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2. Floor and ceiling functions - Wikipedia

   en.wikipedia.org/wiki/Floor_and_ceiling_functions

   Given real numbers x and y, integers m and n and the set of integers, floor and ceiling may be defined by the equations ⌊ ⌋ = {}, ⌈ ⌉ = {}. Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation

3. Euler's formula - Wikipedia

   en.wikipedia.org/wiki/Euler's_formula

   Many texts write φ = tan −1 ⁠ y / x ⁠ instead of φ = atan2(y, x), but the first equation needs adjustment when x ≤ 0. This is because for any real x and y, not both zero, the angles of the vectors (x, y) and (−x, −y) differ by π radians, but have the identical value of tan φ = ⁠ y / x ⁠.

4. Desmos - Wikipedia

   en.wikipedia.org/wiki/Desmos

   Some 50 employees joined Amplify. Desmos Studio was spun off as a separate public benefit corporation focused on building calculator products and other math tools. [7] In May 2023, Desmos released a beta for a remade Geometry Tool. In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra ...

5. Sign function - Wikipedia

   en.wikipedia.org/wiki/Sign_function

   Here, ⁡ is the Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function, tangent. For k > 1 {\displaystyle k>1} , a smooth approximation of the sign function is sgn ⁡ x ≈ tanh ⁡ k x . {\displaystyle \operatorname {sgn} x\approx \tanh kx\,.}

6. Radius of convergence - Wikipedia

   en.wikipedia.org/wiki/Radius_of_convergence

   Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.

7. Sigmoid function - Wikipedia

   en.wikipedia.org/wiki/Sigmoid_function

   A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density , the normal density , and Student's ...

8. Theta function - Wikipedia

   en.wikipedia.org/wiki/Theta_function

   There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has a positive ...

9. Mental calculation - Wikipedia

   en.wikipedia.org/wiki/Mental_calculation

   Mental calculation consists of arithmetical calculations using only the human brain, with no help from any supplies (such as pencil and paper) or devices such as a calculator. People may use mental calculation when computing tools are not available, when it is faster than other means of calculation (such as conventional educational institution ...