# Calculus/Hyperbolic functions

## Contents

# Theory[edit]

## Hyperbolic Functions[edit]

### Definitions[edit]

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

; ;

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

; ;

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### Derivatives of hyperbolic functions[edit]

### Principal values of the main hyperbolic functions[edit]

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.

Sinh: , Cosh: , Tanh:

### Inverse hyperbolic functions[edit]

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

We can define cosech^{-1}, sech^{-1} and coth^{-1} similarly.

We can also write these inverses using the logarithm function,

These identities can simplify some integrals.

### Derivatives of inverse hyperbolic functions[edit]

,

,

,

,

,

## Transcendental Functions[edit]

Transcendental functions are not algebraic. These include trigonometric, inverse trigonometric, logarithmic and exponential functions and many others.