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Coth

gives the hyperbolic cotangent of z.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Cosh[z]/Sinh[z] is automatically converted to Coth[z]. TrigFactorList[expr] does decomposition.
For certain special arguments, Coth automatically evaluates to exact values.
Coth can be evaluated to arbitrary numerical precision.
Coth automatically threads over lists. »
Coth can be used with Interval and CenteredInterval objects. »

Background & Context

Coth is the hyperbolic cotangent function, which is the hyperbolic analogue of the Cot circular function used throughout trigonometry. Coth[α] is defined as the ratio of the corresponding hyperbolic cosine and hyperbolic sine functions via . Coth may also be defined as , where is the base of the natural logarithm Log.
Coth automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Coth may be evaluated to arbitrary numeric precision. TrigFactorList can be used to factor expressions involving Coth into terms containing Sinh, Cosh, Sin, and Cos. Other operations useful for manipulation of symbolic expressions involving Coth include TrigToExp, TrigExpand, Simplify, and FullSimplify.
Coth threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic cotangent of a square matrix (i.e. the power series for the hyperbolic cotangent function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic cotangents of the individual matrix elements.
Coth[x] approaches for small negative x and for large positive x. Coth satisfies an identity similar to the Pythagorean identity satisfied by Cot, namely . The definition of the hyperbolic cotangent function is extended to complex arguments by way of the identities and . Coth has poles at values for an integer and evaluates to ComplexInfinity at these points. Coth[z] has series expansion $sum_(k=0)^infty(2^(2 k) TemplateBox[{{2, , k}}, BernoulliB] )/((2 k)!)z^(2 k-1)$ about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB.
The inverse function of Coth is ArcCoth. Other related mathematical functions include Tanh, Cot, and Cosh.

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

Asymptotic expansion at a singular point:

Scope (47)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

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

Coth can take complex number inputs:

Evaluate Coth efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Coth function using MatrixFunction:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Specific Values (5)

Values of Coth at fixed purely imaginary points:

Values at infinity:

Singular point of Coth:

Find a value of satisfying equation :

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 Coth function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties (12)

Coth is defined for all real values except 0:

Complex domain:

Coth achieves all real values except from the open interval :

Coth is an odd function:

Coth has the mirror property :

Coth is not an analytic function:

However, it is meromorphic:

Coth is neither non-decreasing nor non-increasing:

Coth is injective:

Coth is not surjective:

Coth is neither non-negative nor non-positive:

It has both singularity and discontinuity at zero:

Coth is neither convex nor concave:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (3)

Indefinite integral of Coth:

Definite integral of an odd integrand over the interval centered at the origin is 0:

More integrals:

Series Expansions (3)

Find the Taylor expansion using Series:

Plot the first three approximations for Coth around :

General term in the series expansion of Coth:

Coth can be applied to power series:

Integral Transforms (2)

Compute the Laplace transform using LaplaceTransform:

FourierTransform:

Function Identities and Simplifications (6)

Coth of a double angle:

Convert multiple‐angle expressions:

Coth of a sum:

Convert sums of hyperbolic functions to products:

Expand assuming real variables:

Convert to exponentials:

Function Representations (4)

Representation through Cot:

Representation through Bessel functions:

Representation through Jacobi functions:

Representation through Mathieu functions:

Applications (5)

Plot the absolute value over the complex plane:

Closed form for Newton iterations for square roots of integers:

Compare with explicit iterations:

Sum over bosonic Matsubara frequencies by integrating over a product with Coth:

Temperature‐dependent Brillouin function for dipoles in a magnetic field:

Low‐ and high‐temperature behavior:

Solve a differential equation with the Coth function as inhomogeneous part:

Properties & Relations (11)

Basic parity and periodicity properties of Coth get automatically applied:

Use Simplify and FullSimplify to simplify expressions containing Coth:

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:

Integral transforms:

Obtain Coth from sums and integrals:

Coth appears in special cases of special functions:

Coth 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 Coth has an essential singularity:

In TraditionalForm, parentheses are needed around the argument:

Neat Examples (1)

Plot Coth at infinity:

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