# Csch

Csch[z]

gives the hyperbolic cosecant of z.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• 1/Sinh[z] is automatically converted to Csch[z]. TrigFactorList[expr] does decomposition.
• For certain special arguments, Csch automatically evaluates to exact values.
• Csch can be evaluated to arbitrary numerical precision.
• Csch automatically threads over lists.

# Background & Context

• Csch is the hyperbolic cosecant function, which is the hyperbolic analogue of the Csc circular function used throughout trigonometry. It is defined as the reciprocal of the hyperbolic sine function as . 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 . Csch[α] then represents the reciprocal of the vertical coordinate of the intersection point. Csch may also be defined as , where is the base of the natural logarithm Log.
• Csch automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Csch may be evaluated to arbitrary numeric precision. TrigFactorList can be used to factor expressions involving Csch into terms containing Sinh, Cosh, Sin, and Cos. Other operations useful for manipulation of symbolic expressions involving Csch include TrigToExp, TrigExpand, Simplify, and FullSimplify.
• Csch threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic cosecant of a square matrix (i.e. the power series for the hyperbolic cosecant function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic cosecants of the individual matrix elements.
• Csch[x] decreses exponentially as x approaches . Csch satisfies an identity similar to the Pythagorean identity satisfied by Csc, namely . The definition of the hyperbolic cosecant function is extended to complex arguments by way of the identity . Csch has poles at values for an integer and evaluates to ComplexInfinity at these points. Csch[z] has series expansion about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB.
• The inverse function of Csch is ArcCsch. Other related mathematical functions include Sinh and Sech.

# 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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