Foundations

Scientific relationships · 11 / 11

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

1. Complex numbers → Coordinates and tables
   Uses definition
   Relationship sources · 1

   • Relationship source ↗

   Linked statements

   • A complex number is written z=a+bi with real a and b and i²=−1. Its real part is a and its imaginary part is b. It is represented by (a,b) in the complex plane; addition combines the two components and multiplication uses i²=−1.
2. Fundamental Theorem of Algebra → Complex numbers
   Uses definition
   Relationship sources · 1

   • Abstract Algebra, 6.5 Factoring over C and R ↗

   Linked statements

   • Every nonconstant polynomial with complex coefficients has a complex root. Repeated factorization expresses a degree-n polynomial as a product of n linear factors times its leading coefficient, counting multiplicity.
3. Holomorphic functions and Cauchy–Riemann equations → Complex numbers
   Uses definition
   Relationship sources · 1

   • 18.04 (2018), Topic 2: Analytic functions ↗

   Linked statements

   • A function is holomorphic on an open set when it is complex differentiable at every point. For f=u+iv with C¹ real components, holomorphy is equivalent to uₓ=vᵧ and uᵧ=−vₓ.
4. One-sided Laplace transform and derivatives → Complex numbers
   Uses definition
   Relationship sources · 1

   • 18.04 (2018), Topic 12: Laplace transform ↗

   Linked statements

   • The one-sided Laplace transform is F(s)=∫₀∞e⁻ˢᵗf(t)dt. For piecewise continuous f of exponential type a, it converges absolutely when Re s>a; under corresponding bounds for f′, L{f′}=sF(s)−f(0).
5. Fourier transform and inversion on Schwartz functions → Complex numbers
   Uses definition
   Relationship sources · 1

   • 18.155 lecture notes, chapter 11: Fourier transform ↗

   Linked statements

   • For a smooth rapidly decreasing f whose derivatives also decrease faster than any power, define f̂(ξ)=∫Rⁿe⁻ⁱˣ·ξf(x)dx. Then f̂ has the same property and f(x)=(2π)⁻ⁿ∫Rⁿeⁱˣ·ξf̂(ξ)dξ.
6. Characteristic roots for second-order recurrences → Complex numbers
   Uses definition
   Relationship sources · 1

   • Discrete Mathematics: An Open Introduction, third edition, section 2.4 ↗

   Linked statements

   • For aₙ=saₙ₋₁+taₙ₋₂ with t≠0, distinct roots r₁,r₂ of r²−sr−t give aₙ=Ar₁ⁿ+Br₂ⁿ; a repeated root r gives aₙ=(A+Bn)rⁿ. Determine A,B from a₀,a₁ and verify both recurrence and initial values.
7. Discrete Fourier transform and inverse → Complex numbers
   Uses definition
   Relationship sources · 1

   • 6.046J (Spring 2012), Lecture 5: Fast Fourier transform ↗

   Linked statements

   • For N complex coefficients aⱼ define yₖ=Σ_(j=0)^(N−1)aⱼωᴶᴷ, with ω=e²πⁱ⁄ᴺ. The inverse is aⱼ=N⁻¹Σ_(k=0)^(N−1)yₖω⁻ᴶᴷ. This converts polynomial coefficients to values at the Nth roots of unity.
8. Local conformality of holomorphic maps → Complex numbers
   Uses definition
   Relationship sources · 1

   • 18.04 (2018), Topic 10: Conformal transformations ↗

   Linked statements

   • At a point a where f′(a)≠0, a holomorphic map locally preserves oriented angles: tangent vectors are multiplied by f′(a), rotating by arg f′(a) and scaling by |f′(a)|. A local inverse exists; zero derivatives can destroy conformality.
9. Möbius maps and circles → Complex numbers
   Uses definition
   Relationship sources · 1

   • 18.04 (2018), Topic 10: Conformal transformations ↗

   Linked statements

   • T(z)=(az+b)/(cz+d), ad−bc≠0, extends to a bijection of C∪{∞}, with T(−d/c)=∞ and T(∞)=a/c when c≠0. It maps generalized circles (lines or circles) to generalized circles. For example (z−i)/(z+i) maps the upper half-plane bijectively to the unit disk.
10. Scalar absolute stability and Euler steps → Complex numbers
    Uses definition
    Relationship sources · 1

    • Fundamentals of Numerical Computation: Absolute stability ↗

    Linked statements

    • On y′=λy with fixed h>0, a one-step method y_(k+1)=R(hλ)y_k is absolutely stable for a nonzero start exactly when |R(hλ)|≤1; strict inequality yields decay. Forward Euler has R(z)=1+z and region |1+z|≤1, so real λ<0 requires h≤2/|λ|.
11. Fourier mean-square convergence and Parseval → Complex numbers
    Uses definition
    Relationship sources · 1

    • Basic Analysis, section 11.8: Fourier series ↗

    Linked statements

    • For a complex 2π-periodic Riemann-integrable f, cₙ=(2π)⁻¹∫₋π^πf(x)e⁻ⁱⁿˣdx and s_N=Σ|n|≤N cₙeⁱⁿˣ satisfy ∫|f−s_N|²→0. Parseval gives (2π)⁻¹∫₋π^π|f|²=Σₙ∈Z|cₙ|². This is mean-square convergence, which does not assert pointwise convergence everywhere.