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Elementary Functions > Coth[z] > Introduction to the Hyperbolic Cotangent Function in Mathematica

Automatic evaluations and transformations

Evaluation for exact and machine-number values of arguments

For the exact argument , Mathematica returns exact result.

For a machine‐number argument (numerical argument with a decimal point), 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 cotangent 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 cotangent function at the complex argument .

Mathematica automatically evaluates mathematical functions with machine precision, if the arguments of the function are numerical values and include machine‐number elements. In this case only six digits after the decimal point are shown. 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 cotangent function. Here are some examples.

Mathematica automatically simplifies the composition of the direct and the inverse hyperbolic cotangent 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.

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

Simplification of combinations of hyperbolic cotangent functions

Sometimes simple arithmetic operations containing the hyperbolic cotangent function can automatically generate other equal hyperbolic functions.

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

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