Calculus/Hyperbolic functions

TheoryEdit

 
The unit hyperbola has a sector with an area half of the hyperbolic angle

The independent variable of a hyperbolic function is called a hyperbolic angle. Just as the circular functions sine and cosine can be seen as projections from the unit circle to the axes, so the hyperbolic functions sinh and cosh are projections from a unit hyperbola to the axes.

DefinitionsEdit

The hyperbolic functions are defined in analogy with the trigonometric functions:

 
 
 

The reciprocal functions csch, sech, coth are defined from these functions:

 
 
 

Some simple identitiesEdit

 
 
 
 

Derivatives of hyperbolic functionsEdit

 

 

 

 

 

 

Principal values of the main hyperbolic functionsEdit

There is no problem in defining principal braches for sinh and tanh because they are injective. We choose one of the principal branches for cosh.

 
 
 

Inverse hyperbolic functionsEdit

With the principal values defined above, the definition of the inverse functions is immediate:

 
 
 

We can define   ,   and   similarly.

We can also write these inverses using the logarithm function,

 
 
 

These identities can simplify some integrals.

Derivatives of inverse hyperbolic functionsEdit

 

 

 

 

 

 

Transcendental FunctionsEdit

Hyperbolic functions are examples of transcendental functions -- they are not algebraic functions. They include trigonometric, inverse trigonometric, logarithmic and exponential functions.