Difference between revisions of "Sinh"

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{{:Relationship between sinh and hypergeometric 0F1}}
 
{{:Relationship between sinh and hypergeometric 0F1}}
 
{{:Weierstrass factorization of sinh}}
 
{{:Weierstrass factorization of sinh}}
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{{:Taylor series for sinh}}
<strong>Theorem:</strong> The following formula holds:
 
$$\sinh(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^{2k+1}}{(2k+1)!}.$$
 
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<strong>Proof:</strong> █
 
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{{:Relationship between Bessel I sub 1/2 and sinh}}
 
{{:Relationship between Bessel I sub 1/2 and sinh}}
 
{{:Relationship between sin and sinh}}
 
{{:Relationship between sin and sinh}}

Revision as of 04:51, 7 May 2016

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 following formula holds: $$\sinh(az)=az {}_0F_1 \left( ; \dfrac{3}{2} ; \dfrac{(az)^2}{4} \right),$$ where $\sinh$ denotes the hyperbolic sine and ${}_0F_1$ denotes the hypergeometric pFq.

Proof

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 Taylor series holds for all $z \in \mathbb{C}$: $$\sinh(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^{2k+1}}{(2k+1)!},$$ where $\sinh$ is the hyperbolic sine.

Proof

References

Theorem

The following formula holds: $$I_{\frac{1}{2}}(z)=\sqrt{\dfrac{2}{\pi z}}\sinh(z),$$ where $I_{\frac{1}{2}}$ denotes the modified Bessel function of the first kind and $\sinh$ denotes the hyperbolic sine.

Proof

References

Theorem

The following formula holds: $$\sin(z)=-i \sinh(iz),$$ where $\sin$ denotes the sine and $\sinh$ denotes the hyperbolic sine.

Proof

From the definition of $\sin$ and $\sinh$ and the reciprocal of i, $$-i\sinh(iz) = \dfrac{e^{iz}-e^{-iz}}{2i} =\sin(z),$$ as was to be shown.

References

Theorem

The following formula holds: $$\sinh(z)=-i\sin(iz),$$ where $\sinh$ is the hyperbolic sine and $\sin$ is the sine.

Proof

By definition, $$\sinh(z) = \dfrac{e^{z}-e^{-z}}{2},$$ and so by the definition of $\sin$ and the reciprocal of i, we see $$-i\sinh(iz)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ as was to be shown. █

References

Theorem

The following formula holds: $$\tan(\mathrm{gd}(x))=\sinh(x),$$ where $\tan$ denotes tangent, $\mathrm{gd}$ denotes the Gudermannian, and $\sinh$ denotes the hyperbolic sine.

Proof

References

Theorem

The following formula holds: $$\sinh(\mathrm{gd}^{-1}(x))=\tan(x),$$ where $\sinh$ is the hyperbolic sine, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\tan$ is the tangent.

Proof

References

See Also

Sine
Arcsinh

<center>Hyperbolic trigonometric functions
</center>