Foundations

Every nontrivial finite abelian group is a direct product of cyclic groups of prime-power order.

For any group G, left multiplication λg(x)=gx is a permutation of G.

A nonzero holomorphic function on connected Ω factors near each zero a as f(z)=(z−a)ᵐh(z), with integer m≥1 and holomorphic h(a)≠0.

For f holomorphic on 0<|z−a|<R, its Laurent principal part classifies a: no negative powers means removable; finitely many with lowest exponent −m means a pole of order m; infinitely many means essential.

For f holomorphic near the closed disk |z−a|≤R and |f|≤M on its boundary, |f⁽ⁿ⁾(a)|≤n!M/Rⁿ.

If |f| has a local maximum at an interior point of a connected holomorphic domain, f is constant.

At a pole of order m, H(z)=(z−a)ᵐf(z) extends holomorphically and Res(f,a)=H⁽ᵐ⁻¹⁾(a)/(m−1)!.

A C² real u is harmonic when Δu=uₓₓ+uᵧᵧ=0.

A harmonic u equals its average on every circle whose closed disk lies in its domain.

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)|.

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.

Every nonempty simply connected open proper subset of C admits a bijective holomorphic map with holomorphic inverse to the unit disk.

For meromorphic f, a positively oriented simple closed piecewise C¹ contour γ avoiding its zeros and poles gives (2πi)⁻¹∮γf′/f dz=N−P=Ind(f∘γ,0), counting interior zeros and poles with their multiplicities.

If f,h are holomorphic near the closed interior of a simple closed contour γ and |h(z)|<|f(z)| on γ, then f and f+h have the same number of interior zeros counted with multiplicity.

For continuous real 2π-periodic boundary data g, u(r,θ)=(2π)⁻¹∫₋π^π[(1−r²)/(1−2r cos(θ−α)+r²)]g(α)dα is harmonic on 0≤r<1 and extends continuously to u(1,θ)=g(θ).

With even n≥2 and h=(b−a)/n, Sₙ=(h/3)[f(a)+f(b)+4Σodd i f(a+ih)+2Σeven i f(a+ih)], summing interior indices.

For smooth-data Simpson approximations on N and 2N equal subintervals, (S_(2N)−S_N)/15 estimates the correction to S_(2N) by extrapolation.

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.

Backward Euler sets y_(k+1)=y_k+h f(t_(k+1),y_(k+1)), usually requiring an implicit solve.

A problem is called stiff when stability, rather than desired accuracy, forces severely small steps for a method.

For diagonalizable A with |λ₁|>|λ₂|≥… and a start having nonzero λ₁-eigenvector coefficient, repeated x_(k+1)=Ax_k/‖Ax_k‖₂ approaches the dominant eigendirection, up to phase or sign.

A monotone real sequence converges to a finite limit exactly when it is bounded.

Every bounded real sequence has a convergent subsequence.

For convergent real xₙ→x,yₙ→y, sums and products converge to x+y and xy; quotients tend to x/y if y≠0.

In a metric space, K is compact exactly when every sequence in K has a subsequence converging to a point in K.

Uniform continuity requires one δ>0 for each ε>0 to control d_Y(f(p),f(q))<ε whenever d_X(p,q)<δ, for all p,q.

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.

For a real sequence, lim sup xₙ=lim_N sup{k≥N}xₖ and lim inf xₙ=lim_N inf{k≥N}xₖ.

A space X is simply connected when it is path connected and every continuous loop based at a∈X deforms to the constant loop while keeping its endpoints at a.

A real n×n matrix A is diagonalizable over ℝ exactly when it has n linearly independent real eigenvectors.