Operations carried out by specialized Mathematica functions

Series expansions

Calculating the series expansion of a tangent function to hundreds of terms can be done in seconds.

Mathematica comes with the add‐on package DiscreteMath`RSolve` that allows finding the general terms of series for many functions. After loading this package, and using the package function SeriesTerm, the following term of can be evaluated.

The result can be checked by the following process.

Differentiation

Mathematica can evaluate derivatives of the tangent function of an arbitrary positive integer order.

Finite products

Mathematica can calculate some finite symbolic products that contain the tangent function. Here is an example.

Indefinite integration

Mathematica can calculate a huge number of doable indefinite integrals that contain the tangent function. Here are some examples.

Definite integration

Mathematica can calculate wide classes of definite integrals that contain the tangent function. Here are some examples.

Limit operation

Mathematica can calculate limits that contain the tangent function. Here are some examples.

Solving equations

The next inputs solve two equations that contain the tangent function. Because of the multivalued nature of the inverse tangent function, a printed message indicates that only some of the possible solutions are returned.

A complete solution of the previous equation can be obtained using the function Reduce.

Solving differential equations

Here is a nonlinear differential equation whose independent solutions include the tangent function.

In the previous input, the differential equation was solved for . If the argument is suppressed, the result is returned as a pure function (in the sense of the ‐calculus).

The advantage of such a pure function is that it can be used for different arguments, derivatives, and more.

In carrying out the algorithm to solve the following nonlinear differential equation, Mathematica has to solve a transcendental equation. In doing so, the generically multivariate inverse of a function is encountered, and a message is issued that a solution branch is potentially missed.

Plotting

Mathematica has built‐in functions for 2D and 3D graphics. Here are some examples.