Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Coth






Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Introduction to the Hyperbolic Cotangent Function





Defining the hyperbolic cotangent function


The hyperbolic cotangent function is an old mathematical function. It was first used in the articles by L'Abbe Sauri (1774).

This function is easily defined as the ratio of the hyperbolic sine and cosine functions (or expanded, as the ratio of the half‐sum and half‐difference of two exponential functions in the points and ):

This function can also be defined as reciprocal to the hyperbolic tangent function:

After comparison with the famous Euler formulas for the cosine and sine functions, and , it is easy to derive the following representation of the hyperbolic cotangent through the circular cotangent:

This formula allows for the derivation of all properties and formulas for the hyperbolic cotangent from the corresponding properties and formulas for circular cotangent function.





© 1998-2014 Wolfram Research, Inc.