Science & Education
1/λ = R·Z²(1/n₁² − 1/n₂²) for hydrogen-like ions with reduced-mass correction: wavelength, photon energy, frequency, band, Pickering and Balmer notes.
rydberg-formula-calculatorMath & Numbers
Enumerate primes up to 10⁷ (or within [m, n]) with the classic sieve; shows base primes, crossing-out operations, exact counts, and up to 200 primes.
sieve-of-eratosthenesMath & Numbers
Numeric convergence tests for Σ aₙ: nth-term, ratio, root, p-series comparison, and integral test; alternating series classified by Leibniz with a sum estimate.
series-convergence-testerMath & Numbers
Union, intersection, difference, symmetric difference, and Cartesian product of two comma-separated sets, with cardinalities and capped pair display.
set-operations-calculatorMath & Numbers
Classical fourth-order Runge-Kutta for dy/dx = f(x, y): step table, k1–k4 stage details, final estimate, and optional exact-solution error column.
runge-kutta-rk4-solverMath & Numbers
Solve a·y'' + b·y' + c·y = f(x): characteristic roots, three homogeneous cases, undetermined coefficients with resonance, and C1/C2 from initial conditions.
second-order-ode-solverHealth
Compute Shannon H′ from species counts with H′max and Pielou evenness, in ln, log₂, or log₁₀ base.
shannon-diversity-indexUtilities
One pass replaces bit.ly + Campaign URL Builder + QR generator: GA4-validated UTM parameters, a deterministic self-hosted short link (FNV-1a base62 slug), QR pair, redirect snippets and a tracking CSV row.
short-url-utm-builder-qr-bundleEducation
10-fold series design: step count from a target, per-tube concentration table, and the 1:9 transfer recipe. Educational use only.
serial-dilution-plannerDevelopment
Pixel-accurate Google SERP preview for desktop and mobile, light and dark: current truncation budgets, 2025+ URL line formats, review stars and product rich results, byline dates, meters and diagnostics.
serp-preview-simulatorHealth
Calculate Qs/Qt = (CcO₂ − CaO₂)/(CcO₂ − CvO₂). PAO₂ from alveolar gas equation. Normal <5%; >20% severe shunt refractory to O₂. Defaults SvO₂=0.75, PvO₂=40 when mixed-venous values unavailable. Derived from Berggren 1942, StatPearls, LITFL, West. Not medical advice.
shunt-equationHealth
Calculate serum osmolality from routinely measured solutes. Conventional formula Osm = 2 × Na + Glucose/18 + BUN/2.8 (Na in mmol/L, glucose & BUN in mg/dL), or the SI-unit form Osm = 2 × Na + Glucose + Urea (all mmol/L). An optional ethanol term (EtOH/4.6) is available when a toxic-alcohol ingestion is being considered. Normal range ≈ 275–295 mOsm/kg. Values < 275 suggest hypo-osmolality (SIADH, primary polydipsia, adrenal insufficiency, hypothyroidism); values > 295 suggest hyperosmolality (hyperglycemia/HHS, hypernatremia, uremia, mannitol, toxic alcohols — the latter also raise the osmolar gap, so measure osmolality and compute the gap). Derived from Dorwart 1975, Rasouli 2014, and MDCalc. Not medical advice.
serum-osmolality-calculator