Difference between revisions of "Coth"

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__NOTOC__
 
The hyperbolic cotangent is defined by
 
The hyperbolic cotangent is defined by
$$\mathrm{coth}(z)=\dfrac{1}{\tanh(z)},$$
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$$\mathrm{coth}(z)=\dfrac{1}{\tanh(z)}=\dfrac{\mathrm{cosh}(z)}{\mathrm{sinh}(z)},$$
 
where $\tanh$ denotes the [[Tanh|hyperbolic tangent]] function.
 
where $\tanh$ denotes the [[Tanh|hyperbolic tangent]] function.
  
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<gallery>
 
<gallery>
 
File:Cothplot.png|Plot of $\mathrm{coth}$ on $[-5,5]$.
 
File:Cothplot.png|Plot of $\mathrm{coth}$ on $[-5,5]$.
File:Complex_Coth.jpg|[[Domain coloring]] of $\mathrm{coth}$.
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File:Complexcothplot.png|[[Domain coloring]] of $\mathrm{coth}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
{{:Derivative of coth}}
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[[Derivative of coth]]<br />
{{:Antiderivative of coth}}
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[[Antiderivative of coth]]<br />
{{:Relationship between coth and cot}}
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[[Relationship between coth and csch]]<br />
{{:Relationship between cot and coth}}
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[[Relationship between coth and cot]]<br />
{{:Relationship between csc, Gudermannian, and coth}}
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[[Relationship between cot and coth]]<br />
{{:Relationship between coth, inverse Gudermannian, and csc}}
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[[Relationship between csc, Gudermannian, and coth]]<br />
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[[Relationship between coth, inverse Gudermannian, and csc]]<br />
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[[Pythagorean identity for coth and csch]]<br />
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[[Coth of a sum]]<br />
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[[z coth(z) = 2z/(e^(2z)-1) + z]]<br />
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[[z coth(z) = sum of 2^(2n)B_(2n) z^(2n)/(2n)!]]<br />
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[[z coth(z) = 2 Sum of (-1)^(n+1) zeta(2n) z^(2n)/pi^(2n)]]<br />
  
<center>{{:Hyperbolic trigonometric functions footer}}</center>
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=Videos=
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[https://www.youtube.com/watch?v=Pz7BDxef3HU Calculus I - Derivative of Hyperbolic Cotangent Function coth(x) - Proof]
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=See Also=
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[[Arccoth]]
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=References=
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Sech|next=Relationship between sinh and sin}}: $4.5.6$
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{{:Hyperbolic trigonometric functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 05:53, 4 March 2018

The hyperbolic cotangent is defined by $$\mathrm{coth}(z)=\dfrac{1}{\tanh(z)}=\dfrac{\mathrm{cosh}(z)}{\mathrm{sinh}(z)},$$ where $\tanh$ denotes the hyperbolic tangent function.

Properties

Derivative of coth
Antiderivative of coth
Relationship between coth and csch
Relationship between coth and cot
Relationship between cot and coth
Relationship between csc, Gudermannian, and coth
Relationship between coth, inverse Gudermannian, and csc
Pythagorean identity for coth and csch
Coth of a sum
z coth(z) = 2z/(e^(2z)-1) + z
z coth(z) = sum of 2^(2n)B_(2n) z^(2n)/(2n)!
z coth(z) = 2 Sum of (-1)^(n+1) zeta(2n) z^(2n)/pi^(2n)

Videos

Calculus I - Derivative of Hyperbolic Cotangent Function coth(x) - Proof

See Also

Arccoth

References

Hyperbolic trigonometric functions