Calculus/Hyperbolic functions

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Theory[edit]

Hyperbolic Functions[edit]

Definitions[edit]

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

\sinh x = \frac{1}{2} (e^x-e^{-x}); \cosh x = \frac{1}{2} (e^x+e^{-x}); \tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{\sinh x}{\cosh x}

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

{\rm csch} x = \frac{1}{\sinh x}; {\rm sech} x = \frac{1}{\cosh x}; \coth x = \frac{1}{\tanh x}

Some simple identities[edit]

\cosh^2 x - \sinh^2 x = \frac{}{}1

1 - \tanh^2 x = \frac{}{}\operatorname{sech}^2 x

\sinh 2x = \frac{}{}2\sinh x\cosh x

\cosh 2x = \frac{}{}\cosh^2 x+\sinh^2 x

Derivatives of hyperbolic functions[edit]

\frac{d}{dx} \sinh x = \cosh x

\frac{d}{dx} \cosh x = \sinh x

\frac{d}{dx} \tanh x = \operatorname{sech}^2 x

\frac{d}{dx} \operatorname{cosech} x = -\operatorname{cosech} x \operatorname{coth}x

\frac{d}{dx} \operatorname{sech} x = -\operatorname{sech} x\tanh x

\frac{d}{dx} \operatorname{coth} x = -\operatorname{cosech}^2 x

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: \mathbb{R}\to \mathbb{R}, Cosh: [0,\infty]\to [1,\infty], Tanh: \mathbb{R}\to (-1,1)

Inverse hyperbolic functions[edit]

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

\frac{}{}\sinh ^{-1}: \mathbb{R}\to \mathbb{R}
\frac{}{}\cosh^ {-1}: [1,\infty]\to [0,\infty]
\frac{}{}\tanh^{-1}: (-1,1)\to \mathbb{R}

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

We can also write these inverses using the logarithm function,

\sinh^{-1}z = \ln (z+\sqrt{z^2+1})
\cosh^{-1}z = \ln (z+\sqrt{z^2-1})
\tanh^{-1}z = \ln \sqrt{\frac{1+z}{1-z}}

These identities can simplify some integrals.

Derivatives of inverse hyperbolic functions[edit]

\frac{d}{dx} \operatorname{sinh}^{-1} x = \frac{1}{\sqrt{1+x^2}}

\frac{d}{dx} \operatorname{cosh}^{-1} x = \frac{1}{\sqrt{x^2-1}}, \frac{}{}x > 1

\frac{d}{dx} \operatorname{tanh}^{-1} x = \frac{1}{1-x^2}, \frac{}{}|x| < 1

\frac{d}{dx} \operatorname{cosech}^{-1} x = -\frac{1}{|x|\sqrt{1+x^2}}, \frac{}{}x\ne 0

\frac{d}{dx} \operatorname{sech}^{-1} x = -\frac{1}{x\sqrt{1-x^2}}, \frac{}{}0<x<1

\frac{d}{dx} \operatorname{coth}^{-1} x = \frac{1}{1-x^2}, \frac{}{}|x| > 1

Transcendental Functions[edit]

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