Known limits

What Quantus does not do yet, or does only in part. Each item gives a clear error message in the app.

Calculator (Physics, Algebra, Imaginary)

• No conversion to a unit of your choice. Results stay in the units you typed, or are named when the arithmetic produced a product of units. To get a number in another unit, divide by it: \frac{1\,\mathrm{yr}}{1\,\mathrm{d}} gives $365{,}2425$.
• Plain letters cannot be defined. x = 3 names a line; only subscripted names (x_1 = 3) are stored. Plain letters get a value from \sum, \int, \lim, \frac{d}{dx} or as the unknown of an equation.
• Units win over letters in Physics mode: m, s, g, h and so on cannot be the unknown of an equation. Inside \sum, \int, \lim and \frac{d}{dx} the bound letter wins in the body.
• The unknown of an equation is a plain number. 3x = 6m is a dimension error; 3x\,\mathrm{m} = 6\,\mathrm{m} works.
• One unknown per equation. Degree up to 2 is solved exactly, higher degrees numerically on $[-100, 100]$. Several unknowns need a system of equations.
• Systems must be linear. x^2 + y = 1; x - y = 0 is refused; there is no numerical solver for non-linear systems. A system is always the whole input and cannot be part of a calculation, and its unknowns are plain numbers like those of a single equation.
• Integrals need bounds and are numerical (tanh-sinh), with at most 30 digits. Nested integrals work but are slow; improper integrals of exponentials take about a second. Differentials must stand at the top level of the integrand, not inside \frac{…}{…}.
• Derivatives need a point: \left.\frac{d}{dx} f\right|_{x=2}. No \frac{d^2}{dx^2}, no partial derivatives.
• Limits are numerical; a limit whose two sides disagree is an error, use 0^+ or 0^-.
• Matrices only inside \det, up to 8×8.
• No \operatorname{…} functions beyond those in Syntax.
• siunitx macros such as \kilo\metre are not supported; write \qty{3}{km}.

Complex numbers (Imaginary mode)

Exact: i^n, \sqrt{-a} (principal root $i\sqrt{a}$), sums, products, quotients and integer powers of $a + bi$, so $(3+4i)(1-2i) = 11 - 2i$. Also supported: e^{a+bi}, z^w (principal branch), \ln and \log of negative or complex numbers, the trigonometric and hyperbolic functions of complex arguments, |z|, \Re, \Im, \arg (in $(-\pi, \pi]$) and \overline{z}. Quadratic equations with a negative discriminant have complex roots, and a variable can hold a complex number (z_1 = 3+4i).

Not covered:

• \arcsin, \arccos, \arctan of a complex number
• equations of degree above two with complex roots
• derivatives and integrals of complex functions
• complex units: 2i\,\mathrm{m} works, but $\mathrm{m}$ is not a complex quantity
• odd roots of negative numbers keep their real value: $\sqrt[3]{-8} = -2$
• the simplifier leaves $i$ alone, so (x+i)(x-i) with a free $x$ is shown as typed

Analysis mode

Accepted: a function of one real variable built from numbers, $\pi$, $e$, $+ - \cdot /$, powers, roots, $e^u$, $a^u$, \exp, \ln, \log, \log_b, the six trigonometric functions, their inverses \arcsin, \arccos, \arctan, \sinh, \cosh, \tanh, $|u|$ and degrees, nested in any way.

Rejected with an error that names the reason:

• function families with a second free letter ($x^2 + a$)
• piecewise functions, implicit or parametric curves, complex results
• units, sums, integrals, limits, derivatives, matrices, $n!$ and the other integer functions
• domains made of periodic intervals, such as $\ln(\sin x)$ (periodic gaps like those of $\tan x$ are fine)
• in the area task: improper integrals and areas between two curves

Results that are partial or approximate:

• Equations no school method solves are scanned numerically on $[-100, 100]$ (or in one period); the section is then marked PARTIAL. Such roots are accurate to about 15 digits, polynomial roots to the full precision.
• Antiderivatives come from a table, partial fractions, substitution and integration by parts (up to six nested rounds). Repeated irreducible quadratic factors ($\frac{1}{(x^2+1)^2}$) and integrands without an elementary antiderivative ($e^{x^2}$, $\frac{\sin x}{x}$) give no elementary antiderivative found. Every antiderivative shown has been checked by differentiation.
• Limits that no rule decides are sampled and shown with $\approx$, or as unknown.
• A sketch is cut off after 3 seconds, each task after 2 seconds; the missing sections are marked PARTIAL.
• Simplification treats $\ln x$ or $\sqrt{3}$ as independent symbols, so a result can be correct without being the shortest form.
• Periodic families are listed in $[-P, P]$ (at most 12 members each), or $[-10, 10]$ when $f$ itself is not periodic.