An Org-Mode File for Quick Math Reference

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#+TITLE: Mathematical Reference
#+AUTHOR: Ara
#+STARTUP: overview
#+TAGS: dlmf as nist

* Elementary Functions
** Exponential & Logarithm
*** Fundamental definitions                                     :dlmf4_2:as4_1:
    e^x = lim_{n→∞} (1 + x/n)^n
    e = 2.71828 18284 59045 23536...
    ln(x) = ∫₁ˣ dt/t,  x > 0

*** Exponential laws                                            :dlmf4_2:as4_1:
    e^{a+b} = e^a · e^b
    e^{a·b} = (e^a)^b
    e^0 = 1
    e^{ln x} = x,  x > 0
    ln(e^x) = x

*** Logarithm laws                                              :dlmf4_2:as4_1:
    ln(ab) = ln a + ln b
    ln(a/b) = ln a - ln b
    ln(a^r) = r ln a
    log_b(x) = ln(x) / ln(b)

*** Exponential limits & asymptotics                            :dlmf4_2:
    lim_{x→0} (e^x - 1)/x = 1
    lim_{x→∞} x^n / e^x = 0  for any fixed n
    lim_{x→∞} ln(x) / x^ε = 0  for any ε > 0

** Trigonometric Functions
*** Euler's formula                                             :dlmf4_14:
    e^{iθ} = cos θ + i sin θ
    e^{-iθ} = cos θ - i sin θ
    cos θ = (e^{iθ} + e^{-iθ}) / 2
    sin θ = (e^{iθ} - e^{-iθ}) / (2i)

*** Pythagorean identities                                      :dlmf4_14:as4_3_10:
    sin²θ + cos²θ = 1
    1 + tan²θ = sec²θ
    1 + cot²θ = csc²θ

*** Sum/difference formulas                                     :dlmf4_21:as4_3_16:
    sin(α ± β) = sin α cos β ± cos α sin β
    cos(α ± β) = cos α cos β ∓ sin α sin β
    tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)

*** Double-angle                                                :dlmf4_21:as4_3_24:
    sin 2α = 2 sin α cos α
    cos 2α = cos²α - sin²α = 2cos²α - 1 = 1 - 2sin²α
    tan 2α = 2 tan α / (1 - tan²α)

*** Half-angle                                                  :dlmf4_21:as4_3_20:
    sin²(α/2) = (1 - cos α)/2
    cos²(α/2) = (1 + cos α)/2
    tan(α/2) = sin α / (1 + cos α) = (1 - cos α) / sin α

*** Product-to-sum                                              :dlmf4_21:as4_3_31:
    cos α cos β = ½[cos(α-β) + cos(α+β)]
    sin α sin β = ½[cos(α-β) - cos(α+β)]
    sin α cos β = ½[sin(α+β) + sin(α-β)]
    Proof (first): take Re(e^{i(α-β)}) = Re(e^{iα} · e^{-iβ}), expand

*** Sum-to-product                                              :dlmf4_21:as4_3_34:
    sin α + sin β = 2 sin((α+β)/2) cos((α-β)/2)
    sin α - sin β = 2 cos((α+β)/2) sin((α-β)/2)
    cos α + cos β = 2 cos((α+β)/2) cos((α-β)/2)
    cos α - cos β = -2 sin((α+β)/2) sin((α-β)/2)

*** Power-reduction                                             :as4_3_29:
    sin²θ = (1 - cos 2θ)/2
    cos²θ = (1 + cos 2θ)/2
    sin³θ = (3 sin θ - sin 3θ)/4
    cos³θ = (3 cos θ + cos 3θ)/4

*** Chebyshev expansion (cos nθ)                                :dlmf4_21:
    cos 2θ = 2cos²θ - 1
    cos 3θ = 4cos³θ - 3cos θ
    cos nθ = T_n(cos θ)  where T_n is Chebyshev polynomial

*** Inverse trig principal values                               :dlmf4_23:as4_4:
    arcsin: [-π/2, π/2],  domain [-1,1]
    arccos: [0, π],  domain [-1,1]
    arctan: (-π/2, π/2),  domain ℝ
    arctan(x) + arctan(1/x) = π/2  for x > 0

*** Inverse trig relations                                      :dlmf4_23:as4_4:
    arcsin x + arccos x = π/2
    arctan x + arccot x = π/2
    arctan x + arctan y = arctan((x+y)/(1-xy))  when xy < 1
    2 arctan x = arctan(2x/(1-x²))  when |x| < 1

** Hyperbolic Functions
*** Definitions                                                 :dlmf4_28:as4_5:
    sinh x = (e^x - e^{-x})/2
    cosh x = (e^x + e^{-x})/2
    tanh x = sinh x / cosh x = (e^{2x} - 1)/(e^{2x} + 1)

*** Fundamental identity                                        :dlmf4_28:as4_5_1:
    cosh²x - sinh²x = 1
    1 - tanh²x = sech²x
    coth²x - 1 = csch²x

*** Sum formulas                                                :dlmf4_35:as4_5_7:
    sinh(α ± β) = sinh α cosh β ± cosh α sinh β
    cosh(α ± β) = cosh α cosh β ± sinh α sinh β
    tanh(α ± β) = (tanh α ± tanh β)/(1 ± tanh α tanh β)

*** Connection to trig                                          :dlmf4_28:
    sinh(ix) = i sin x
    cosh(ix) = cos x
    sin(ix) = i sinh x
    cos(ix) = cosh x

*** Inverse hyperbolic                                          :dlmf4_37:as4_6:
    arsinh x = ln(x + √(x²+1)),  x ∈ ℝ
    arcosh x = ln(x + √(x²-1)),  x ≥ 1
    artanh x = ½ ln((1+x)/(1-x)),  |x| < 1

* Series & Approximations
** Taylor / Maclaurin series                                    :dlmf4_6:as4_1:
*** Exponential and logarithm
    e^x = Σ_{n=0}^∞ x^n/n!  for all x
    ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ...  |x| ≤ 1, x ≠ -1
    ln((1+x)/(1-x)) = 2(x + x³/3 + x⁵/5 + ...)  |x| < 1

*** Trigonometric                                               :dlmf4_19:as4_3_65:
    sin x = x - x³/3! + x⁵/5! - x⁷/7! + ...
    cos x = 1 - x²/2! + x⁴/4! - x⁶/6! + ...
    tan x = x + x³/3 + 2x⁵/15 + 17x⁷/315 + ...  |x| < π/2
    arctan x = x - x³/3 + x⁵/5 - x⁷/7 + ...  |x| ≤ 1

*** Hyperbolic                                                  :dlmf4_33:as4_5_62:
    sinh x = x + x³/3! + x⁵/5! + x⁷/7! + ...
    cosh x = 1 + x²/2! + x⁴/4! + x⁶/6! + ...
    tanh x = x - x³/3 + 2x⁵/15 - 17x⁷/315 + ...  |x| < π/2

*** Binomial series                                             :dlmf1_2:as3_6_10:
    (1+x)^α = Σ_{n=0}^∞ C(α,n) x^n  where C(α,n) = α(α-1)...(α-n+1)/n!
    Valid for |x| < 1 (any α), or |x| = 1 if α > 0

*** Geometric series                                            :as3_6_2:
    1/(1-x) = 1 + x + x² + x³ + ...  |x| < 1
    1/(1-x)^2 = 1 + 2x + 3x² + 4x³ + ...  |x| < 1
    1/(1-x)^n = Σ_{k=0}^∞ C(n+k-1,k) x^k  |x| < 1

** Asymptotic expansions
*** Stirling's approximation                                    :dlmf5_11:as6_1_37:
    n! ~ √(2πn) (n/e)^n [1 + 1/(12n) + 1/(288n²) - 139/(51840n³) + ...]
    ln(n!) = n ln n - n + ½ ln(2πn) + 1/(12n) - 1/(360n³) + ...

*** Euler-Maclaurin formula                                     :dlmf2_10:as23_1_30:
    Σ_{k=a}^b f(k) = ∫_a^b f(x)dx + (f(a)+f(b))/2
        + Σ_{j=1}^p B_{2j}/(2j)! [f^{(2j-1)}(b) - f^{(2j-1)}(a)] + R_p
    where B_{2j} are Bernoulli numbers

*** Laplace method (saddle point)                               :dlmf2_4:
    ∫_a^b f(x) e^{λg(x)} dx ~ f(x₀) e^{λg(x₀)} √(2π/(λ|g''(x₀)|))
    as λ → ∞, where g'(x₀) = 0, g''(x₀) < 0

** Summation formulas
*** Bernoulli numbers                                           :dlmf24_2:as23_1:
    B₀=1, B₁=-½, B₂=1/6, B₄=-1/30, B₆=1/42, B₈=-1/30, B₁₀=5/66
    B_{2n+1} = 0 for n ≥ 1
    Generating function: t/(e^t - 1) = Σ_{n=0}^∞ B_n t^n/n!

*** Euler numbers                                               :dlmf24_2:as23_1:
    E₀=1, E₂=-1, E₄=5, E₆=-61, E₈=1385
    Generating function: sech t = Σ_{n=0}^∞ E_n t^n/n!

*** Riemann zeta at integers                                    :dlmf25_6:as23_2:
    ζ(2) = π²/6
    ζ(4) = π⁴/90
    ζ(6) = π⁶/945
    ζ(2n) = (-1)^{n+1} (2π)^{2n} B_{2n} / (2(2n)!)
    ζ(3) = 1.20205 69031 59594... (Apéry's constant, no closed form)

*** Partial fractions for trig                                  :dlmf4_22:
    π cot(πz) = 1/z + Σ_{n=1}^∞ [1/(z-n) + 1/(z+n)]
    π csc(πz) = 1/z + Σ_{n=1}^∞ (-1)^n [1/(z-n) + 1/(z+n)]
    π²/sin²(πz) = Σ_{n=-∞}^∞ 1/(z-n)²

* Combinatorics & Number Theory
** Binomial coefficients                                        :dlmf26_3:as24_1_1:
*** Basic identities
    C(n,k) = n! / (k!(n-k)!)
    C(n,k) = C(n, n-k)                     (symmetry)
    C(n,k) = C(n-1,k-1) + C(n-1,k)         (Pascal)
    Σ_{k=0}^n C(n,k) = 2^n
    Σ_{k=0}^n (-1)^k C(n,k) = 0

*** Vandermonde convolution                                     :dlmf26_3:as24_1_1:
    Σ_{k=0}^r C(m,k) C(n,r-k) = C(m+n, r)

*** Chu-Vandermonde                                             :dlmf15_4:
    ₂F₁(-n, b; c; 1) = (c-b)_n / (c)_n
    Reduces to Vandermonde when b,c are integers

*** Upper negation
    C(-n, k) = (-1)^k C(n+k-1, k)

*** Multinomial
    C(n; k₁,k₂,...,k_m) = n! / (k₁! k₂! ... k_m!)
    where k₁ + k₂ + ... + k_m = n

** Partition formulas
*** Euler's pentagonal theorem                                  :dlmf27_14:
    Π_{n=1}^∞ (1-x^n) = Σ_{k=-∞}^∞ (-1)^k x^{k(3k-1)/2}

*** Partition generating function
    Σ_{n=0}^∞ p(n) x^n = Π_{k=1}^∞ 1/(1-x^k)
    p(0)=1, p(1)=1, p(2)=2, p(3)=3, p(4)=5, p(5)=7, p(10)=42

** Number theory
*** Möbius function μ(n)                                        :dlmf27_2:
    μ(1) = 1
    μ(n) = (-1)^k if n = p₁p₂...p_k (distinct primes)
    μ(n) = 0 if n has a squared prime factor

*** Möbius inversion                                            :dlmf27_5:
    If g(n) = Σ_{d|n} f(d)  then  f(n) = Σ_{d|n} μ(n/d) g(d)

*** Euler totient φ(n)                                          :dlmf27_2:as24_3_2:
    φ(n) = n Π_{p|n} (1 - 1/p)
    Σ_{d|n} φ(d) = n
    φ(mn) = φ(m)φ(n) · d/φ(d)  where d = gcd(m,n)

*** Quadratic reciprocity
    For odd primes p ≠ q:
    (p/q)(q/p) = (-1)^{(p-1)(q-1)/4}
    where (·/·) is the Legendre symbol

* Linear Algebra
** Matrix identities
*** Woodbury identity                                           :as_matrix:
    (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}

*** Sherman-Morrison (rank-1 case)
    (A + uv^T)^{-1} = A^{-1} - A^{-1}u v^T A^{-1} / (1 + v^T A^{-1} u)

*** Matrix determinant lemma
    det(A + uv^T) = (1 + v^T A^{-1} u) det(A)

*** Push-through identity
    (I + AB)^{-1} A = A (I + BA)^{-1}
    Useful: maps between m×m and n×n inversions

*** Block matrix inverse (2×2)
    [A B; C D]^{-1}, Schur complement S = D - CA^{-1}B:
    top-left block = A^{-1} + A^{-1}B S^{-1} C A^{-1}
    top-right = -A^{-1}B S^{-1}
    bottom-left = -S^{-1} C A^{-1}
    bottom-right = S^{-1}

** Determinant & trace
*** Properties
    det(AB) = det(A) det(B)
    det(A^T) = det(A)
    det(cA) = c^n det(A)  for n×n matrix
    det(A^{-1}) = 1/det(A)
    tr(AB) = tr(BA)
    tr(A+B) = tr(A) + tr(B)
    tr(A^T) = tr(A)

*** Jacobi's formula
    d/dt det(A(t)) = det(A) tr(A^{-1} dA/dt)

*** Matrix exponential
    e^A = Σ_{n=0}^∞ A^n/n!
    det(e^A) = e^{tr(A)}
    e^{A+B} = e^A e^B  only if AB = BA

** Eigenvalue bounds
*** Gershgorin circle theorem
    Every eigenvalue of A lies in at least one disk
    D_i = {z : |z - a_{ii}| ≤ Σ_{j≠i} |a_{ij}|}

*** Courant-Fischer (min-max)
    λ_k = min_{dim(V)=k} max_{x∈V, x≠0} (x^T A x)/(x^T x)
    for A symmetric, eigenvalues λ₁ ≤ λ₂ ≤ ... ≤ λ_n

*** Weyl's inequality
    For Hermitian A, B:
    λ_i(A) + λ_1(B) ≤ λ_i(A+B) ≤ λ_i(A) + λ_n(B)

** Decompositions
*** SVD
    A = UΣV^T,  U^TU = I, V^TV = I, Σ diagonal ≥ 0
    Best rank-k approx: A_k = Σ_{i=1}^k σ_i u_i v_i^T (Eckart-Young)

*** Cholesky
    A = LL^T for A symmetric positive definite, L lower triangular

*** QR
    A = QR, Q orthogonal, R upper triangular

*** Spectral
    A = QΛQ^T for A symmetric, Q orthogonal, Λ = diag(λ₁,...,λ_n)

* Calculus & Analysis
** Differentiation identities
*** Leibniz rule (product, nth derivative)                      :as3_3_8:
    (fg)^{(n)} = Σ_{k=0}^n C(n,k) f^{(k)} g^{(n-k)}

*** Faà di Bruno (chain rule, nth derivative)
    (f∘g)^{(n)} = Σ f^{(k)}(g(x)) · B_{n,k}(g'(x), g''(x), ..., g^{(n-k+1)}(x))
    where B_{n,k} are partial Bell polynomials

*** L'Hôpital's rule
    lim f(x)/g(x) = lim f'(x)/g'(x)
    when both → 0 or both → ±∞, provided RHS exists

** Standard integrals                                           :as_chapter5:
*** Gaussian integral
    ∫_{-∞}^∞ e^{-x²} dx = √π
    ∫_{-∞}^∞ e^{-ax²+bx} dx = √(π/a) e^{b²/(4a)},  Re(a) > 0
    ∫_0^∞ x^{2n} e^{-x²} dx = √π (2n)! / (n! 4^n · 2)

*** Dirichlet integral
    ∫_0^∞ sin(x)/x dx = π/2

*** Fresnel integrals                                           :dlmf7_2:
    ∫_0^∞ sin(x²) dx = ∫_0^∞ cos(x²) dx = √(π/2)/2

*** Beta integral                                               :dlmf5_12:as6_2_1:
    ∫_0^1 x^{a-1}(1-x)^{b-1} dx = B(a,b) = Γ(a)Γ(b)/Γ(a+b)

*** Integration by parts (tabular method)
    ∫ u dv = uv - ∫ v du
    For repeated application: alternate signs, D column, I column

** Inequalities
*** AM-GM
    (a₁+a₂+...+a_n)/n ≥ (a₁a₂...a_n)^{1/n}  for a_i ≥ 0
    Equality iff all a_i equal

*** Cauchy-Schwarz
    |⟨u,v⟩|² ≤ ⟨u,u⟩ · ⟨v,v⟩
    (Σ a_i b_i)² ≤ (Σ a_i²)(Σ b_i²)

*** Hölder's inequality
    Σ |a_i b_i| ≤ (Σ |a_i|^p)^{1/p} · (Σ |b_i|^q)^{1/q}
    where 1/p + 1/q = 1, p > 1

*** Minkowski's inequality
    (Σ |a_i+b_i|^p)^{1/p} ≤ (Σ |a_i|^p)^{1/p} + (Σ |b_i|^p)^{1/p}
    (triangle inequality in l^p)

*** Jensen's inequality
    f(E[X]) ≤ E[f(X)]  for f convex
    f(E[X]) ≥ E[f(X)]  for f concave

*** Young's inequality
    ab ≤ a^p/p + b^q/q  for a,b ≥ 0, 1/p + 1/q = 1

** Integral inequalities
*** Cauchy-Schwarz (integral form)
    (∫ fg dx)² ≤ (∫ f² dx)(∫ g² dx)

*** Hölder (integral form)
    ∫ |fg| dx ≤ (∫ |f|^p dx)^{1/p} · (∫ |g|^q dx)^{1/q}

*** Grönwall's inequality
    If u'(t) ≤ β(t)u(t) then u(t) ≤ u(0) exp(∫_0^t β(s)ds)

* Gamma, Beta & Related                                         :dlmf5:as6:
** Gamma function                                               :dlmf5_2:as6_1:
*** Definition
    Γ(z) = ∫_0^∞ t^{z-1} e^{-t} dt,  Re(z) > 0
    Γ(z+1) = z Γ(z)
    Γ(n+1) = n!  for n ∈ ℕ

*** Special values                                              :dlmf5_4:as6_1_5:
    Γ(1/2) = √π
    Γ(1) = 1
    Γ(3/2) = √π/2
    Γ(n+½) = (2n)! √π / (4^n n!)

*** Reflection formula                                          :dlmf5_5:as6_1_17:
    Γ(z)Γ(1-z) = π / sin(πz)

*** Duplication formula (Legendre)                              :dlmf5_5:as6_1_18:
    Γ(z)Γ(z+½) = √π / 2^{2z-1} · Γ(2z)

*** Multiplication formula (Gauss)                              :dlmf5_5:as6_1_20:
    Π_{k=0}^{n-1} Γ(z+k/n) = (2π)^{(n-1)/2} n^{1/2-nz} Γ(nz)

*** Stirling (asymptotic)                                       :dlmf5_11:as6_1_37:
    Γ(z) ~ √(2π/z) (z/e)^z [1 + 1/(12z) + 1/(288z²) + ...]
    as |z| → ∞ in |arg(z)| < π

** Beta function                                                :dlmf5_12:as6_2:
    B(a,b) = Γ(a)Γ(b)/Γ(a+b)
    B(a,b) = ∫_0^1 t^{a-1}(1-t)^{b-1} dt
    B(a,b) = B(b,a)
    B(a,b) = 2∫_0^{π/2} sin^{2a-1}θ cos^{2b-1}θ dθ

** Digamma & polygamma                                          :dlmf5_2:as6_3:
*** Digamma
    ψ(z) = Γ'(z)/Γ(z) = d/dz ln Γ(z)
    ψ(1) = -γ  (Euler-Mascheroni, γ = 0.57721 56649...)
    ψ(n+1) = -γ + Σ_{k=1}^n 1/k = -γ + H_n
    ψ(z+1) = ψ(z) + 1/z
    ψ(1/2) = -γ - 2 ln 2
    ψ(1-z) - ψ(z) = π cot(πz)

*** Polygamma
    ψ^{(n)}(z) = (-1)^{n+1} n! Σ_{k=0}^∞ 1/(z+k)^{n+1}
    ψ^{(n)}(z+1) = ψ^{(n)}(z) + (-1)^n n!/z^{n+1}

** Pochhammer symbols                                           :dlmf5_2:as6_1_22:
    Rising factorial: (a)_n = a(a+1)(a+2)...(a+n-1) = Γ(a+n)/Γ(a)
    (a)_0 = 1
    (1)_n = n!
    Falling factorial: a^{(n)} = a(a-1)(a-2)...(a-n+1) = n! C(a,n)

** Incomplete gamma                                             :dlmf8_2:as6_5:
    γ(a,x) = ∫_0^x t^{a-1} e^{-t} dt        (lower)
    Γ(a,x) = ∫_x^∞ t^{a-1} e^{-t} dt        (upper)
    γ(a,x) + Γ(a,x) = Γ(a)
    Regularized: P(a,x) = γ(a,x)/Γ(a),  Q(a,x) = Γ(a,x)/Γ(a)

* Error Function & Related                                      :dlmf7:as7:
** Error function                                               :dlmf7_2:as7_1:
*** Definition
    erf(z) = (2/√π) ∫_0^z e^{-t²} dt
    erfc(z) = 1 - erf(z) = (2/√π) ∫_z^∞ e^{-t²} dt
    erf(∞) = 1,  erf(0) = 0,  erf(-z) = -erf(z)

*** Relation to Gaussian CDF
    Φ(x) = ½[1 + erf(x/√2)]
    erfc(x) = 2[1 - Φ(x√2)]
    Q(x) = ½ erfc(x/√2)  (tail probability)

*** Relation to incomplete gamma
    erf(z) = P(½, z²) = γ(½, z²)/√π

*** Series expansion                                            :dlmf7_6:as7_1_5:
    erf(z) = (2/√π) Σ_{n=0}^∞ (-1)^n z^{2n+1} / (n!(2n+1))
           = (2/√π)(z - z³/3 + z⁵/10 - z⁷/42 + ...)

*** Asymptotic expansion (large z)                              :dlmf7_12:as7_1_23:
    erfc(z) ~ (e^{-z²}/(z√π)) Σ_{n=0}^∞ (-1)^n (2n-1)!! / (2z²)^n
            = (e^{-z²}/(z√π))[1 - 1/(2z²) + 3/(4z⁴) - ...]
    as |z|→∞ in |arg z| < 3π/4

*** Bounds
    (2/√π) · x/(x²+1) · e^{-x²} < erfc(x) < (2/√π) · (1/x) · e^{-x²}  for x>0
    Mills ratio: R(x) = erfc(x)√π e^{x²}/2 ~ 1/x - 1/x³ + 3/x⁵ - ...

** Dawson function                                              :dlmf7_2:
    F(z) = e^{-z²} ∫_0^z e^{t²} dt
    F(z) = (√π/2) e^{-z²} erfi(z)  where erfi(z) = -i erf(iz)
    F(z) ~ 1/(2z) + 1/(4z³) + 3/(8z⁵) + ...  as z→∞
    F(0) = 0, max at z ≈ 0.9241, F(0.9241) ≈ 0.5411

** Fresnel integrals                                            :dlmf7_2:as7_3:
    S(z) = ∫_0^z sin(πt²/2) dt
    C(z) = ∫_0^z cos(πt²/2) dt
    S(∞) = C(∞) = ½
    S(z) = (√π/2) Σ_{n=0}^∞ (-1)^n (πz²/2)^{2n+1} / ((2n+1)!(4n+3))
    C(z) = (√π/2) Σ_{n=0}^∞ (-1)^n (πz²/2)^{2n} / ((2n)!(4n+1))

** Voigt function (plasma/spectral line)                        :dlmf7_19:
    H(a,u) = (a/π) ∫_{-∞}^∞ e^{-t²} / (a² + (u-t)²) dt
    H(a,u) = Re[w(u+ia)]  where w is Faddeeva function
    Faddeeva: w(z) = e^{-z²} erfc(-iz) = e^{-z²}(1 + (2i/√π)∫_0^z e^{t²}dt)

** Exponential integral & relatives                             :dlmf6:as5_1:
*** Exponential integral
    Ei(x) = -PV ∫_{-x}^∞ e^{-t}/t dt = PV ∫_{-∞}^x e^t/t dt  (x>0)
    E₁(z) = ∫_z^∞ e^{-t}/t dt = ∫_1^∞ e^{-zt}/t dt  (Re z>0 or z≠0)
    E₁(z) = -γ - ln z - Σ_{n=1}^∞ (-z)^n/(n·n!)
    Ei(x) = γ + ln x + Σ_{n=1}^∞ x^n/(n·n!)  for x>0

*** Logarithmic integral
    li(x) = ∫_0^x dt/ln t = Ei(ln x)
    li(x) ~ x/ln x  (prime number theorem: π(x) ~ li(x))

*** Sine and cosine integrals                                   :dlmf6_2:as5_2:
    Si(x) = ∫_0^x sin(t)/t dt,  Si(∞) = π/2
    Ci(x) = -∫_x^∞ cos(t)/t dt = γ + ln x + ∫_0^x (cos t - 1)/t dt
    si(x) = Si(x) - π/2 = -∫_x^∞ sin(t)/t dt

* Orthogonal Polynomials                                        :dlmf18:as22:
** General Rodrigues formula
    p_n(x) = (c_n / w(x)) · (d/dx)^n [w(x) s(x)^n]

** Legendre polynomials P_n(x)                                  :dlmf18_3:as22_2:
    Weight: w=1 on [-1,1]
    Rodrigues: P_n(x) = 1/(2^n n!) (d/dx)^n (x²-1)^n
    P₀=1, P₁=x, P₂=(3x²-1)/2, P₃=(5x³-3x)/2
    Recurrence: (n+1)P_{n+1} = (2n+1)x P_n - n P_{n-1}
    Orthogonality: ∫_{-1}^1 P_m P_n dx = 2δ_{mn}/(2n+1)
    Generating fn: 1/√(1-2xh+h²) = Σ P_n(x) h^n

** Chebyshev polynomials T_n(x)                                 :dlmf18_3:as22_3:
    Weight: w = 1/√(1-x²) on [-1,1]
    T_n(cos θ) = cos(nθ)
    T₀=1, T₁=x, T₂=2x²-1, T₃=4x³-3x
    Recurrence: T_{n+1} = 2x T_n - T_{n-1}
    Orthogonality: ∫_{-1}^1 T_m T_n / √(1-x²) dx = (π/2)δ_{mn} (n≥1), π (n=m=0)

** Hermite polynomials H_n(x)                                   :dlmf18_3:as22_5:
    Weight: w = e^{-x²} on (-∞,∞)  [physicist convention]
    Rodrigues: H_n(x) = (-1)^n e^{x²} (d/dx)^n e^{-x²}
    H₀=1, H₁=2x, H₂=4x²-2, H₃=8x³-12x
    Recurrence: H_{n+1} = 2x H_n - 2n H_{n-1}
    Orthogonality: ∫_{-∞}^∞ H_m H_n e^{-x²} dx = √π 2^n n! δ_{mn}

** Laguerre polynomials L_n(x)                                  :dlmf18_3:as22_11:
    Weight: w = e^{-x} on [0,∞)
    Rodrigues: L_n(x) = e^x/n! (d/dx)^n (x^n e^{-x})
    L₀=1, L₁=1-x, L₂=(x²-4x+2)/2
    Recurrence: (n+1)L_{n+1} = (2n+1-x)L_n - n L_{n-1}
    Generalized: L_n^{(α)}(x) with weight x^α e^{-x}

** Jacobi polynomials P_n^{(α,β)}(x)                           :dlmf18_3:as22_1:
    Weight: w = (1-x)^α (1+x)^β on [-1,1]
    Includes Legendre (α=β=0), Chebyshev (α=β=-½), Gegenbauer (α=β)
    Orthogonality: ∫_{-1}^1 P_m^{(α,β)} P_n^{(α,β)} w(x)dx = h_n δ_{mn}

* Bessel Functions                                              :dlmf10:as9:
** Bessel equation
    x² y'' + x y' + (x² - ν²) y = 0
    Solutions: J_ν(x), Y_ν(x) (Bessel 1st/2nd kind)

** Modified Bessel equation
    x² y'' + x y' - (x² + ν²) y = 0
    Solutions: I_ν(x), K_ν(x)

** J_ν definitions & properties                                 :dlmf10_2:as9_1_10:
    J_ν(x) = Σ_{m=0}^∞ (-1)^m (x/2)^{ν+2m} / (m! Γ(ν+m+1))
    J_{-n}(x) = (-1)^n J_n(x)  for integer n

** Recurrence relations                                         :dlmf10_6:as9_1_27:
    J_{ν-1}(z) + J_{ν+1}(z) = (2ν/z) J_ν(z)
    J_{ν-1}(z) - J_{ν+1}(z) = 2 J'_ν(z)
    Same recurrences hold for Y_ν, I_ν (with sign change for K_ν)

** Asymptotic forms (large x)                                   :dlmf10_17:as9_2:
    J_ν(x) ~ √(2/(πx)) cos(x - νπ/2 - π/4)
    Y_ν(x) ~ √(2/(πx)) sin(x - νπ/2 - π/4)
    I_ν(x) ~ e^x / √(2πx)
    K_ν(x) ~ √(π/(2x)) e^{-x}

** Special cases                                                :dlmf10_16:as9_1_1:
    J_{1/2}(x) = √(2/(πx)) sin x
    J_{-1/2}(x) = √(2/(πx)) cos x
    I_{1/2}(x) = √(2/(πx)) sinh x
    K_{1/2}(x) = √(π/(2x)) e^{-x}

** Integral representations                                     :dlmf10_9:as9_1_20:
    J_n(x) = (1/π) ∫_0^π cos(nθ - x sin θ) dθ  for integer n
    J_0(x) = (1/π) ∫_0^π cos(x sin θ) dθ

** Hankel functions                                             :dlmf10_2:as9_1_3:
    H_ν^{(1)}(z) = J_ν(z) + i Y_ν(z)
    H_ν^{(2)}(z) = J_ν(z) - i Y_ν(z)

** Zeros of J_ν                                                 :dlmf10_21:as9_5:
    J₀ zeros: 2.4048, 5.5201, 8.6537, 11.7915, ...
    J₁ zeros: 3.8317, 7.0156, 10.1735, 13.3237, ...
    McMahon expansion: j_{ν,s} ~ β - (4ν²-1)/(8β) - ...
    where β = (s + ν/2 - 1/4)π

* Hypergeometric Functions                                      :dlmf15:as15:
** Gauss hypergeometric ₂F₁                                    :dlmf15_2:as15_1:
    ₂F₁(a,b;c;z) = Σ_{n=0}^∞ (a)_n(b)_n / ((c)_n n!) z^n  |z|<1
    ₂F₁(a,b;c;1) = Γ(c)Γ(c-a-b) / (Γ(c-a)Γ(c-b))  Re(c-a-b) > 0

** Euler's integral                                             :dlmf15_6:as15_3_1:
    ₂F₁(a,b;c;z) = Γ(c)/(Γ(b)Γ(c-b)) ∫_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a} dt

** Kummer's transformation                                      :dlmf15_8:as15_3_3:
    ₂F₁(a,b;c;z) = (1-z)^{c-a-b} ₂F₁(c-a, c-b; c; z)

** Pfaff's transformation                                       :dlmf15_8:as15_3_4:
    ₂F₁(a,b;c;z) = (1-z)^{-a} ₂F₁(a, c-b; c; z/(z-1))

** Confluent hypergeometric (Kummer) ₁F₁                       :dlmf13_2:as13_1:
    M(a,b,z) = ₁F₁(a;b;z) = Σ_{n=0}^∞ (a)_n z^n / ((b)_n n!)
    Kummer's 2nd: U(a,b,z) (irregular solution)
    Relation: ₁F₁(a;b;z) = e^z ₁F₁(b-a;b;-z)

** Generalized hypergeometric                                   :dlmf16:as_gen:
    _pF_q(a₁,...,a_p; b₁,...,b_q; z) = Σ_{n=0}^∞ (a₁)_n...(a_p)_n / ((b₁)_n...(b_q)_n) z^n/n!
    Converges for all z when p ≤ q; for |z|<1 when p = q+1

** Special cases expressed as ₂F₁
    P_n(x) = ₂F₁(-n, n+1; 1; (1-x)/2)           (Legendre)
    T_n(x) = ₂F₁(-n, n; 1/2; (1-x)/2)            (Chebyshev)
    ln(1+z) = z · ₂F₁(1, 1; 2; -z)
    arcsin z = z · ₂F₁(1/2, 1/2; 3/2; z²)

* Elliptic Integrals & Functions                                :dlmf19:as17:
** Incomplete elliptic integrals                                :dlmf19_2:as17_2:
    F(φ,k) = ∫_0^φ dθ/√(1 - k²sin²θ)             (1st kind)
    E(φ,k) = ∫_0^φ √(1 - k²sin²θ) dθ             (2nd kind)
    Π(n;φ,k) = ∫_0^φ dθ/((1-n sin²θ)√(1-k²sin²θ)) (3rd kind)

** Complete elliptic integrals                                  :dlmf19_2:as17_3:
    K(k) = F(π/2, k) = ∫_0^{π/2} dθ/√(1 - k²sin²θ)
    E(k) = E(π/2, k) = ∫_0^{π/2} √(1 - k²sin²θ) dθ
    K(0)=π/2, E(0)=π/2
    K(k) ~ -½ ln(1-k²) as k→1
    Legendre relation: E K' + E' K - K K' = π/2

** Series expansions                                            :dlmf19_5:as17_3_11:
    K(k) = (π/2)[1 + (1/2)²k² + (1·3/(2·4))²k⁴ + ...]
    E(k) = (π/2)[1 - (1/2)²k²/1 - (1·3/(2·4))²k⁴/3 - ...]

** AGM relation                                                 :dlmf19_8:
    K(k) = π / (2 AGM(1, k'))  where k' = √(1-k²)
    AGM(a,b): iterate a_{n+1} = (a_n+b_n)/2, b_{n+1} = √(a_n b_n)

** Jacobi elliptic functions                                    :dlmf22_2:as16_1:
    sn(u,k), cn(u,k), dn(u,k)
    sn²u + cn²u = 1
    k²sn²u + dn²u = 1
    Period: 4K(k) for sn,cn; 2K(k) for dn

*** Addition formulas                                           :dlmf22_8:as16_17:
    sn(u+v) = (sn u cn v dn v + sn v cn u dn u) / (1 - k²sn²u sn²v)
    cn(u+v) = (cn u cn v - sn u dn u sn v dn v) / (1 - k²sn²u sn²v)
    dn(u+v) = (dn u dn v - k²sn u cn u sn v cn v) / (1 - k²sn²u sn²v)

*** Derivatives                                                 :dlmf22_13:as16_16:
    d/du sn u = cn u dn u
    d/du cn u = -sn u dn u
    d/du dn u = -k² sn u cn u

*** Limiting cases
    k → 0: sn → sin, cn → cos, dn → 1, K → π/2
    k → 1: sn → tanh, cn → sech, dn → sech, K → ∞

** Jacobi theta functions                                       :dlmf20_2:
*** Definitions (nome q = e^{iπτ}, |q| < 1)
    θ₁(z,q) = 2 Σ_{n=0}^∞ (-1)^n q^{(n+1/2)²} sin((2n+1)z)
    θ₂(z,q) = 2 Σ_{n=0}^∞ q^{(n+1/2)²} cos((2n+1)z)
    θ₃(z,q) = 1 + 2 Σ_{n=1}^∞ q^{n²} cos(2nz)
    θ₄(z,q) = 1 + 2 Σ_{n=1}^∞ (-1)^n q^{n²} cos(2nz)

*** Quasi-periodicity
    θ₁(z+π) = -θ₁(z),  θ₁(z+πτ) = -q⁻¹e^{-2iz} θ₁(z)
    θ₃(z+π) = θ₃(z),   θ₃(z+πτ) = q⁻¹e^{-2iz} θ₃(z)

*** Theta function at z=0 (theta constants)
    θ₂ = θ₂(0,q),  θ₃ = θ₃(0,q),  θ₄ = θ₄(0,q)
    θ₁'(0,q) = πθ₂θ₃θ₄
    Jacobi identity: θ₃⁴ = θ₂⁴ + θ₄⁴

*** Relation to elliptic functions                              :dlmf22_2:
    k = (θ₂/θ₃)²,  k' = (θ₄/θ₃)²
    K = πθ₃²/2
    sn(u) = θ₃/θ₂ · θ₁(v)/θ₄(v)  where v = u/(2K/π)

*** Jacobi triple product                                       :dlmf20_5:
    θ₃(z,q) = Π_{n=1}^∞ (1-q^{2n})(1 + q^{2n-1}e^{2iz})(1 + q^{2n-1}e^{-2iz})

** Modular forms (brief)                                        :dlmf23_15:
*** Dedekind eta function
    η(τ) = e^{iπτ/12} Π_{n=1}^∞ (1 - e^{2πinτ})
    η(τ) = q^{1/24} Π_{n=1}^∞ (1-q^n)  where q = e^{2πiτ}

*** Modular discriminant
    Δ(τ) = η(τ)^{24} = q Π_{n=1}^∞ (1-q^n)^{24}

*** Eisenstein series
    G_{2k}(τ) = Σ_{(m,n)≠(0,0)} 1/(mτ+n)^{2k}  for k ≥ 2
    G₄ and G₆ generate all modular forms
    g₂ = 60 G₄,  g₃ = 140 G₆  (Weierstrass invariants)
    Δ = g₂³ - 27g₃²
    j-invariant: j(τ) = 1728 g₂³/Δ

** Weierstrass ℘-function                                      :dlmf23_2:
    ℘(z) = 1/z² + Σ_{ω≠0} [1/(z-ω)² - 1/ω²]
    (℘')² = 4℘³ - g₂℘ - g₃
    ℘(z) is doubly periodic with periods ω₁, ω₂

* Probability & Statistics
** Discrete distributions
*** Binomial(n,p)
    P(X=k) = C(n,k) p^k (1-p)^{n-k}
    E[X] = np, Var(X) = np(1-p)
    MGF: M(t) = (pe^t + 1-p)^n

*** Poisson(λ)
    P(X=k) = e^{-λ} λ^k / k!
    E[X] = λ, Var(X) = λ
    MGF: M(t) = exp(λ(e^t - 1))

*** Geometric(p)
    P(X=k) = (1-p)^{k-1} p,  k=1,2,...
    E[X] = 1/p, Var(X) = (1-p)/p²

*** Negative Binomial(r,p)
    P(X=k) = C(k-1,r-1) p^r (1-p)^{k-r},  k=r,r+1,...
    E[X] = r/p, Var(X) = r(1-p)/p²

** Continuous distributions
*** Normal(μ,σ²)
    f(x) = (1/(σ√(2π))) exp(-(x-μ)²/(2σ²))
    E[X] = μ, Var(X) = σ²
    MGF: M(t) = exp(μt + σ²t²/2)
    Φ(x) = CDF of N(0,1);  Φ(-x) = 1 - Φ(x)

*** Exponential(λ)
    f(x) = λ e^{-λx},  x ≥ 0
    E[X] = 1/λ, Var(X) = 1/λ²
    CDF: F(x) = 1 - e^{-λx}
    Memoryless: P(X > s+t | X > s) = P(X > t)

*** Gamma(α,β)
    f(x) = β^α x^{α-1} e^{-βx} / Γ(α),  x > 0
    E[X] = α/β, Var(X) = α/β²
    Includes Exponential (α=1) and χ² (α=n/2, β=1/2)

*** Beta(α,β)
    f(x) = x^{α-1}(1-x)^{β-1} / B(α,β),  x ∈ (0,1)
    E[X] = α/(α+β), Var(X) = αβ/((α+β)²(α+β+1))

*** Student-t(ν)
    f(x) = Γ((ν+1)/2)/(√(νπ)Γ(ν/2)) (1+x²/ν)^{-(ν+1)/2}
    E[X] = 0 (ν>1), Var(X) = ν/(ν-2) (ν>2)
    Approaches N(0,1) as ν→∞

*** χ²(k)
    f(x) = x^{k/2-1} e^{-x/2} / (2^{k/2} Γ(k/2)),  x > 0
    E[X] = k, Var(X) = 2k
    Sum of k independent N(0,1)² variables

*** F(d₁,d₂)
    Ratio (χ²_{d₁}/d₁) / (χ²_{d₂}/d₂)
    E[X] = d₂/(d₂-2) for d₂ > 2

** Concentration inequalities
*** Markov
    P(X ≥ a) ≤ E[X]/a  for X ≥ 0, a > 0

*** Chebyshev
    P(|X-μ| ≥ kσ) ≤ 1/k²

*** Chernoff bound
    P(X ≥ a) ≤ inf_{t>0} e^{-ta} M_X(t)

*** Hoeffding
    P(S_n - E[S_n] ≥ t) ≤ exp(-2t²/Σ(b_i-a_i)²)
    where X_i ∈ [a_i, b_i] independent, S_n = ΣX_i

** Bayesian conjugate pairs
    Likelihood → Prior → Posterior:
    Bernoulli → Beta(α,β) → Beta(α+Σx, β+n-Σx)
    Poisson → Gamma(α,β) → Gamma(α+Σx, β+n)
    Normal (known σ²) → Normal(μ₀,σ₀²) → Normal(weighted mean, ...)
    Normal (known μ) → Inverse-Gamma → Inverse-Gamma
    Multinomial → Dirichlet → Dirichlet

** Multivariate normal                                          :stats:
*** Definition
    X ~ N(μ, Σ) where μ ∈ ℝⁿ, Σ is n×n positive definite
    f(x) = (2π)^{-n/2} |Σ|^{-1/2} exp(-½(x-μ)^T Σ^{-1} (x-μ))

*** Properties
    Marginals are normal: X_i ~ N(μ_i, Σ_{ii})
    Conditionals are normal:
      X₁|X₂=x₂ ~ N(μ₁ + Σ₁₂Σ₂₂⁻¹(x₂-μ₂), Σ₁₁ - Σ₁₂Σ₂₂⁻¹Σ₂₁)
    Affine transform: AX + b ~ N(Aμ+b, AΣA^T)
    Uncorrelated ⟹ independent (only for multivariate normal)

*** Mahalanobis distance
    d²(x) = (x-μ)^T Σ^{-1} (x-μ)
    d²(X) ~ χ²(n) when X ~ N(μ,Σ)

** Fisher information & estimation                              :stats:
*** Score function
    s(θ) = ∂/∂θ ln f(x;θ)
    E[s(θ)] = 0

*** Fisher information
    I(θ) = E[s(θ)²] = -E[∂²/∂θ² ln f(x;θ)]
    n observations: I_n(θ) = n I(θ) for iid data
    Matrix form: I_{ij}(θ) = -E[∂²/∂θ_i∂θ_j ln f(x;θ)]

*** Cramér-Rao lower bound
    Var(T) ≥ [∂/∂θ E[T]]² / I(θ)
    For unbiased T: Var(T) ≥ 1/I(θ)
    Multivariate: Cov(T) ≥ I(θ)⁻¹  (matrix inequality)
    Efficiency: T is efficient if it attains the bound

*** Sufficient statistics (Fisher-Neyman)
    T(X) sufficient for θ iff f(x;θ) = g(T(x),θ) h(x)

** Asymptotic theory                                            :stats:
*** Central limit theorem
    √n(X̄_n - μ)/σ →_d N(0,1)  for iid X_i with E[X]=μ, Var(X)=σ²

*** Multivariate CLT
    √n(X̄_n - μ) →_d N(0, Σ)

*** Delta method
    If √n(T_n - θ) →_d N(0, σ²), then
    √n(g(T_n) - g(θ)) →_d N(0, σ²[g'(θ)]²)
    Multivariate: √n(g(T_n) - g(θ)) →_d N(0, ∇g^T Σ ∇g)

*** Berry-Esseen bound
    sup_x |P(√n(X̄-μ)/σ ≤ x) - Φ(x)| ≤ Cρ/(σ³√n)
    where ρ = E[|X-μ|³], C ≤ 0.4748

*** MLE asymptotics
    θ̂_MLE →_p θ  (consistent)
    √n(θ̂ - θ) →_d N(0, I(θ)⁻¹)  (asymptotically efficient)

** Maximum entropy distributions                                :stats:
    Given constraints, max entropy dist is:
    Constraint E[X]=μ, X∈[a,b] → Uniform
    Constraint E[X]=μ, X≥0 → Exponential
    Constraint E[X]=μ, E[X²]=σ²+μ², X∈ℝ → Normal
    Constraint E[ln X]=ψ, X>0 → Gamma

** Information theory                                           :stats:
*** Entropy
    H(X) = -Σ p(x) ln p(x)          (discrete)
    h(X) = -∫ f(x) ln f(x) dx       (differential)
    H(X,Y) = H(X) + H(Y|X)          (chain rule)
    H(X) ≤ ln|support|               (max for uniform)

*** KL divergence
    D_{KL}(P||Q) = Σ p(x) ln(p(x)/q(x)) ≥ 0
    D_{KL}(P||Q) = 0 iff P = Q
    Not symmetric: D_{KL}(P||Q) ≠ D_{KL}(Q||P) in general

*** Mutual information
    I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X) = D_{KL}(P_{XY}||P_X P_Y)
    I(X;Y) ≥ 0, equality iff X,Y independent

* Fourier Analysis                                              :dlmf1_14:
** Fourier series                                               :as_fourier:
    f(x) = a₀/2 + Σ_{n=1}^∞ [a_n cos(nx) + b_n sin(nx)]
    a_n = (1/π) ∫_{-π}^π f(x) cos(nx) dx
    b_n = (1/π) ∫_{-π}^π f(x) sin(nx) dx
    Complex form: f(x) = Σ_{n=-∞}^∞ c_n e^{inx}
    c_n = (1/2π) ∫_{-π}^π f(x) e^{-inx} dx

** Fourier transform                                            :dlmf1_14:
    F̂(ω) = ∫_{-∞}^∞ f(t) e^{-iωt} dt
    f(t) = (1/2π) ∫_{-∞}^∞ F̂(ω) e^{iωt} dω
    Note: conventions vary (2π placement, sign)

** Standard transform pairs
    e^{-a|t|} ↔ 2a/(a²+ω²)
    e^{-at²} ↔ √(π/a) e^{-ω²/(4a)}
    rect(t) ↔ sinc(ω/2)  [sinc(x) = sin(x)/x]
    δ(t) ↔ 1
    1 ↔ 2π δ(ω)
    e^{iω₀t} ↔ 2π δ(ω-ω₀)

** Convolution theorems
    f*g ↔ F̂·Ĝ                 (convolution → multiplication)
    f·g ↔ (1/2π) F̂*Ĝ          (multiplication → convolution)

** Parseval / Plancherel                                        :dlmf1_14:
    ∫ |f(t)|² dt = (1/2π) ∫ |F̂(ω)|² dω

** Sampling theorem (Shannon-Nyquist)
    If f is bandlimited to [-W,W]:
    f(t) = Σ_{n=-∞}^∞ f(n/(2W)) sinc(2W(t - n/(2W)))
    Minimum sample rate: 2W

** Poisson summation                                            :dlmf1_8:
    Σ_{n=-∞}^∞ f(n) = Σ_{k=-∞}^∞ F̂(2πk)

* Laplace & Z-Transforms
** Laplace transform                                            :as_laplace:
    F(s) = ∫_0^∞ f(t) e^{-st} dt
    f'(t) ↔ sF(s) - f(0)
    f''(t) ↔ s²F(s) - sf(0) - f'(0)
    ∫_0^t f(τ)dτ ↔ F(s)/s
    e^{at}f(t) ↔ F(s-a)
    t^n f(t) ↔ (-1)^n F^{(n)}(s)

** Standard Laplace pairs
    1 ↔ 1/s
    t^n ↔ n!/s^{n+1}
    e^{at} ↔ 1/(s-a)
    sin(ωt) ↔ ω/(s²+ω²)
    cos(ωt) ↔ s/(s²+ω²)
    t e^{at} ↔ 1/(s-a)²
    δ(t) ↔ 1

** Z-transform
    X(z) = Σ_{n=0}^∞ x[n] z^{-n}
    Standard pairs:
    δ[n] ↔ 1
    u[n] ↔ z/(z-1)
    a^n u[n] ↔ z/(z-a)
    n a^n u[n] ↔ az/(z-a)²

** Final/initial value theorems
    Laplace: lim_{t→∞} f(t) = lim_{s→0} sF(s)  (if limit exists)
    Laplace: lim_{t→0+} f(t) = lim_{s→∞} sF(s)
    Z: lim_{n→∞} x[n] = lim_{z→1} (z-1)X(z)

* Additional Integral Transforms
** Mellin transform
    M{f}(s) = ∫_0^∞ x^{s-1} f(x) dx
    Inverse: f(x) = (1/2πi) ∫_{c-i∞}^{c+i∞} x^{-s} M{f}(s) ds
    Convolution: M{f*_M g}(s) = M{f}(s) · M{g}(s)
    where (f *_M g)(x) = ∫_0^∞ f(x/t) g(t) dt/t

*** Standard Mellin pairs
    e^{-x} ↔ Γ(s),  Re(s) > 0
    (1+x)^{-a} ↔ Γ(s)Γ(a-s)/Γ(a),  0 < Re(s) < Re(a)
    x^a e^{-x} ↔ Γ(s+a),  Re(s+a) > 0

*** Connection to other transforms
    Mellin of f at s = Laplace of f(e^{-t}) at s (via x=e^{-t})
    Mellin of f at s relates to Fourier of f(e^t) at Im(s)

** Hilbert transform
    H{f}(x) = (1/π) PV ∫_{-∞}^∞ f(t)/(x-t) dt
    H{H{f}} = -f  (involutory up to sign)
    F{H{f}}(ω) = -i sgn(ω) F̂(ω)  (acts as -i sgn in frequency)
    Parseval: ∫ f(x) H{g}(x) dx = -∫ H{f}(x) g(x) dx

*** Standard Hilbert pairs
    H{cos ωt} = sin ωt
    H{sin ωt} = -cos ωt
    H{1/(x²+a²)} = x/(a(x²+a²))
    H{χ_{[a,b]}(x)} = (1/π) ln|(x-a)/(x-b)|

** Hankel transform (Fourier-Bessel)
    H_ν{f}(k) = ∫_0^∞ f(r) J_ν(kr) r dr
    Self-reciprocal: H_ν{H_ν{f}} = f
    Transforms radial parts of 2D/3D Fourier transforms

* Table of Definite Integrals                                   :as_chapter5:
** Algebraic & rational
    ∫_0^∞ dx/(1+x²) = π/2
    ∫_0^∞ x^{p-1}/(1+x) dx = π/sin(pπ),  0 < p < 1
    ∫_0^∞ x^{p-1}/(1+x^n) dx = (π/n)/sin(pπ/n),  0 < p < n
    ∫_0^1 x^{a-1}(1-x)^{b-1} dx = B(a,b)

** Logarithmic
    ∫_0^1 ln x / (1-x) dx = -π²/6 = -ζ(2)
    ∫_0^1 ln x / (1+x) dx = -π²/12
    ∫_0^∞ ln x / (1+x²) dx = 0
    ∫_0^1 x^{n-1} ln^m(1/x) dx = m!/n^{m+1}
    ∫_0^1 ln Γ(x) dx = ½ ln(2π)

** Trigonometric definite integrals
    ∫_0^{π/2} sin^n x dx = ∫_0^{π/2} cos^n x dx = √π Γ((n+1)/2) / (2Γ(n/2+1))
    ∫_0^π sin^n x dx = 2 ∫_0^{π/2} sin^n x dx  (n even or odd)
    ∫_0^{2π} dθ/(a+b cos θ) = 2π/√(a²-b²),  a > |b|
    ∫_0^∞ sin²(x)/x² dx = π/2
    ∫_0^∞ sin(ax)/sinh(πx) dx = ½ tanh(a/2)

** Exponential & Gaussian
    ∫_0^∞ x^n e^{-ax} dx = n!/a^{n+1},  a > 0
    ∫_0^∞ x^{2n} e^{-ax²} dx = (2n-1)!!/(2^{n+1} a^n) · √(π/a)
    ∫_0^∞ x^{2n+1} e^{-ax²} dx = n!/(2a^{n+1})
    ∫_0^∞ e^{-ax²-b/x²} dx = ½√(π/a) e^{-2√(ab)},  a,b > 0
    ∫_0^∞ e^{-x²} cos(2bx) dx = (√π/2) e^{-b²}
    ∫_{-∞}^∞ e^{-(ax²+bx+c)} dx = √(π/a) e^{(b²-4ac)/(4a)}

** Bessel-related integrals                                     :dlmf10_22:as11_4:
    ∫_0^∞ J_0(ax) e^{-px} dx = 1/√(p²+a²)
    ∫_0^∞ J_ν(at) t^{ν+1} e^{-pt} dt = (2a)^ν Γ(ν+3/2) / (√π(p²+a²)^{ν+3/2})
    ∫_0^∞ J_0(at) J_0(bt) t dt = δ(a-b)/a  (orthogonality, a,b>0)
    Weber-Schafheitlin: ∫_0^∞ J_μ(at) J_ν(bt) t^{-λ} dt  (see DLMF 10.22.63)

* Differential Equations
** Standard ODE solutions
*** Constant coefficients
    ay'' + by' + cy = 0
    Characteristic eqn: ar² + br + c = 0
    Distinct real roots r₁,r₂: y = C₁e^{r₁x} + C₂e^{r₂x}
    Repeated root r: y = (C₁ + C₂x)e^{rx}
    Complex r = α ± βi: y = e^{αx}(C₁cos βx + C₂sin βx)

*** Euler-Cauchy equation
    x²y'' + bxy' + cy = 0
    Try y = x^m: m(m-1) + bm + c = 0

*** Airy equation                                               :dlmf9_2:as10_4:
    y'' - xy = 0
    Solutions: Ai(x), Bi(x)
    Ai(0) = 1/(3^{2/3}Γ(2/3)), Bi(0) = 1/(3^{1/6}Γ(2/3))

** Green's functions
*** Definition
    Ly = f(x) where L is a linear operator
    G(x,ξ): LG = δ(x-ξ), then y(x) = ∫ G(x,ξ) f(ξ) dξ

*** Free-space Green's functions
    -∇²G = δ in ℝ²: G = -ln|r|/(2π)
    -∇²G = δ in ℝ³: G = 1/(4π|r|)
    (∂_t - κ∇²)G = δ in ℝ³: G = (4πκt)^{-3/2} exp(-|r|²/(4κt))

** Sturm-Liouville theory                                      :dlmf1_18:
    d/dx[p(x)y'] + [q(x) + λw(x)]y = 0
    Self-adjoint form; eigenvalues real, eigenfunctions orthogonal:
    ∫_a^b y_m y_n w(x) dx = 0 for m ≠ n
    Completeness: f(x) = Σ c_n y_n(x) with c_n = ∫ f y_n w dx / ∫ y_n² w dx

** Partial Differential Equations
*** Classification (2nd order, 2 variables)
    Au_{xx} + 2Bu_{xy} + Cu_{yy} + ... = 0
    B²-AC > 0: hyperbolic (wave-like)
    B²-AC = 0: parabolic (diffusion-like)
    B²-AC < 0: elliptic (steady-state)

*** Heat equation
    u_t = κ ∇²u
    Fundamental solution (ℝⁿ): G(x,t) = (4πκt)^{-n/2} exp(-|x|²/(4κt))
    Separation on [0,L]: u(x,t) = Σ B_n sin(nπx/L) exp(-κn²π²t/L²)
    Maximum principle: max of u on domain occurs on boundary or at t=0

*** Wave equation
    u_{tt} = c² ∇²u
    d'Alembert (1D): u(x,t) = f(x-ct) + g(x+ct)
    Separation on [0,L]: u = Σ [A_n cos(nπct/L) + B_n sin(nπct/L)] sin(nπx/L)
    Fundamental solution (3D): G = δ(|x|-ct)/(4πc²t)  (sharp wavefront)
    Fundamental solution (2D): G = H(ct-|x|)/(2πc√(c²t²-|x|²))  (no sharp tail)

*** Laplace & Poisson equations
    ∇²u = 0 (Laplace),  ∇²u = f (Poisson)
    Mean value property: u(x₀) = average of u on any sphere centered at x₀
    Maximum principle: harmonic functions attain extrema on boundary
    Separation in sphere: u(r,θ) = Σ (A_n r^n + B_n r^{-(n+1)}) P_n(cos θ)

*** Helmholtz equation
    ∇²u + k²u = 0
    Separation in cylindrical: Bessel functions J_m(kr), Y_m(kr)
    Separation in spherical: spherical Bessel j_n(kr), spherical harmonics Y_n^m(θ,φ)

*** Schrödinger equation (time-independent)                     :quantum:
    -ℏ²/(2m) ∇²ψ + V(x)ψ = Eψ
    Free particle: ψ = e^{ikx}, E = ℏ²k²/(2m)
    Harmonic oscillator: E_n = ℏω(n+½), ψ_n ∝ H_n(ξ) e^{-ξ²/2}, ξ=x√(mω/ℏ)

*** Boundary conditions
    Dirichlet: u = g on ∂Ω
    Neumann: ∂u/∂n = g on ∂Ω
    Robin: αu + β ∂u/∂n = g on ∂Ω
    Sommerfeld radiation: lim_{r→∞} r^{(n-1)/2} (∂u/∂r - iku) = 0

*** Weak formulation (variational)                              :fenics:
    Strong: -∇·(a∇u) = f in Ω, u = 0 on ∂Ω
    Weak: find u ∈ H¹₀(Ω) s.t. ∫ a∇u·∇v dx = ∫ fv dx ∀v ∈ H¹₀(Ω)
    Lax-Milgram: unique solution if bilinear form is continuous and coercive
    Céa's lemma: ||u-u_h|| ≤ (C/α) inf_{v_h} ||u-v_h||

* Vector Calculus & Differential Geometry
** Div, grad, curl identities
*** Operator identities
    ∇×(∇f) = 0                (curl of gradient = 0)
    ∇·(∇×F) = 0               (div of curl = 0)
    ∇×(∇×F) = ∇(∇·F) - ∇²F
    ∇·(fF) = f(∇·F) + F·(∇f)
    ∇×(fF) = f(∇×F) + (∇f)×F
    ∇(F·G) = (F·∇)G + (G·∇)F + F×(∇×G) + G×(∇×F)

** Integral theorems
*** Divergence (Gauss)
    ∫_V (∇·F) dV = ∮_S F·dA

*** Stokes
    ∮_C F·dr = ∫_S (∇×F)·dA

*** Green's theorems
    First:  ∫_V (f∇²g + ∇f·∇g) dV = ∮_S f(∇g·n̂) dA
    Second: ∫_V (f∇²g - g∇²f) dV = ∮_S (f∇g - g∇f)·n̂ dA

** Coordinate systems
*** Cylindrical (r,φ,z)
    ∇f = (∂f/∂r)r̂ + (1/r)(∂f/∂φ)φ̂ + (∂f/∂z)ẑ
    ∇²f = (1/r)∂/∂r(r ∂f/∂r) + (1/r²)∂²f/∂φ² + ∂²f/∂z²
    ∇·F = (1/r)∂(rF_r)/∂r + (1/r)∂F_φ/∂φ + ∂F_z/∂z

*** Spherical (r,θ,φ)
    ∇f = (∂f/∂r)r̂ + (1/r)(∂f/∂θ)θ̂ + (1/(r sinθ))(∂f/∂φ)φ̂
    ∇²f = (1/r²)∂/∂r(r²∂f/∂r) + (1/(r²sinθ))∂/∂θ(sinθ ∂f/∂θ)
           + (1/(r²sin²θ))∂²f/∂φ²

** Tensor Calculus & Index Notation
*** Einstein summation convention
    Repeated upper/lower index implies summation:
    a^i b_i ≡ Σ_i a^i b_i
    A^i_j x^j ≡ Σ_j A^i_j x^j

*** Kronecker delta & Levi-Civita
    δ^i_j = 1 if i=j, 0 otherwise
    δ^i_j δ^j_k = δ^i_k
    ε_{ijk} = +1 (even perm), -1 (odd perm), 0 (repeated index)
    ε_{ijk} ε_{ilm} = δ_{jl}δ_{km} - δ_{jm}δ_{kl}
    ε_{ijk} ε_{ijm} = 2δ_{km}
    ε_{ijk} ε_{ijk} = 6

*** Vector operations in index notation
    (A×B)_i = ε_{ijk} A_j B_k
    A·B = A_i B_i
    (A×B)·C = ε_{ijk} A_i B_j C_k  (scalar triple product)
    [∇×F]_i = ε_{ijk} ∂_j F_k
    ∇·F = ∂_i F_i

*** Metric tensor & covariant derivative
    ds² = g_{ij} dx^i dx^j
    g^{ik} g_{kj} = δ^i_j
    Christoffel symbols: Γ^k_{ij} = ½g^{kl}(∂_i g_{jl} + ∂_j g_{il} - ∂_l g_{ij})
    Covariant derivative of vector: ∇_i V^j = ∂_i V^j + Γ^j_{ik} V^k
    Covariant derivative of covector: ∇_i V_j = ∂_i V_j - Γ^k_{ij} V_k
    Covariant derivative of tensor: ∇_k T^{ij} = ∂_k T^{ij} + Γ^i_{kl}T^{lj} + Γ^j_{kl}T^{il}

*** Riemann curvature tensor
    R^l_{ijk} = ∂_j Γ^l_{ik} - ∂_k Γ^l_{ij} + Γ^l_{jm}Γ^m_{ik} - Γ^l_{km}Γ^m_{ij}
    Ricci tensor: R_{ij} = R^k_{ikj}
    Scalar curvature: R = g^{ij} R_{ij}
    Bianchi identity: ∇_l R^i_{jkm} + ∇_k R^i_{jml} + ∇_m R^i_{jlk} = 0

*** Differential forms (brief)
    Exterior derivative: d(dω) = 0
    Stokes (general): ∫_M dω = ∫_{∂M} ω
    Hodge star in ℝ³: *(dx∧dy) = dz,  *(dx) = dy∧dz, etc.
    div F = *d(*F̃),  curl F = (*dF̃)^♯  where F̃ is the 1-form dual to F

*** Volume elements
    Cartesian: dV = dx dy dz
    Cylindrical: dV = r dr dφ dz
    Spherical: dV = r² sinθ dr dθ dφ
    General: dV = √|g| dx¹ dx² ... dx^n  where g = det(g_{ij})

* Numerical Methods
** Floating-point Arithmetic                                    :ieee754:
*** IEEE 754 basics
    Single (float32): 1 sign + 8 exp + 23 mantissa, ε = 2^{-23} ≈ 1.19e-7
    Double (float64): 1 sign + 11 exp + 52 mantissa, ε = 2^{-52} ≈ 2.22e-16
    Machine epsilon ε: smallest ε s.t. fl(1+ε) ≠ 1

*** Ranges
    float32: ±1.18e-38 to ±3.40e+38
    float64: ±2.23e-308 to ±1.80e+308
    Subnormals extend lower range at cost of precision

*** Rounding model
    fl(a ⊕ b) = (a ⊕ b)(1+δ),  |δ| ≤ ε  for ⊕ ∈ {+,-,×,÷}
    Catastrophic cancellation: a - b when a ≈ b loses relative precision
    Fix: rewrite (e.g., use (a²-b²) = (a-b)(a+b), or Taylor for small args)
    log1p(x), expm1(x): use these instead of log(1+x), exp(x)-1 for small x

*** Condition number
    cond(f,x) = |x f'(x)/f(x)|
    Relative error amplification: |Δf/f| ≈ cond(f,x) · |Δx/x|
    cond(A) = ||A|| · ||A⁻¹|| = σ_max/σ_min (for 2-norm)
    Rule of thumb: lose log₁₀(cond) decimal digits in solve

** Finite Difference Stencils
*** First derivative
    Forward:  f'(x) ≈ [f(x+h) - f(x)] / h                         O(h)
    Backward: f'(x) ≈ [f(x) - f(x-h)] / h                         O(h)
    Central:  f'(x) ≈ [f(x+h) - f(x-h)] / (2h)                    O(h²)
    4th order: f'(x) ≈ [-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)]/(12h)  O(h⁴)

*** Second derivative
    Central:  f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h²            O(h²)
    4th order: f''(x) ≈ [-f(x+2h)+16f(x+h)-30f(x)+16f(x-h)-f(x-2h)]/(12h²)  O(h⁴)

*** Laplacian (2D, grid spacing h)
    5-point: ∇²f ≈ [f(i+1,j)+f(i-1,j)+f(i,j+1)+f(i,j-1)-4f(i,j)]/h²   O(h²)
    9-point: include diagonal neighbors, O(h⁴) with appropriate weights

*** Optimal step size (balancing truncation vs roundoff)
    Forward diff: h_opt ~ ε^{1/2} ≈ 1.5e-8 (float64)
    Central diff: h_opt ~ ε^{1/3} ≈ 6e-6 (float64)
    Complex step: f'(x) ≈ Im[f(x+ih)]/h, works for any small h (no cancellation)

** Interpolation
*** Lagrange interpolation
    p(x) = Σ_{i=0}^n y_i Π_{j≠i} (x-x_j)/(x_i-x_j)
    Error: f(x)-p(x) = f^{(n+1)}(ξ)/(n+1)! · Π(x-x_i)

*** Newton divided differences
    p(x) = f[x₀] + f[x₀,x₁](x-x₀) + f[x₀,x₁,x₂](x-x₀)(x-x₁) + ...
    f[x_i,...,x_{i+k}] = (f[x_{i+1},...,x_{i+k}] - f[x_i,...,x_{i+k-1}]) / (x_{i+k}-x_i)

*** Runge phenomenon
    Equispaced nodes on [-1,1]: interpolation error grows near endpoints
    Fix: Chebyshev nodes x_k = cos((2k+1)π/(2n+2)), k=0,...,n
    Chebyshev gives near-optimal |Π(x-x_i)| bound: 2^{-n}

*** Cubic spline
    Piecewise cubic, C² continuous
    Natural: S''(a)=S''(b)=0
    Not-a-knot: S''' continuous at x₁ and x_{n-1}
    Error: O(h⁴) for natural spline with smooth f

** Quadrature
*** Newton-Cotes
    Trapezoidal: ∫_a^b f dx ≈ h[f(a)/2 + Σ f(a+kh) + f(b)/2], error O(h²)
    Simpson's: ∫_a^b f dx ≈ (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + ... + f_n], error O(h⁴)
    Simpson's 3/8: (3h/8)[f₀ + 3f₁ + 3f₂ + 2f₃ + 3f₄ + ...], error O(h⁴)

*** Gaussian quadrature
    ∫_{-1}^1 f(x) dx ≈ Σ_{i=1}^n w_i f(x_i)
    Exact for polynomials of degree ≤ 2n-1
    Nodes x_i are zeros of P_n(x) (Legendre)
    Gauss-Laguerre: weight e^{-x} on [0,∞)
    Gauss-Hermite: weight e^{-x²} on (-∞,∞)

*** Low-order Gauss-Legendre nodes & weights
    n=1: x=0, w=2
    n=2: x=±1/√3, w=1
    n=3: x=0,±√(3/5), w=8/9, 5/9

*** Romberg integration (Richardson extrapolation)
    T(h), T(h/2), ... from trapezoidal rule
    R_{k+1} = (4^k R_k(h/2) - R_k(h)) / (4^k - 1)
    Successively eliminates error terms: O(h²)→O(h⁴)→O(h⁶)→...

*** Adaptive quadrature
    Compare Q(h) and Q(h/2); subdivide where |Q(h)-Q(h/2)| > tol
    Gauss-Kronrod: embed n-point Gauss in (2n+1)-point rule for error est

** Root-finding
*** Newton's method
    x_{n+1} = x_n - f(x_n)/f'(x_n)
    Convergence: quadratic near simple root
    Modified for multiple roots: x_{n+1} = x_n - m·f(x_n)/f'(x_n)

*** Secant method
    x_{n+1} = x_n - f(x_n)(x_n - x_{n-1})/(f(x_n) - f(x_{n-1}))
    Convergence order: golden ratio φ ≈ 1.618

*** Bisection
    Always converges, linear rate, gain 1 bit per step
    After k steps: interval width = (b-a)/2^k

*** Brent's method
    Combines bisection, secant, inverse quadratic interpolation
    Guaranteed convergence with superlinear speed in practice

** Iterative Linear Solvers
*** Conjugate gradient (CG)
    For A symmetric positive definite: Ax = b
    Convergence: ||e_k||_A ≤ 2((√κ-1)/(√κ+1))^k ||e_0||_A  where κ=cond(A)
    With preconditioner M: solve M⁻¹Ax = M⁻¹b, effective κ = cond(M⁻¹A)
    Cost per iteration: one matrix-vector multiply + O(n)

*** GMRES
    For general (nonsymmetric) systems
    Minimizes ||b-Ax_k|| over Krylov subspace K_k(A,r₀)
    No short recurrence: storage grows with iteration count
    Restart after m steps (GMRES(m)) to bound memory

*** Preconditioning strategies
    Jacobi: M = diag(A), cheap, often weak
    ILU(0): incomplete LU, good general-purpose
    Algebraic multigrid (AMG): near-optimal for elliptic PDEs
    Rule of thumb: good preconditioner makes κ(M⁻¹A) = O(1) or O(log n)

** Numerical Linear Algebra
*** Factorization costs (n×n dense)
    LU: (2/3)n³ flops
    Cholesky: (1/3)n³ flops (SPD only)
    QR (Householder): (4/3)n³ flops
    SVD: ~11n³ flops
    Eigenvalues (symmetric): ~(4/3)n³ flops (QR algorithm)

*** Sparse direct solvers
    Band matrix (bandwidth b): O(nb²)
    2D FEM (n unknowns): O(n^{3/2}) fill-in with nested dissection
    3D FEM: O(n²) fill-in — often prefer iterative

*** Stability
    Backward stability: computed x̂ solves (A+δA)x̂ = b+δb with ||δA||/||A|| = O(ε)
    Partial pivoting (PA=LU): backward stable in practice
    Normal equations (A^TAx = A^Tb): cond² — avoid; use QR or SVD instead

** ODE Solvers
*** Euler (explicit)
    y_{n+1} = y_n + h f(t_n, y_n)
    Error: O(h), 1 function eval/step

*** RK4 (classical)
    k₁ = f(t, y)
    k₂ = f(t+h/2, y+hk₁/2)
    k₃ = f(t+h/2, y+hk₂/2)
    k₄ = f(t+h, y+hk₃)
    y_{n+1} = y_n + (h/6)(k₁+2k₂+2k₃+k₄)
    Error: O(h⁴), 4 evals/step

*** Dormand-Prince (RK45)
    Embedded 4th/5th order pair for adaptive stepping
    Default in most ODE libraries (e.g., ode45, DOP853)
    Local error est from difference of 4th and 5th order

*** BDF (backward differentiation formulas)
    Implicit, A-stable (BDF1,BDF2), good for stiff problems
    BDF1 (backward Euler): y_{n+1} = y_n + h f(t_{n+1}, y_{n+1})
    BDF2: y_{n+1} = (4/3)y_n - (1/3)y_{n-1} + (2h/3)f(t_{n+1}, y_{n+1})
    BDF≥7 unstable; BDF3-6 A(α)-stable

*** Stiffness detection (rule of thumb)
    Stiff if max|Re(λ)|/min|Re(λ)| >> 1 for Jacobian eigenvalues
    Stiff → use implicit (BDF, implicit RK, SDIRK)
    Non-stiff → use explicit (RK45, Adams-Bashforth)

*** Symplectic integrators (Hamiltonian systems)
    Störmer-Verlet (leapfrog): O(h²), preserves phase-space volume
    q_{n+½} = q_n + (h/2)p_n/m
    p_{n+1} = p_n + h F(q_{n+½})
    q_{n+1} = q_{n+½} + (h/2)p_{n+1}/m
    Energy error bounded (no drift), crucial for long-time integration

** FEM Basics                                                   :fenics:
*** Standard element families
    Lagrange P_k: continuous, piecewise degree-k polynomials
    P1 (linear): 3 DOF/triangle, 4 DOF/tet
    P2 (quadratic): 6 DOF/triangle, 10 DOF/tet
    Convergence: ||u-u_h||_{L²} = O(h^{k+1}), ||u-u_h||_{H¹} = O(h^k)

*** Assembly pattern
    K_{IJ} = Σ_e ∫_{Ω_e} a(x) ∇φ_I · ∇φ_J dx  (stiffness)
    F_I = Σ_e ∫_{Ω_e} f(x) φ_I dx               (load)
    Map to reference element, use quadrature

* Statistical Tables
** Normal distribution (two-sided z_α)
    α=0.10: z=1.645
    α=0.05: z=1.960
    α=0.02: z=2.326
    α=0.01: z=2.576
    α=0.001: z=3.291
    Quick: P(|Z|>1)≈0.317, P(|Z|>2)≈0.046, P(|Z|>3)≈0.0027

** Normal tail probabilities (one-sided)
    P(Z>1.0) = 0.1587
    P(Z>1.5) = 0.0668
    P(Z>2.0) = 0.0228
    P(Z>2.5) = 0.0062
    P(Z>3.0) = 0.0013
    P(Z>4.0) = 3.17e-5
    P(Z>5.0) = 2.87e-7
    Quick approximation: P(Z>x) ≈ φ(x)/x for large x, φ = density

** Student-t critical values (two-tailed)
    ν\α    0.10    0.05    0.01
    1     6.314   12.706  63.657
    2     2.920    4.303   9.925
    3     2.353    3.182   5.841
    5     2.015    2.571   4.032
    10    1.812    2.228   3.169
    20    1.725    2.086   2.845
    30    1.697    2.042   2.750
    60    1.671    2.000   2.660
    120   1.658    1.980   2.617
    ∞     1.645    1.960   2.576

** χ² critical values (upper tail α)
    ν\α    0.10    0.05    0.01
    1     2.706    3.841   6.635
    2     4.605    5.991   9.210
    3     6.251    7.815  11.345
    5     9.236   11.070  15.086
    10   15.987   18.307  23.209
    20   28.412   31.410  37.566
    30   40.256   43.773  50.892
    Quick: for large ν, χ²_α ≈ ν(1 - 2/(9ν) + z_α√(2/(9ν)))³

** F-distribution critical values (α=0.05)
    F_{d1,d2}:
    d1\d2     5      10      20      ∞
    1      6.608   4.965   4.351   3.842
    2      5.786   4.103   3.493   2.996
    5      5.050   3.326   2.711   2.214
    10     4.735   2.978   2.348   1.831
    20     4.558   2.774   2.124   1.571
    ∞      4.365   2.538   1.843   1.000

** Kolmogorov-Smirnov critical values
    D_α for n observations (two-sided):
    α=0.10: D = 1.224/√n
    α=0.05: D = 1.358/√n
    α=0.01: D = 1.628/√n

** Correlation significance (two-tailed, α=0.05)
    Test: t = r√(n-2)/√(1-r²) with ν=n-2
    Quick critical |r| values:
    n=10: 0.632,  n=20: 0.444,  n=30: 0.361
    n=50: 0.279,  n=100: 0.197, n=200: 0.139

* Series Convergence Tests
** Ratio test
    L = lim |a_{n+1}/a_n|
    L < 1: converges absolutely
    L > 1: diverges
    L = 1: inconclusive

** Root test
    L = lim sup |a_n|^{1/n}
    Same criteria as ratio test; sometimes stronger

** Comparison test
    0 ≤ a_n ≤ b_n and Σb_n converges ⟹ Σa_n converges
    Limit comparison: lim a_n/b_n = c > 0 ⟹ both converge or both diverge

** Integral test
    f(x) ≥ 0 decreasing: Σ f(n) converges iff ∫_1^∞ f(x)dx converges
    Remainder bound: ∫_{n+1}^∞ f(x)dx ≤ R_n ≤ ∫_n^∞ f(x)dx

** Alternating series test (Leibniz)
    Σ(-1)^n b_n with b_n > 0, b_n decreasing, b_n → 0: converges
    Error bound: |R_n| ≤ b_{n+1}

** p-series
    Σ 1/n^p: converges for p > 1, diverges for p ≤ 1

** Absolute vs conditional
    Absolute convergence ⟹ convergence (not converse)
    Riemann rearrangement: conditionally convergent series can be
    rearranged to sum to any value (or diverge)

** Radius of convergence
    R = 1/lim sup |a_n|^{1/n} = lim |a_n/a_{n+1}|  (when limit exists)
    Power series converges absolutely for |z| < R, diverges for |z| > R

* Quick Reference & Back-of-Envelope
** Small-angle & common approximations
    sin x ≈ x - x³/6          (|x| < 0.5: error < 3e-4)
    cos x ≈ 1 - x²/2          (|x| < 0.5: error < 3e-3)
    tan x ≈ x + x³/3          (|x| < 0.5)
    e^x ≈ 1 + x + x²/2       (|x| < 0.5: error < 2e-2)
    ln(1+x) ≈ x - x²/2       (|x| < 0.5)
    (1+x)^n ≈ 1 + nx          (|nx| << 1)
    √(1+x) ≈ 1 + x/2 - x²/8 (|x| < 1)
    1/(1+x) ≈ 1 - x + x²     (|x| < 1)

** Rule of 72
    Doubling time ≈ 72 / (rate in %)
    At r% per period, doubles in ~72/r periods
    More precise: 69.3/r (continuous), 70/r (quick mental math)

** Stirling for small n (exact n! and approximation)
    0!=1  1!=1  2!=2  3!=6  4!=24  5!=120  6!=720
    7!=5040  8!=40320  9!=362880  10!=3628800
    12!=479001600  15!=1.307e12  20!=2.432e18
    Stirling error: n! = √(2πn)(n/e)^n · e^{θ/(12n)}, 0<θ<1

** Powers of 2
    2⁰=1  2¹=2  2²=4  2³=8  2⁴=16  2⁵=32  2⁶=64  2⁷=128
    2⁸=256  2⁹=512  2¹⁰=1024  2¹¹=2048  2¹²=4096
    2¹⁶=65536  2²⁰=1048576≈10⁶  2³⁰≈10⁹  2³²≈4.29×10⁹
    2⁴⁰≈10¹²  2⁵⁰≈10¹⁵  2⁶⁴≈1.84×10¹⁹
    Rule: 2¹⁰ ≈ 10³  (1024 ≈ 1000, error ~2.4%)

** Powers of 10 and logs
    log₁₀(2) = 0.30103    log₁₀(3) = 0.47712    log₁₀(7) = 0.84510
    ln(2) = 0.6931   ln(3) = 1.0986   ln(5) = 1.6094   ln(7) = 1.9459
    ln(10) = 2.3026
    Number of digits of n: ⌊log₁₀(n)⌋ + 1
    Number of bits of n: ⌊log₂(n)⌋ + 1

** Small primes & prime-counting
    2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
    73 79 83 89 97 101 103 107 109 113 127 131 137 139 149
    π(10)=4, π(100)=25, π(1000)=168, π(10⁴)=1229, π(10⁶)=78498
    PNT: π(x) ~ x/ln(x),  better: π(x) ~ li(x)

** Common Jacobians (change of variables)
    Polar (r,θ) → (x,y):        |J| = r
    Cylindrical (r,φ,z) → (x,y,z): |J| = r
    Spherical (r,θ,φ) → (x,y,z):   |J| = r² sinθ
    u=x+y, v=x-y → (x,y):       |J| = ½
    u=xy, v=x/y → (x,y):        |J| = 1/(2v)  [or equivalently |∂(x,y)/∂(u,v)| = 1/(2v)]

** Algorithm complexity (common operations)
*** Sorting
    Comparison-based lower bound: Ω(n log n)
    Merge sort: O(n log n) always, O(n) extra space
    Quicksort: O(n log n) average, O(n²) worst
    Heapsort: O(n log n) worst, in-place
    Radix sort: O(nk) for k-digit keys (not comparison-based)

*** Searching
    Binary search (sorted array): O(log n)
    Hash table lookup: O(1) average, O(n) worst
    Balanced BST (AVL, red-black): O(log n)

*** Matrix operations
    Dense multiply (n×n): O(n³), Strassen O(n^{2.81}), best known O(n^{2.37})
    Dense solve (LU): O(n³)
    Dense SVD: O(n³)
    Sparse multiply (m nonzeros): O(m)
    FFT: O(n log n)
    Matrix-vector (sparse, m nonzeros): O(m)

** Dimensional analysis (Buckingham π theorem)
    n variables, k independent dimensions → n-k dimensionless groups
    Procedure: choose k repeating variables, form π groups with remaining
    All physical laws expressible in terms of dimensionless π groups

** Probability rules of thumb
    Normal 68-95-99.7 rule: P(|X-μ| < kσ) for k=1,2,3
    Poisson: if λ>20, approximate by N(λ,λ)
    Binomial: if np>5 and n(1-p)>5, approximate by N(np, np(1-p))
    Sample size for proportion: n ≈ (z/ε)² p(1-p); conservative p=0.5: n ≈ (z/(2ε))²
    Sample size for mean: n ≈ (zσ/ε)²

* Constants & Conversions
** Mathematical constants
    π  = 3.14159 26535 89793 23846...
    e  = 2.71828 18284 59045 23536...
    γ  = 0.57721 56649 01532 86060...  (Euler-Mascheroni)
    φ  = 1.61803 39887 49894 84820...  (golden ratio, (1+√5)/2)
    √2 = 1.41421 35623 73095 04880...
    √3 = 1.73205 08075 68877 29353...
    √5 = 2.23606 79774 99789 69640...
    ln 2 = 0.69314 71805 59945 30941...
    ln 10 = 2.30258 50929 94045 68402...
    log₁₀(e) = 0.43429 44819 03251 82765...
    Catalan G = 0.91596 55941 77219 01505...
    Apéry ζ(3) = 1.20205 69031 59594 28540...

** Powers of π
    π² = 9.86960 44010 89358...
    π³ = 31.00627 66802 99820...
    π⁴ = 97.40909 10340 02437...
    1/π = 0.31830 98861 83790 67154...
    √π = 1.77245 38509 05516 02730...
    2π = 6.28318 53071 79586 47692...
    π/2 = 1.57079 63267 94896 61923...
    π/4 = 0.78539 81633 97448 30961...

** Trigonometric exact values
    sin(π/6) = 1/2         cos(π/6) = √3/2       tan(π/6) = 1/√3
    sin(π/4) = √2/2        cos(π/4) = √2/2       tan(π/4) = 1
    sin(π/3) = √3/2        cos(π/3) = 1/2         tan(π/3) = √3
    sin(π/12) = (√6-√2)/4  cos(π/12) = (√6+√2)/4
    sin(π/5) = √(10-2√5)/4 cos(π/5) = (1+√5)/4
    sin(π/8) = √(2-√2)/2   cos(π/8) = √(2+√2)/2

** Degrees ↔ radians
    1° = π/180 ≈ 0.01745 rad
    1 rad = 180/π ≈ 57.296°
    30°=π/6  45°=π/4  60°=π/3  90°=π/2  180°=π  360°=2π

** Unit conversions (for dimensional analysis)
    1 inch = 2.54 cm (exact)
    1 mile = 1.60934 km
    1 lb = 0.45359 kg
    1 gallon (US) = 3.78541 L
    °C = (°F - 32) × 5/9
    1 atm = 101325 Pa = 760 mmHg
    1 eV = 1.60218e-19 J
    1 cal = 4.184 J (exact, thermochemical)
		    
Ara / アラ