Operations under special Mathematica functions

Series expansions

Calculating the series expansion of a hyperbolic secant 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.

Here is a quick check of the last result.

This series should be evaluated to , which can be concluded from the following relation.

Differentiation

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

Indefinite integration

Mathematica can calculate a huge set of doable indefinite integrals that contain the hyperbolic secant function. The results can contain special functions. Here are some examples.

Definite integration

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

Limit operation

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

Solving equations

The next inputs solve two equations that contain the hyperbolic secant function. Because of the multivalued nature of the inverse hyperbolic secant function, a message is printed indicating 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 linear first-order differential equation that is obeyed by the hyperbolic secant function.

Here is a nonlinear second-order differential equation that is obeyed by the hyperbolic secant function. Mathematica solves the differential equation as a rational function of . But it is straightforward to show that a hyperbolic secant function is also a solution.

Plotting

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