Cosh

Cosh[z]

gives the hyperbolic cosine of z.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • For certain special arguments, Cosh automatically evaluates to exact values.
  • Cosh can be evaluated to arbitrary numerical precision.
  • Cosh automatically threads over lists.

Background & Context

  • Cosh is the hyperbolic cosine function, which is the hyperbolic analogue of the Cos circular function used throughout trigonometry. It is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola . Cosh[α] then represents the horizontal coordinate of the intersection point. Cosh may also be defined as , where is the base of the natural logarithm Log.
  • Cosh automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numerical expressions as arguments, Cosh may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Cosh include TrigToExp, TrigExpand, Simplify, and FullSimplify.
  • Cosh threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic cosine of a square matrix (i.e. the power series for the hyperbolic cosine function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic cosines of the individual matrix elements.
  • Cosh[x] increases exponentially as x approaches . Cosh satisfies an identity similar to the Pythagorean identity satisfied by Cos, namely . The definition of the hyperbolic sine function is extended to complex arguments by way of the identity . The hyperbolic cosine function is entire, meaning it is complex differentiable at all finite points of the complex plane. Cosh[z] has series expansion about the origin.
  • The inverse function of Cosh is ArcCosh. Additional related mathematical functions include Sinh and Sech.

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

Scope  (41)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

Cosh can take complex number inputs:

Evaluate Cosh efficiently at high precision:

Cosh can deal with realvalued intervals:

Cosh threads elementwise over lists and matrices:

Specific Values  (4)

Values of Cosh at fixed, purely imaginary points:

Values at infinity:

Minimum of Cosh:

Find the minimum as the root of :

Substitute in the value:

Visualize the result:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Visualization  (3)

Plot the Cosh function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (5)

Cosh is defined for all real and complex values:

Cosh achieves all real values greater than or equal to 1:

The range for complex values is the whole plane:

Cosh is an even function:

Cosh has the mirror property cosh(TemplateBox[{z}, Conjugate])=TemplateBox[{{cosh, (, z, )}}, Conjugate]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (4)

Indefinite integral of Cosh:

Indefinite integral of a trigonometric product involving Cosh:

Definite integral of Cosh over the interval centered at the origin:

This is twice the integral over half the interval:

More integrals:

Series Expansions  (4)

Find the Taylor expansion using Series:

Plot the first three approximations for Cosh around :

General term in the series expansion of Cosh:

The first few terms in the Fourier series of Cosh:

Cosh can be applied to power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:

HankelTransform:

Function Identities and Simplifications  (6)

Cosh of a double angle:

Cosh of a sum:

Convert multipleangle expressions:

Convert sums of hyperbolic functions to products:

Expand assuming real variables and :

Convert to exponentials:

Function Representations  (4)

Representation through Cos:

Representation through Bessel functions:

Representation in terms of MeijerG:

Cosh can be represented as a DifferentialRoot:

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 η:

Nonrelativistic limit:

Special solution of the sineGordon equation:

Check the solution:

Pythagorean theorem in hyperbolic geometry:

The small t limit gives the ordinary Pythagorean theorem:

Properties & Relations  (12)

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:

The generating function for Cosh:

The exponential generating function for Cosh:

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 highprecision results:

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

In TraditionalForm, parentheses are needed around the argument:

Neat Examples  (1)

Nested hyperbolic cosine over the complex plane:

Introduced in 1988
 (1.0)