Difference between revisions of "Sinh"

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(Properties)
(Properties)
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{{:Derivative of sinh}}
 
{{:Derivative of sinh}}
 
{{:Weierstrass factorization of sinh}}
 
{{:Weierstrass factorization of sinh}}
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<strong>Theorem:</strong> The following formula holds:
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$$\sinh(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^{2k+1}}{(2k+1)!}.$$
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<strong>Proof:</strong> █
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<center>{{:Hyperbolic trigonometric functions footer}}</center>
 
<center>{{:Hyperbolic trigonometric functions footer}}</center>

Revision as of 05:32, 16 May 2015

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

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

Theorem: The following formula holds: $$\sinh(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^{2k+1}}{(2k+1)!}.$$

Proof:

<center>Hyperbolic trigonometric functions
</center>