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Elementary Functions > Tanh[z] > Introduction to the Hyperbolic Tangent Function in Mathematica

Automatic evaluations and transformations

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

For the exact argument , Mathematica returns an exact result.

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

The next inputs calculate 100‐digit approximations at and .

It is possible to calculate thousands of digits for the hyperbolic tangent function within a second. The next input calculates 10000 digits for and analyzes the frequency of the digit in the resulting decimal number.

Here is a 50‐digit approximation to the hyperbolic tangent function at the complex argument .

Mathematica automatically evaluates mathematical functions with machine precision, if the arguments of the function are machine‐number elements. 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 knows the symmetry and periodicity of the hyperbolic tangent function. Here are some examples.

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

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

If the argument has the structure or , and or with integer , the hyperbolic tangent function can be automatically transformed into hyperbolic or trigonometric tangent or cotangent functions.

Simplification of simple expressions containing the hyperbolic tangent function

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

The hyperbolic tangent function arising as special cases from more general functions

The hyperbolic tangent function can be treated as a particular case of other, more general special functions. For example, can appear automatically from Bessel, Mathieu, Jacobi, hypergeometric, and Meijer functions or their ratios for appropriate parameters.