definition 7.62 Real powers

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1428 · p. 224

ground object -- no derivation owed

Rests on

depends_on proposition 7.61 The logarithm ¶
- 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 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 equation 7.42 eq:ana-exp-derivative ¶
- depends_on lemma 7.60 Functional equation of the exponential ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on definition 7.59 The real exponential ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.4 Convergence ¶
    - depends_on proposition 7.48 Ratio test ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on proposition 7.46 Geometric series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-29 ¶
    - 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 ¶
  - 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 ¶
  - proves proof ch:05-real-analysis@proof-38 ¶
- depends_on proposition 7.32 Derivative of the inverse function ¶
  - depends_on definition 7.14 Inverse function ¶
    - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.47 Bijective map ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - 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 equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - proves proof ch:05-real-analysis@proof-8 ¶
  - proves proof ch:05-real-analysis@proof-16 ¶
- 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 ¶ ↺
  - proves proof ch:05-real-analysis@proof-11 ¶
- depends_on theorem 7.23 Intermediate value theorem ¶ ↺
- proves proof ch:05-real-analysis@proof-39 ¶

Supports

depends_on definition 7.90 The golden ratio ¶
- depends_on proposition 7.91 The golden ratio ¶
  - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
    - depends_on remark 7.95 Where $\varphi$ is used ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on remark 125.3 What the theorem does and does not forbid ¶
  - depends_on remark 7.95 Where $\varphi$ is used ¶ ↺
  - depends_on remark 7.94 Three numbers, two kinds ¶
depends_on proposition 7.63 Laws of real powers ¶
- depends_on definition 7.90 The golden ratio ¶ ↺
- depends_on lemma 7.67 The limit over real exponents ¶
  - depends_on proposition 7.68 The natural base ¶
    - depends_on remark 7.70 What the condition says ¶
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.91 The golden ratio ¶ ↺
- depends_on proposition 7.89 Irrationality of square roots ¶
  - depends_on proposition 7.91 The golden ratio ¶ ↺
depends_on proposition 7.89 Irrationality of square roots ¶ ↺

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`	→	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1433
`depends_on`	←	The golden ratio	declared	parts/02-mathematical-methods/05-real-analysis.tex:2681
`depends_on`	←	Laws of real powers	declared	parts/02-mathematical-methods/05-real-analysis.tex:1449
`depends_on`	←	Irrationality of square roots	declared	parts/02-mathematical-methods/05-real-analysis.tex:2649