BUILT-IN MATHEMATICA SYMBOL

Tanh

Tanh[z]
gives the hyperbolic tangent of

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

Sinh[z]/Cosh[z] is automatically converted to Tanh[z]. TrigFactorList[expr] does decomposition.

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

Tanh can be evaluated to arbitrary numerical precision.

Tanh automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

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Scope (11)

Evaluate to high precision:

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

Tanh threads element-wise over lists and matrices:

Tanh can take complex number inputs:

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)

Tanh can deal with real-valued intervals:

Infinite arguments give symbolic results:

Tanh can be applied to power series:

Applications (4)

Plot a tractrix pursuit curve:

Plot a pseudosphere:

Calculate the finite area of the surface extending to infinity:

Velocity of a relativistic body in a constant force field:

Solution of the Burgers' equation using the tanh method:

Properties & Relations (13)

Basic parity and periodicity properties of Tanh are automatically applied:

Expressions containing hyperbolic functions do not automatically simplify:

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

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 Tanh from sums and integrals:

Tanh appears in special cases of special functions:

Tanh is a numeric function:

Possible Issues (4)

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:

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

In traditional form parentheses are needed around the argument:

Neat Examples (1)

Continued fraction expansion: