$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic

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 $(a-bi)/(a^2+b^2)$

Used by

• A finite-dimensional normed subspace is closed Corollary
• Every finite-dimensional normed space is Banach Corollary
• Holomorphic functions are real analytic and smooth in their two real coordinates Corollary
• Path integral of a holomorphic differential on a Riemann surface Definition
• The edgewise Riemann integral around a complex triangle for an integrable pullback Definition
• The Wirtinger derivatives ∂_z f and ∂_z̄f, and antiholomorphic functions Definition
• A circle representation with an averaged orthogonal weight form Example
• A complex torus has a lattice of parabolic deck translations Example
• Moving the boundary of a fundamental parallelogram Example
• Real-time unitary and imaginary-time finite-matrix propagation Example
• Rectangular lattices, real mapping, and inverse elliptic integrals Example
• The complex line through (1,i) has a two-vector real basis Example
• The complex torus as a Riemann surface Example
• FALSE: real differentiability as a map ℝ²→ℝ² implies complex differentiability; conjugation is the counterexample False statement
• A complex linear functional is recovered from its real part by f(x)=u(x)-iu(ix) Lemma
• A complex normed subspace and its finite basis realify Lemma
• A skew-Hermitian matrix exponential is unitary Lemma
• Certified action bounds control one exponential-integrator step Lemma
• Exponential Euler has a second-order local defect Lemma
• Integral representations of the first two phi functions Lemma
• Local holomorphic charts on nonsingular complex algebraic curves Lemma
• One-step stability after removing the exact linear flow Lemma
• Plane similarities are complex or conjugate-complex multiplications; the orientation-preserving ones are exactly the nonzero complex multiplications Lemma
• The cut surface, primitives of closed forms, and their boundary jumps Lemma
• The multiplicative unit circle is a compact metrizable topological abelian group Lemma
• The period pairing is well defined and computed by integration Lemma
• The real-valued part of a point-separating self-adjoint complex function algebra is separating and has the same common zeros Lemma
• The two-stage ETD2 method has a third-order local defect Lemma
• Variation of constants for a finite-matrix action defect Lemma
• Variation of constants for a finite-matrix semilinear system Lemma
• Vector addition and scalar multiplication are continuous in a normed space Lemma
• ℂ=ℝ[x]/(x²+1) as the Euclidean plane and as a normed real algebra: what the identification preserves Remark
• A Banach space has no countably infinite Hamel basis Theorem
• A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space Theorem
• A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic Theorem
• A normed space is locally compact if and only if it is finite-dimensional Theorem
• Addition formula for ℘ Theorem
• All norms on a finite-dimensional complex normed space are equivalent Theorem
• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations Theorem
• Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions Theorem

…and 9 more results.

Dependency tree · two levels

6 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

• R. K. Srivastava, Complex Analysis lecture notes (standard reference, not scraped)