Functions of a complex variable Basic notions Concept
Complex plane
Summary
The complex plane is the plane in which the point represents the complex number . The horizontal axis is the real axis and the vertical axis is the imaginary axis. In this picture, addition of complex numbers is addition of vectors. The modulus is the distance from to , and the argument of is the angle from the positive real axis to .
Remarks
Multiplication by a complex number scales the plane by and turns it by the argument of . The complex conjugate is the reflection in the real axis.
Where this leads
81 pages build on this one.
- Residue at a simple pole
- Cauchy's integral theorem
- Riemann mapping theorem
- Residue theorem
- Open mapping theorem
- Uniform limits of holomorphic functions are holomorphic
- Maximum modulus principle
- Cross-ratio
- Existence of Laurent expansions on an annulus
- Domain in the complex plane
- Annulus
- Möbius transformation
- Riemann sphere
- Residue
- Schwarz lemma
- Principal part
- Holomorphic functions with nonzero derivative are conformal
- Hurwitz's theorem
- Möbius transformations map circles and lines to circles and lines
- Conformal map
- Cayley transform
- Identity theorem
- Arakelyan's theorem
- Phragmén–Lindelöf principle for subharmonic functions
- Joukowsky transform
- Schwarzian derivative
- Lyusternik's theorem on tangent spaces
- Hardy's uncertainty principle
- Schwarz–Christoffel mapping
- Uniformization theorem
- High indices theorem
- Stein interpolation theorem
- Kobayashi metric
- Isoperimetric problem of the calculus of variations
- Gamma function as Mellin transform of the exponential
- Fundamental recurrence formulas
- Splitting circle method
- Identity theorem in several variables
- Catenary problem
- Maximum modulus principle in several variables
- Carathéodory's theorem on boundary extension
- Green's function and the Riemann map
- Blaschke product
- Euler's elastica
- Brennan conjecture
- Hadamard gap theorem
- Conformal radius
- Harmonic measure
- Borel–Carathéodory theorem
- Dido's problem
- Phragmén–Lindelöf principle
- Kramers–Kronig relations
- Second main theorem of Nevanlinna theory
- Nevanlinna theory
- Conformal modulus
- Śleszyński–Pringsheim theorem
- Carathéodory metric
- Schwarz reflection principle
- Nörlund–Rice integral
- Littlewood subordination theorem
- Lagrange multipliers on Banach spaces
- Ahlfors theory
- Liouville's theorems on elliptic functions
- Stereographic projection
- Hadamard three-circle theorem for subharmonic functions
- Lindelöf's maximum principle
- Hadamard three-circle theorem
- Schwarz–Pick theorem
- Natural boundary
- Hadamard three-lines theorem
- Analytic capacity
- Lindelöf's theorem
- Branch point
- Hadamard's two-constants theorem
- Schramm–Loewner evolution
- Automorphism group of a bounded domain is a Lie group
- Loewner differential equation
- Euler integral for the gamma function
- Faber polynomials
- Montel's theorem
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Complex_plane, lead section
- Web https://en.wikipedia.org/wiki/Complex_plane, Argand diagram
About this page
| Tag | 0CTE7A31XQ |
|---|---|
| URI | m2://0CTE7A31XQ |
| Aliases | argand-diagram, argand-plane, gauss-plane |
| Kind | concept |
| Writing | W1 stated |
| MSC | 30A99 |
| Same as |
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What this page needs for W2
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- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
- edge specializes both: the more general notions and the special cases (0 of 1)
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