Operations carried out by specialized Mathematica functions

Series expansions

Calculating the series expansion of trigonometric 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 trigonometric functions can be evaluated.

Differentiation

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

Finite summation

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

Infinite summation

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

Finite products

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

Infinite products

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

Indefinite integration

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

Definite integration

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

Limit operation

Mathematica can calculate limits that contain trigonometric functions.

Solving equations

The next input solves equations that contain trigonometric 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 trigonometric functions. The solutions of the simplest second-order linear ordinary differential equation with constant coefficients can be represented through and .

In the last 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.

All trigonometric 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 that contain classical or generalized functions. Here are some transforms of trigonometric functions.

Plotting

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