends_on proposition 7.79 The polygon recursion ¶ ↺
depends_on proposition 7.83 Viète's product ¶ ↺
depends_on proposition 125.2 Crystallographic restriction ¶
- depends_on remark 125.3 What the theorem does and does not forbid ¶
depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	$\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2077
`depends_on`	→	eq:ana-half-angle	declared	parts/02-mathematical-methods/05-real-analysis.tex:2077
`depends_on`	→	The integer part	declared	parts/02-mathematical-methods/05-real-analysis.tex:2077
`depends_on`	→	The first quadrant	declared	parts/02-mathematical-methods/05-real-analysis.tex:2077
`depends_on`	→	Addition theorems	declared	parts/02-mathematical-methods/05-real-analysis.tex:2077
`depends_on`	←	Chord, arc, tangent	declared	parts/02-mathematical-methods/05-real-analysis.tex:2150
`depends_on`	←	Irrationality of $\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	←	The polygon recursion	declared	parts/02-mathematical-methods/05-real-analysis.tex:2198
`depends_on`	←	Viète's product	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	←	Crystallographic restriction	declared	parts/12-statistical-mechanics/08-phonons-lattices.tex:127
`depends_on`	←	$\pi$ elsewhere in this treatise	declared	parts/02-mathematical-methods/05-real-analysis.tex:2623
`proves`	←	ch:05-real-analysis@proof-51	declared	parts/02-mathematical-methods/05-real-analysis.tex:2081