Operations performed by specialized Mathematica functions

Series expansions

Calculating the series expansion of hyperbolic functions to hundreds of terms can be done in seconds. Here are some examples.

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 for odd hyperbolic functions 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 hyperbolic functions of an arbitrary positive integer order.

Finite summation

Mathematica can calculate finite sums that contain hyperbolic functions. Here are two examples.

Infinite summation

Mathematica can calculate infinite sums that contain hyperbolic functions. Here are some examples.

Finite products

Mathematica can calculate some finite symbolic products that contain the hyperbolic functions. Here are two examples.

Infinite products

Mathematica can calculate infinite products that contain hyperbolic functions. Here are some examples.

Indefinite integration

Mathematica can calculate a huge set of doable indefinite integrals that contain hyperbolic functions. Here are some examples.

Definite integration

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

Limit operation

Mathematica can calculate limits that contain hyperbolic functions. Here are some examples.

Solving equations

The next input solves equations that contain hyperbolic functions. The message indicates that the multivalued functions are used to express the result and that some solutions might be absent.

Complete solutions can be obtained by using the function Reduce.

Solving differential equations

Here are differential equations whose linear‐independent solutions are hyperbolic functions. The solutions of the simplest second-order linear ordinary differential equation with constant coefficients can be represented through and .

All hyperbolic functions satisfy first-order nonlinear differential equations. In carrying out the algorithm to solve the 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.

Integral transforms

Mathematica supports the main integral transforms like direct and inverse Fourier, Laplace, and Z transforms that can give results containing classical or generalized functions. Here are some transforms of hyperbolic functions.

Plotting

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