Mathematics Quick Revision

Complex numbers — advanced

Functional equations & inequalities

• f(x+y)=f(x)+f(y) ⇒ f(x)=kx (Cauchy, with continuity).
• AM ≥ GM ≥ HM.
• Cauchy–Schwarz: (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²).
• Jensen: for convex f, f(Σλᵢxᵢ) ≤ Σλᵢf(xᵢ).

Calculus — techniques

• L'Hopital: 0/0 or ∞/∞ ⇒ lim = lim f'/g'.
• Series expansions: sinx = x − x³/6 + …; eˣ = Σxⁿ/n!; ln(1+x) = x − x²/2 + ….
• Leibniz nth derivative of product.
• Definite integral properties: ∫₀ᵃ f(x) = ∫₀ᵃ f(a−x); if f(2a−x)=f(x) then ∫₀^{2a} = 2∫₀ᵃ.
• Reduction formulae for ∫sinⁿx, ∫cosⁿx.
• Beta/Gamma: Γ(n+1) = n·Γ(n); Γ(½)=√π.

Differential equations — advanced

• Bernoulli: dy/dx + Py = Qyⁿ → sub v = y¹⁻ⁿ.
• Exact: M dx + N dy = 0 with ∂M/∂y = ∂N/∂x.
• 2nd order linear (constant coeffs): auxiliary equation roots (real distinct / equal / complex).
• Particular integral by inspection or method of undetermined coeffs.

Vectors, 3D & geometry

Matrices — advanced

• Rank of matrix; consistency of AX=B (unique / infinite / no solution).
• Cayley–Hamilton: A satisfies its characteristic eqn.
• Eigenvalues: |A − λI|=0; sum = trace(A); product = |A|.
• Symmetric ↔ A^T = A; Skew A^T = −A (diagonal 0).
• Orthogonal AAᵀ = I; unitary AA* = I.

Probability — advanced

• Total probability + Bayes for tree diagrams.
• Independence: P(A∩B) = P(A)P(B).
• Geometric distribution: P(X=k) = q^(k−1)p; mean 1/p, var q/p².
• Poisson: P(X=k) = e⁻λ λᵏ/k!; mean = var = λ.
• Normal ~ N(μ, σ²): 68-95-99.7 rule.