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# Coth

 Coth[z]gives the hyperbolic cotangent of z.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, Coth automatically evaluates to exact values.
• Coth can be evaluated to arbitrary numerical precision.
• Coth automatically threads over lists.
 Scope   (11)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Coth threads element-wise over lists and matrices:
Evaluate for complex arguments:
Simple exact purely imaginary values are generated automatically:
Convert multiple-angle expressions:
Find factors of decomposition:
Convert sums of hyperbolic functions to products:
Expand assuming real variables:
Convert to exponentials:
Coth can deal with real-valued intervals:
Infinite arguments give symbolic results:
Coth can be applied to power series:
Coth threads element-wise over sparse arrays as well as lists:
 Applications   (4)
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:
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:
Integrals:
Integral transforms:
Obtain Coth from sums and integrals:
Coth appears in special cases of special functions:
Coth is a numeric function:
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 traditional form parentheses are needed around the argument:
Plot Coth at infinity: