Real and imaginary parts, complex conjugation, and modulus

Depends on

• $\mathbb C=\mathbb R[x]/(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)/(a^2+b^2)$
• Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$
• The Cauchy-sequence reals have the least-upper-bound property

Used by

• Certified complex-polynomial root isolation in exact algebraic arithmetic Algorithm
• A holomorphic function with continuous complex derivative has C¹ real and imaginary components Corollary
• A positive reaction term restores coercivity without Poincar'e Corollary
• An inverse-error chain obeys Minkowski in L²; root-sum-square is equivalent to aggregate cross-term cancellation Corollary
• Eigenfunctions for distinct symmetric elliptic eigenvalues are L²-orthogonal Corollary
• Symmetric Lax--Milgram is energy minimisation Corollary
• The induced length is a norm Corollary
• The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• The winding number is the increment of a continuous argument divided by 2π Corollary
• A bounded form without coercivity need not be solvable Counterexample
• A coercive form need not be symmetric Counterexample
• Boundary-scale derivative blowup despite bounded ball data Counterexample
• The absolute-value extension gives a zero complex-step quotient despite its real derivative Counterexample
• The complement property is not sufficient for complex phase retrieval Counterexample
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• The integers do not have property (T) Counterexample
• The Neumann Poisson problem is not coercive on all of H¹ Counterexample
• Admissible polynomial pair for Hermitian-reflection QSP Definition
• Balls, polydiscs and the distinguished boundary in ℂᵐ Definition
• Blaschke factors and Blaschke products Definition
• Bounded, coercive and symmetric sesquilinear forms Definition
• C star algebra Definition
• Closed sectorial form and its associated operator Definition
• Complex associative algebra with involution Definition
• Complex L^∞ space of a locally compact group Definition
• Complex Lp classes and Euclidean test-function conventions Definition
• Complex norm and complex normed space Definition
• Continuous logarithms and continuous arguments along a contour Definition
• Dirac Notation Conversion Definition
• Direct integral of a measurable Hilbert field Definition
• Finite complex-array norms used by the NUFFT certificates Definition
• Green function with a pole at infinity Definition
• Holomorphic and antiholomorphic discrete-series models Definition
• Integrable real and complex functions, and their integrals Definition
• ℓ2 sample/query access to vectors and matrices Definition
• Left-invariant means on L^∞ of a locally compact group Definition
• Linear multistep methods, consistency, order, and the root condition Definition
• Logarithmic embedding of a number field Definition
• QSP signal matrix and ordered phase sequence Definition

…and 160 more results.

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)