Functions of a complex variable Definition
Modulus of a complex number
Builds on Complex numbersReal and imaginary partsSquare root
Definition
Let be a complex number with real part and imaginary part . The modulus, or absolute value, of is the nonnegative real number
Remarks
For a real number , the modulus is , the absolute value of . In the complex plane, is the distance from to , by the Pythagorean theorem. The distance between and is . For example, .
Reading path
To understand this page from the ground up, read these 9 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Complex numbers object
- Real numbers object
- Set bounded above definition
- Real and imaginary parts definition
- Supremum definition
- Completeness axiom statement
- Existence of square roots of nonnegative reals statement
- Square root definition
- Modulus of a complex number definition
Where this leads
236 pages build on this one.
- Modulus of the conjugate
- Residue at a simple pole
- Derivative of the complex exponential
- Cauchy's integral theorem
- Arguments add under multiplication
- Holomorphic functions on a disc have primitives
- Casorati–Weierstrass theorem
- Riemann mapping theorem
- The real part of a holomorphic function is harmonic
- Residue theorem
- Formula for the modulus
- Open mapping theorem
- Entire function
- Uniform limits of holomorphic functions are holomorphic
- Maximum modulus principle
- The nth roots of unity
- Order of a zero
- Branch cut
- Existence of Laurent expansions on an annulus
- Holomorphic functions are analytic
- Picard's great theorem
- Domain in the complex plane
- Power series of the complex logarithm
- Annulus
- Goursat's theorem
- Polar form
- Riemann's theorem on removable singularities
- Harmonic conjugate
- Zero of a holomorphic function
- exp(2 pi i/n) is a primitive root of unity
- Complex differentiability
- Rouché's theorem
- Holomorphic function
- Cauchy's integral formula
- Number of nth roots of unity
- Liouville's theorem
- Cauchy's integral formula for derivatives
- Inverse of a complex number
- The complex exponential is entire
- Modulus is multiplicative
- Mean value property
- Residue
- Cauchy's estimates
- Fundamental theorem of algebra
- Isolated zeros theorem
- Quadratic formula
- Schwarz lemma
- exp(1/z) has an essential singularity at 0
- Principal part
- Meromorphic function
- Removable singularity
- Argument principle
- Holomorphic functions with nonzero derivative are conformal
- Holomorphic functions are infinitely differentiable
- Branch of the logarithm
- Principal value of the argument
- Hurwitz's theorem
- Cauchy–Riemann equations
- Isolated singularity
- Morera's theorem
- Pole
- Complex logarithm
- The principal logarithm is holomorphic off the branch cut
- Identity theorem
- Order of a pole
- Essential singularity
- Principal branch
- Existence of nth roots of complex numbers
- Power series are holomorphic in the disc of convergence
- Fundamental theorem of contour integration
- The complex conjugate is not holomorphic
- A complex number times its conjugate
- Complex power
- Argument of a complex number
- Function with no Laplace transform
- Weierstrass factorization theorem
- Univalent function
- Riesz projector
- Mahler measure
- Arakelyan's theorem
- Conformal invariance of harmonic functions
- Shell theorem
- Phragmén–Lindelöf principle for subharmonic functions
- Nevanlinna characteristic
- Cauchy–Pompeiu formula
- Weierstrass elliptic function
- Weierstrass zeta and sigma functions
- Joukowsky transform
- Cheng–Yau gradient estimate
- Schwarzian derivative
- Removable singularities for harmonic functions
- Minimum principle for harmonic functions
- Lyusternik's theorem on tangent spaces
- Durand–Kerner method
- Hardy's uncertainty principle
- Schwarz–Christoffel mapping
- Jensen's formula
- Teichmüller space
- Nevanlinna–Pick interpolation
- Modular lambda function
- Monodromy theorem
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Elliptic function
- Uniformization theorem
- Mergelyan's theorem
- High indices theorem
- Stein interpolation theorem
- Kobayashi metric
- Wirtinger derivatives
- Runge's theorem
- Carlson's theorem
- Isoperimetric problem of the calculus of variations
- Harmonic majorant
- Gamma function as Mellin transform of the exponential
- Analyticity of the Laplace transform in its half-plane
- Splitting circle method
- Identity theorem in several variables
- Log-modulus of a holomorphic function is subharmonic
- Holomorphic map between Riemann surfaces
- Cauchy-type integral
- Catenary problem
- Maximum modulus principle in several variables
- Weierstrass product
- Weyl's lemma
- Zeros and poles
- Bromwich integral
- Poisson–Jensen formula
- Carathéodory's theorem on boundary extension
- Smoothness and analyticity of harmonic functions
- Green's function and the Riemann map
- Band-limited function
- Landau's constants
- Sub-mean value property
- Blaschke product
- Borel–Ritt theorem
- Euler's elastica
- Kakutani's solution of the Dirichlet problem
- Brennan conjecture
- Legendre duplication formula
- Converse of the mean value property
- Local existence of harmonic conjugates
- Pólya's theorem on integer-valued entire functions
- Post's inversion formula
- Hadamard gap theorem
- Earnshaw's theorem
- Conformal radius
- Nehari's theorem on univalence
- Euler's sine product
- Nontangential limit
- Hartogs's theorem on separate holomorphicity
- Harmonic measure
- Superharmonic potential
- Borel–Carathéodory theorem
- Dido's problem
- Cauchy integral formula on a polydisc
- Phragmén–Lindelöf principle
- Strip of convergence of the bilateral transform
- Kramers–Kronig relations
- Second main theorem of Nevanlinna theory
- Nevanlinna theory
- Iterated Laplace transform equals Stieltjes transform
- Order of a zero or pole
- Wiman–Valiron theory
- Conformal modulus
- Koebe's theorem on the mean value property
- Bloch's theorem
- Hadamard factorization theorem
- Wiener–Hopf method
- Compactness of families of harmonic functions
- Resurgent function
- Gradient estimates for harmonic functions
- Osgood's lemma
- Carathéodory metric
- Brody hyperbolicity
- Schwarz reflection principle
- Laguerre–Pólya class
- Nörlund–Rice integral
- Littlewood subordination theorem
- Fatou's theorem
- Lagrange multipliers on Banach spaces
- Ahlfors theory
- Poles of the gamma function
- First main theorem of Nevanlinna theory
- Liouville's theorem for harmonic functions
- Cauchy–Riemann equations in several variables
- Normal family
- Picard's little theorem
- Liouville's theorems on elliptic functions
- Cauchy–Riemann operator
- Harmonic functions on Riemann surfaces
- Denjoy–Carleman–Ahlfors theorem
- Euler–Gauss limit formula
- Liouville's theorem for planar harmonic functions
- Maximum principle for harmonic functions
- Inner function
- Outer function
- Paley–Wiener theorem
- Nevanlinna class
- Hadamard three-circle theorem for subharmonic functions
- Lindelöf's maximum principle
- Hadamard three-circle theorem
- Schwarz–Pick theorem
- Differentiation in the transform variable
- Jury stability criterion
- Grunsky matrix
- Exponential type
- Gamma function has no zeros
- Harmonic measure as exit distribution
- Natural boundary
- Reflection formula for the gamma function
- Riemann surface
- Hadamard three-lines theorem
- Corona theorem
- Parabolic and hyperbolic Riemann surfaces
- Analytic capacity
- Mellin's inverse formula
- Mittag-Leffler's theorem
- Lindelöf's theorem
- Branch point
- Hadamard's two-constants theorem
- Disk algebra
- Bôcher's theorem
- Gauss multiplication formula
- de Branges space
- Reciprocal gamma function
- Automorphism group of a bounded domain is a Lie group
- Circle average
- Multiplication theorem
- Order of an entire function
- Euler integral for the gamma function
- Functional equation of the gamma function
- Hartogs lemma for subharmonic functions
- Faber polynomials
- Montel's theorem
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Absolute_value, Complex numbers
- Web https://en.wikipedia.org/wiki/Complex_number, Complex conjugate, absolute value, argument and division
About this page
| Tag | 0KW2X0TFBX |
|---|---|
| URI | m2://0KW2X0TFBX |
| Aliases | absolute-value-of-a-complex-number |
| Kind | definition |
| Writing | W1 stated |
| MSC | 30A99 |
| Defines | Absolute value |
| Uses | Complex numbers, Real and imaginary parts, Square root |
Markdown · JSON · Edit on GitHub
What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.