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Elementary Functions > Sech[z] > Introduction to the Hyperbolic Functions in Mathematica

Automatic evaluations and transformations

Evaluation for exact, machine-number, and high-precision arguments

For a simple exact argument, Mathematica returns an exact result. For instance, for the argument , the Sinh function evaluates to .

For a generic machine‐number argument (a numerical argument with a decimal point and not too many digits), a machine number is returned.

The next inputs calculate 100‐digit approximations of the six hyperbolic functions at .

Within a second, it is possible to calculate thousands of digits for the hyperbolic functions. The next input calculates 10000 digits for , , , , , and and analyzes the frequency of the occurrence of the digit in the resulting decimal number.

Here are 50‐digit approximations to the six hyperbolic functions at the complex argument .

Mathematica always evaluates mathematical functions with machine precision, if the arguments are machine numbers. In this case, only six digits after the decimal point are shown in the results. The remaining digits are suppressed, but can be displayed using the function InputForm.

Simplification of the argument

Mathematica uses symmetries and periodicities of all the hyperbolic functions to simplify expressions. Here are some examples.

Mathematica automatically simplifies the composition of the direct and the inverse hyperbolic functions into the argument.

Mathematica also automatically simplifies the composition of the direct and any of the inverse hyperbolic functions into algebraic functions of the argument.

In cases where the argument has the structure or , e or with integer , trigonometric functions can be automatically transformed into other trigonometric or hyperbolic functions. Here are some examples.

Simplification of simple expressions containing hyperbolic functions

Sometimes simple arithmetic operations containing hyperbolic functions can automatically produce other hyperbolic functions.

Hyperbolic functions as special cases of more general functions

All hyperbolic functions can be treated as particular cases of some more advanced special functions. For example, and are sometimes the results of auto-simplifications from Bessel, Mathieu, Jacobi, hypergeometric, and Meijer functions (for appropriate values of their parameters).