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

• A holomorphic function with continuous complex derivative has C¹ real and imaginary components 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
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• Balls, polydiscs and the distinguished boundary in ℂᵐ Definition
• C star algebra Definition
• Complex Lp classes and Euclidean test-function conventions Definition
• Continuous logarithms and continuous arguments along a contour Definition
• Integrable real and complex functions, and their integrals Definition
• Real and complex inner product spaces, with the inner product linear in the first argument Definition
• Real and complex inner-product spaces and their induced length Definition
• Real Givens rotations and complex Givens transformations Definition
• Self-adjoint complex function algebras, unitality, and point separation Definition
• Square-summable families on an arbitrary index set and the space ℓ²(I) Definition
• Stolz approach regions at the boundary point 1 of the unit disc Definition
• The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral Definition
• The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• The standard inner product on cf(G) Definition
• The unit disc, the upper half-plane, and Blaschke factors Definition
• Wirtinger operators in ℂᵐ Definition
• {1+i, 1-i} is a normal basis of ℂ/ℝ while {1,i} is not Example
• A continuous argument computed along a spiralling contour Example
• Continuous kernel integral operator is compact on c of an interval Example
• Fredholm alternative for an integral equation Example
• The power series of exp(z₀+z₁) on every bidisc Example
• The standard inner products make K n, ell two and quotient L two Hilbert spaces Example
• The unit circle traversed three times has index 3 at every interior point Example
• Trigonometric polynomials are uniformly dense on the unit circle Example
• Unitary and special unitary Lie groups Example
• FALSE: real differentiability as a map ℝ²→ℝ² implies complex differentiability; conjugation is the counterexample False statement
• A disc missing p carries a holomorphic logarithm of z-p Lemma
• A real-linear functional on ℂᵐ is complex linear exactly when its antiholomorphic part vanishes Lemma
• An algebraic-integer average of roots of unity is either 0 or a common root of unity Lemma
• Canonical Banach complexification of a real Banach space Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive Lemma
• Equality in the unit-complex finite-sum bound Lemma

…and 21 more results.

Dependency tree · two levels

18 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)