Foundations

If real functions fₙ on one set A satisfy |fₙ(x)|≤Mₙ for all x∈A, with Mₙ≥0 and ΣMₙ<∞, then Σfₙ converges uniformly and absolutely on A.

An improper integral is the finite limit of proper integrals as a singular endpoint or an infinite endpoint is approached.

For positive continuous decreasing f on [1,∞), Σₙ₌₁∞f(n) and ∫₁∞f(x)dx converge or diverge together.

If bₙ≥0 decreases to zero, Σ(−1)ⁿbₙ converges; the absolute error after N terms is at most the next omitted magnitude b_(N+1).

For increasing nonnegative measurable fₙ→f, ∫fₙdμ→∫f dμ.

If measurable real fₙ→f pointwise and |fₙ|≤g for one integrable g, then ∫fₙdμ→∫f dμ.

A subgroup H⊆G is a group under G's restricted operation.

Left cosets gH={gh:h∈H} partition G.

A subgroup N is normal when gNg⁻¹=N for every g∈G.

A two-sided ideal I of R is an additive subgroup absorbing multiplication by R on either side.

A function is holomorphic on an open set when it is complex differentiable at every point.

For a piecewise C¹ path γ, ∫γf(z)dz=∫f(γ(t))γ′(t)dt.

If f is holomorphic on a neighborhood of a closed disk and z lies inside its positively oriented boundary γ, f⁽ⁿ⁾(z)=n!/(2πi)∮γ f(w)/(w−z)ⁿ⁺¹dw for n≥0.

A holomorphic function has a convergent Taylor expansion within a disk free of singularities.

The residue at an isolated singularity is the (z−a)⁻¹ coefficient of its Laurent series.

Floating-point numbers use a sign, finite significand and exponent.

Conditioning measures sensitivity of the exact answer to data perturbations.

Forward error compares a computed answer with the exact answer.

Newton iteration uses xₖ₊₁=xₖ−f(xₖ)/f′(xₖ).

An LU factorization writes A=LU with unit lower-triangular L and upper-triangular U.

Partial pivoting swaps rows to choose a largest-magnitude available entry in the pivot column before eliminating below it.

A real orthogonal square matrix Q satisfies QᵀQ=I and preserves Euclidean lengths.

For invertible A and an induced matrix norm, κ(A)=‖A‖‖A⁻¹‖.

For finite G and subgroup H, |G|=[G:H]|H|.

For a group homomorphism φ:G→H, ker φ is normal and G/ker φ is isomorphic to im φ by g ker φ↦φ(g).

Swapping rows, scaling a row by a nonzero scalar and adding a multiple of another row preserve a linear system's solution set.

With Aᵢⱼ formed by deleting row i and column j, cofactor expansion gives det A=Σⱼ(−1)ⁱ⁺ʲaᵢⱼdet Aᵢⱼ, starting from det[a]=a.

For x,y∈Rⁿ, ⟨x,y⟩=Σxᵢyᵢ and ‖x‖=√⟨x,x⟩.

The minimizers of ‖Ax−b‖₂² satisfy Aᵀ(Ax−b)=0, so their residual is orthogonal to the column space.

Interpolation chooses p with p(tᵢ)=yᵢ at specified distinct nodes.