definition 7.59 The real exponential

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:1319 · p. 223

ground object -- no derivation owed

Rests on

depends_on definition 7.45 Series ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.4 Convergence ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
depends_on proposition 7.48 Ratio test ¶
- depends_on proposition 7.47 Comparison; absolute convergence ¶
  - 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 $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶ ↺
  - depends_on theorem 7.8 Cauchy criterion ¶
    - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on theorem A.40 Least-upper-bound property ¶ ↺
      - proves proof app:A-long-proofs@proof-34 ¶
    - 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 7.46 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 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 ¶

Supports

depends_on lemma 7.60 Functional equation of the exponential ¶
- depends_on proposition 7.68 The natural base ¶
  - depends_on remark 7.70 What the condition says ¶
- depends_on proposition 7.61 The logarithm ¶
  - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
    - depends_on remark 7.70 What the condition says ¶ ↺
  - depends_on definition 7.62 Real powers ¶
    - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.91 The golden ratio ¶
    - 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 proposition 7.91 The golden ratio ¶ ↺
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
  - depends_on lemma 7.67 The limit over real exponents ¶ ↺
  - depends_on proposition 7.68 The natural base ¶ ↺
  - depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on proposition 7.63 Laws of real powers ¶ ↺

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`	→	Series	declared	parts/02-mathematical-methods/05-real-analysis.tex:1328
`depends_on`	→	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:1328
`depends_on`	→	Power series; radius of convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:1328
`depends_on`	←	Functional equation of the exponential	declared	parts/02-mathematical-methods/05-real-analysis.tex:1361