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] decreases 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 sum_(k=0)^infty(2(2^(2 k-1)-1) 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 Csch is ArcCsch. Other related mathematical functions include Sinh and Sech.

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 the origin:

Asymptotic expansion at a singular point:

Scope  (40)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate Csch efficiently at high precision:

Csch can deal with realvalued intervals:

Csch threads elementwise over lists and matrices:

Specific Values  (5)

Values of Csch at fixed purely imaginary points:

Values at infinity:

Singular point of Csch:

Find the 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 Csch function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (5)

Domain of Csch over reals:

Complex domain:

Csch achieves all real values except 0:

Csch is an odd function:

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

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of Csch:

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 Csch around :

General term in the series expansion of Csch:

Csch can be applied to power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:

FourierTransform:

Function Identities and Simplifications  (6)

Csch of a double angle:

Csch 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 Sin:

Representation through Bessel functions:

Representation through Jacobi functions:

Representation in terms of MeijerG:

Applications  (2)

Plot absolute value over the complex plane:

Plot Poinsot's spirals:

Properties & Relations  (9)

Basic parity and periodicity properties of Csch get automatically applied:

Use Simplify and FullSimplify to simplify expressions containing Csch:

Use FunctionExpand to express special values in radicals:

Compose with inverse functions:

Solve a hyperbolic equation:

Numerically find a root of a transcendental equation:

Obtain Csch from sums, products, and differential equations:

Csch appears in special cases of special functions:

Csch is a numeric function:

Possible Issues  (5)

Machine-precision input is insufficient to give a correct answer:

Use arbitrary-precision evaluation instead:

A larger setting for $MaxExtraPrecision may be needed:

The inverse of Csch evaluates to Sinh:

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

In TraditionalForm, parentheses are needed around the argument:

Neat Examples  (1)

Plot Csch at infinity:

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