theorem 8.6 Cauchy–Riemann equations

open in the book · parts/02-mathematical-methods/06-complex-analysis.tex:124 · p. 266

Rests on

depends_on definition 7.98 Functions of class $C^{1}$ ¶
- 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 definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on definition 7.97 Partial derivative; gradient ¶
  - 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 definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
depends_on equation 8.4 eq:cpx-derivative ¶
proves proof ch:06-complex-analysis@proof-3 ¶

Supports

depends_on definition A.734 The Joukowski map ¶
- depends_on lemma A.735 Transfer of the flow ¶
  - depends_on theorem A.737 The lift-curve slope of the flat plate ¶
depends_on example 8.7 ex:cpx-zbar ¶
depends_on lemma A.722 The basic lattice sum ¶
depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
- depends_on lemma A.607 A zero-free entire function is an exponential ¶
  - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on proposition A.611 The constraint on $\Gamma$ ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
depends_on lemma A.607 A zero-free entire function is an exponential ¶ ↺
depends_on lemma A.599 One variable, complex coefficient ¶
- depends_on lemma A.601 Several variables, complex symmetric matrix ¶
  - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
  - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
      - depends_on proposition A.609 The exponent is a quadratic polynomial ¶ ↺
depends_on lemma 31.67 Blasius' force formula ¶
- depends_on theorem 31.68 Kutta–Joukowski ¶
  - depends_on proposition 31.70 The lift-curve slope of a thin aerofoil ¶
  - depends_on remark 31.69 The Kutta condition, and where the circulation comes from ¶
  - depends_on theorem A.737 The lift-curve slope of the flat plate ¶ ↺
depends_on theorem A.783 Riemann mapping with boundary correspondence, quoted ¶
- depends_on corollary A.788 The two parameters are the same parameter ¶
  - depends_on theorem A.789 Kirchhoff's contraction coefficient ¶

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`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/06-complex-analysis.tex:135
`depends_on`	→	eq:cpx-derivative	declared	parts/02-mathematical-methods/06-complex-analysis.tex:135
`depends_on`	←	The Joukowski map	declared	appendices/A-long-proofs.tex:35667
`depends_on`	←	ex:cpx-zbar	declared	parts/02-mathematical-methods/06-complex-analysis.tex:173
`depends_on`	←	The basic lattice sum	declared	appendices/A-long-proofs.tex:34992
`depends_on`	←	$F$ is entire, and is the overlap in disguise	declared	appendices/A-long-proofs.tex:29014
`depends_on`	←	A zero-free entire function is an exponential	declared	appendices/A-long-proofs.tex:29067
`depends_on`	←	One variable, complex coefficient	declared	appendices/A-long-proofs.tex:28699
`depends_on`	←	Blasius' force formula	declared	parts/03-classical-mechanics/14-fluid-dynamics.tex:2114
`depends_on`	←	Riemann mapping with boundary correspondence, quoted	declared	appendices/A-long-proofs.tex:38426
`proves`	←	ch:06-complex-analysis@proof-3	declared	parts/02-mathematical-methods/06-complex-analysis.tex:138