Automatic evaluations and transformations
Evaluation for exact and machine-number values of arguments
For the exact argument , Mathematica returns an 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 secant function in less than 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 secant 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 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 secant function. Here are some examples.
Mathematica automatically simplifies the composition of the direct and the inverse hyperbolic secant functions to the inner 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 , and or with integer , the hyperbolic secant function can be automatically transformed into hyperbolic or trigonometric secant or cosecant functions.
Simplification of combination of hyperbolic secant functions
Sometimes simple arithmetic operations containing the hyperbolic secant function can automatically generate other equal hyperbolic functions.
The hyperbolic secant function arising as special cases from more general functions
The hyperbolic secant function can be treated as a particular case of other, more general special functions. For example, appears automatically from Bessel, Struve, Mathieu, Jacobi, hypergeometric, and Meijer functions or their reciprocals for appropriate parameters.