The hyperbolic cosine function is defined by $$\mathrm{cosh}(z)=\dfrac{e^z + e^{-z}}{2}$$

Complex Cosh.jpg

Domain coloring of analytic continuation of $\cosh$.

Properties

Theorem

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

Proof

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

References

Theorem

The Weierstrass factorization of $\cosh(x)$ is $$\cosh x = \displaystyle\prod_{k=1}^{\infty} 1 + \dfrac{4x^2}{(2k-1)^2\pi^2}.$$

Proof

References

Theorem

The following formula holds: $$\cosh(az)=az {}_0F_1 \left( ; \dfrac{1}{2}; \dfrac{(az)^2}{4} \right),$$ where $\cosh$ denotes the hyperbolic cosine and ${}_0F_1$ denotes the hypergeometric pFq.

Proof

References

Theorem

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