definition 8.3 Exponential, sine, cosine

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:67 · p. 265

ground object -- no derivation owed

Rests on

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 definition 7.2 Absolute value ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
  - 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 A.786 From the half-strip to the half plane ¶
- depends_on corollary A.788 The two parameters are the same parameter ¶
  - depends_on theorem A.789 Kirchhoff's contraction coefficient ¶
- depends_on lemma A.787 From the potential strip to the half plane ¶
  - depends_on corollary A.788 The two parameters are the same parameter ¶ ↺
depends_on lemma A.787 From the potential strip to the half plane ¶ ↺
depends_on lemma A.785 The hodograph image ¶
- depends_on lemma A.786 From the half-strip to the half plane ¶ ↺

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`	→	Ratio test	declared	parts/02-mathematical-methods/06-complex-analysis.tex:77
`depends_on`	→	Power series; radius of convergence	declared	parts/02-mathematical-methods/06-complex-analysis.tex:77
`depends_on`	←	From the half-strip to the half plane	declared	appendices/A-long-proofs.tex:38603
`depends_on`	←	From the potential strip to the half plane	declared	appendices/A-long-proofs.tex:38667
`depends_on`	←	The hodograph image	declared	appendices/A-long-proofs.tex:38540