# Coth

Coth[z]

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.

# 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 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(3)

Evaluate numerically:

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Plot over a subset of the reals:

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Series expansion:

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# Tutorials

Introduced in 1988
(1.0)
| Updated in 1996
(3.0)