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