Science & Education
φ = artanh(β) with γ = cosh φ, γβ = sinh φ, the e^φ Doppler factor, and the tanh(2φ) additivity check — where boosts just add.
rapidity-calculatorScience & Education
E_n = n²h²/(8mL²) for preset or custom particles: zero-point energy, level ladder to n = 12, λ_dB = 2L/n standing-wave check, transition photon wavelengths.
particle-in-a-boxMath & Numbers
Smallest k with a^k ≡ 1 (mod n) via φ(n) reduction: power table, minimality proof, cyclic subgroup <a>, primitive-root and maximal-order flags.
order-of-element-mod-nMath & Numbers
Decompose N(x)/D(x) into A/(x−r)^j + (Bx + C)/((x−α)² + β²) with long division for improper fractions and numeric residual verification.
partial-fraction-decomposerMath & Numbers
Partial-fraction inverse Laplace transform for proper rational functions: root factorization (real + conjugate pairs), exact coefficient system, term-by-term inversion.
inverse-laplace-calculatorDevelopment
Turn an OpenAPI 3.x or Swagger 2.0 spec (JSON/YAML) into an importable Postman Collection v2.1.0 with folders, variables, auth and example responses.
openapi-to-postman-collectionMath & Numbers
Rescale any recipe's ingredient list to a new serving count, converting cups/tbsp/tsp ⇄ grams/ml with per-ingredient densities and kitchen-friendly fraction output.
recipe-ingredient-scalerEducation
Compute Wright's F = Σ (1/2)^(n1+n2+1)(1+F_A) over common-ancestor paths, with reference values (full sib 0.25, half sib 0.125, first cousins 0.0625) and elevated recessive risk q² + F·q·(1−q). Derived from Wright 1922, Falconer & Mackay, Hartl & Clark 2007, StatPearls. Educational use only.
inbreeding-coefficientEducation
Compute Punnett square and affected/carrier probabilities for X-linked recessive and dominant traits. Carrier mother × unaffected father → 50% of sons affected; affected father → all daughters affected, no sons. Derived from Morgan 1910, Griffiths, Hartl & Clark 2007, StatPearls. Educational use only.
x-linked-inheritance-probabilityHealth
Convert serum/plasma insulin between µIU/mL and pmol/L. Factor 6.945 (WHO 1st IR 83/500, specific activity ~24 IU/nmol). Derived from WHO 83/500, Young 1987, StatPearls, Matthews 1985. Not medical advice.
insulin-unit-converterHealth
Calculate expected respiratory compensation PaCO₂ for metabolic acidosis: 1.5×HCO₃⁻ + 8 ± 2 mmHg. Classifies compensation as appropriate or flags mixed disorders. Derived from Albert 1967, Emmett 2016, LITFL, MDCalc. Not medical advice.
winters-formulaDesign
Capture a public web page and export observed colors, typography, spacing and components with a screenshot.
website-design-style-extractor