# Difference between revisions of "Sinh"

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## Revision as of 01:47, 17 April 2015

The hyperbolic sine function is defined by $$\mathrm{sinh}(z)=\dfrac{e^z-e^{-z}}{2}.$$

- Complex Sinh.jpg
Domain coloring of analytic continuation of $\sinh$.

# Properties

## Theorem

The following formula holds:
$$\dfrac{\mathrm{d}}{\mathrm{d}z} \sinh(z) = \cosh(z),$$
where $\sinh$ denotes the **hyperbolic sine** and $\cosh$ denotes the hyperbolic cosine.

## Proof

From the definition, $$\sinh(z) = \dfrac{e^z-e^{-z}}{2},$$ and so using the derivative of the exponential function, the linear property of the derivative, the chain rule, and the definition of the hyperbolic cosine, $$\dfrac{\mathrm{d}}{\mathrm{d}z} \sinh(z)=\dfrac{e^z + e^{-z}}{2}=\cosh(z),$$ as was to be shown. █

## References

## Theorem

The Weierstrass factorization of **$\sinh(x)$** is
$$\sinh(x)=x\displaystyle\prod_{k=1}^{\infty} 1 + \dfrac{x^2}{k^2\pi^2}.$$

## Proof

## References

**Hyperbolic trigonometric functions**