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Operations performed by specialized Mathematica functions

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.

Mathematica can evaluate derivatives of hyperbolic functions of an arbitrary positive integer order.

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

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

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

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

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

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

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

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.

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.

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.

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