This does an integral over the interior of the unit circle. Here is an equivalent form. Even though an integral may be straightforward over a simple rectangular region, it ...

Representations of real intervals. This represents all numbers between -2 and +5. The square of any number between -2 and +5 is always between 0 and 25.

Numerical solution of differential equations. This generates a numerical solution to the equation y^′(x)y(x) with 0<x<2. The result is given in terms of an ...

Nested logical and piecewise functions can be expanded out much like nested arithmetic functions. You can do this using LogicalExpand and PiecewiseExpand. Expanding out ...

Mathieu and related functions. The Mathieu functions MathieuC[a,q,z] and MathieuS[a,q,z] are solutions to the equation y^′′+[a-2qcos(2z)]y0. This equation appears in many ...

It is often useful to carry out a numerical integration using fixed step sizes. For example, certain methods such as "DoubleStep" and "Extrapolation" carry out a sequence of ...

Numerical root finding. NSolve gives you numerical approximations to all the roots of a polynomial equation. You can also use NSolve to solve sets of simultaneous equations ...

Numerical sums and products. This gives a numerical approximation to ∑_(i=1)^∞((1)/(i^3+i!)). There is no exact result for this sum, so Mathematica leaves it in a symbolic ...

Functions relating real numbers and integers. Extracting integer and fractional parts. IntegerPart[x] and FractionalPart[x] can be thought of as extracting digits to the left ...

One of the important features of Mathematica is its ability to give you exact, symbolic, results for computations. There are, however, computations where it is just ...