Built-in Mathematica Symbol

Sech

Sech[z]
gives the hyperbolic secant of z.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

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1/Cosh[z] is automatically converted to Sech[z]. TrigFactorList[expr] does decomposition.

For certain special arguments, Sech automatically evaluates to exact values.

Sech can be evaluated to arbitrary numerical precision.

Sech automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Scope (11)

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Sech threads element-wise over lists and matrices:

Evaluate for complex arguments:

Simple exact purely imaginary values are generated automatically:

Convert multiple-angle expressions:

Find factors of decomposition:

Convert sums of hyperbolic functions to products:

Expand assuming real variables:

Convert to exponentials:

TraditionalForm formatting:

Generalizations & Extensions (3)

Sech can deal with real-valued intervals:

Infinite arguments give symbolic results:

Sech can be applied to power series:

Applications (5)

Plot a tractrix pursuit curve:

Plot a pseudosphere:

Calculate the finite area of the surface extending to infinity:

A soliton in the Korteweg-de Vries equation:

A Schrödinger equation with a zero energy solution:

Calculate the CDF of the hyperbolic secant PDF:

Plot the PDF and CDF:

Properties & Relations (13)

Basic parity and periodicity properties of Sech get automatically applied:

Expressions containing hyperbolic functions do not automatically simplify:

Use Refine, Simplify, and FullSimplify to simplify expressions containing Sech:

Use FunctionExpand to express special values in radicals:

Compose with inverse functions:

Solve a hyperbolic equation:

Numerically find a root of a transcendental equation:

Reduce a hyperbolic equation:

Integrals:

Integral transforms:

Obtain Sech from sums, products, and integrals:

Sech appears in special cases of special functions:

Sech is a numeric function:

Possible Issues (5)

Machine-precision input is insufficient to give a correct answer:

With exact input, the answer is correct:

A larger setting for $MaxExtraPrecision can be needed:

The inverse of Sech evaluates to Cosh:

No power series exists at infinity, where Sech has an essential singularity:

In traditional form parentheses are needed around the argument: