Built-in Mathematica Symbol

Cosh

Cosh[z]
gives the hyperbolic cosine of z.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

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

Cosh can be evaluated to arbitrary numerical precision.

Cosh automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Scope (10)

Cosh can take complex number inputs:

Evaluate to high precision:

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

Cosh threads element-wise over lists and matrices:

Simple exact values are generated automatically:

Convert multiple-angle expressions:

Convert sums of hyperbolic functions to products:

Expand assuming real variables:

Convert to exponentials:

TraditionalForm formatting:

Generalizations & Extensions (3)

Cosh can deal with real-valued intervals:

Infinite arguments give symbolic results:

Cosh can be applied to power series:

Applications (6)

Draw a hyperbola:

Build rotation matrix in hyperbolic space from infinitesimal transformations:

Relativistic boost matrix:

The matrix is orthogonal with respect to the Minkowski metric:

Construct a relativistic coordinate transformation for rapidity

:

Nonrelativistic limit:

Special solution of the Sine-Gordon equation:

Check the solution:

Pythagorean theorem in hyperbolic geometry:

The small t limit gives the ordinary Pythagorean theorem:

Properties & Relations (10)

Basic parity and periodicity properties of Cosh get automatically applied:

Expressions containing hyperbolic functions do not automatically simplify:

Use Simplify:

Compose with inverse functions:

Solve a hyperbolic equation:

Numerically find a root of a transcendental equation:

Reduce a hyperbolic equation:

Integrals:

Integral transforms:

Cosh appears in special cases of many mathematical functions:

Cosh 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:

Machine-number inputs can give high-precision results:

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

In traditional form, parentheses are needed around the argument:

Neat Examples (1)

Nested hyperbolic cosine over the complex plane: