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

Proof: █

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. █