# Coshc

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$$u+\mathrm i v = \mathrm{coshc}(x+\mathrm i y)$$

Coshc or coshc is elementary function, defined with

$$\displaystyle \mathrm{coshc}(z)=\frac{\cosh(z)}{z}$$

where cosh is hyperbolic cosine,

$$\displaystyle \cosh(z)=\cos(\mathrm i z) = \frac{\mathrm e^z+\mathrm e^{-z} }{2}$$

The derivative of coshc, id est, cosec', can be expressed with

$$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \cosh'(z)$$ = $$\frac{\sinh(z)}{z}-\frac{\cosh(z)}{z^2}$$ = $$\mathrm{sinhc}(z) - \frac{\mathrm{coshc}(z)}{z}$$

Functions $$~y\!=\!\mathrm{coshc}(x)~$$ and $$~y\!=\!\mathrm{coshc}'(x)~$$ are shown in the $$x$$,$$y$$ plane.

Coshc is related with the cosc function with the simple relations

$$\displaystyle \mathrm{coshc}(z) = \mathrm i ~ \mathrm{cosc} ( \mathrm i z )$$ $$= \mathrm i \frac{\cos(\mathrm i z)}{\mathrm i z}$$
$$\displaystyle \mathrm{cosc}(z) = \mathrm i ~ \mathrm{coshc} ( \mathrm i z )$$ $$= \mathrm i \frac{\cosh(\mathrm i z)}{\mathrm i z}$$

quite analogous to the relation between cos and cosh.

Minimum of $$\mathrm{coshc}(x)$$ is realized at

$$x = H\approx 1.199678640257734 ~$$ ; $$~\mathrm{coshc}'(H)=0$$.
$$J = \mathrm{coshc}(H) \approx 1.50887956153832~$$ .

These values are marked in the figure.

Cosch is odd function,

$$\mathrm{coshc}(z)=-\mathrm{coshc}(-z)$$

Coshc is holomorphic function with the only singularity, namely, pole at zero. As the real argument increases from zero to infinity, the cosch decreases until its mimumim $$J$$ at $$H$$, and again increase, almost exponentially, to infinity.

## Asymptotic expansions

At small values of the argument, coshc can be expanded as follows:

$$\displaystyle \mathrm{coshc}(z)=\frac{1}{z}+ \frac{1}{2!} z + \frac{1}{4!}z^3 + \frac{1}{6!}z^5+ ..$$

the series converges in the whole compelex plane except zero. This series can be inverted giving the expansion for the inverse function ArcCoshc:

$$\displaystyle \mathrm{ArcCoshc}(z)=\frac{1}{z}+ \frac{1}{2} \frac{1}{z^3}+ \frac{13}{24} \frac{1}{z^5}+ \frac{541}{720} \frac{1}{z^7}+..$$

With Mathematica software one can calculate many coefficients of this expansion. The series seems to converge while $$~\left|\frac{1}{z}\right|\!<\!\frac{1}{J}~$$, id est, $$|z|>J$$.

In vicinity of $$H$$, coshc can be expanded as follows:

$$\displaystyle \mathrm{coshc}(H+t)=J+\frac{\mathrm{coshc}''(H)}{2!} t^2 + \frac{\mathrm{coshc}'''(H)}{3!} t^3+..$$

which can be simplified to

$$\displaystyle \mathrm{coshc}(H+t)=J+\frac{J}{2} {t^2} - \frac{J}{3H}{t^3}+ \frac{(8\!+\!H^2)J}{24 H^3}{t^4} -\frac{(10\!+\!H^2)J}{30 H^3}{t^5}+..$$

The replacement of $$J$$ and $$H$$ to their approximations gives
$$\displaystyle \mathrm{coshc}(H\!+\!t)\approx$$ $$1.50887956153832$$ $$+ 0.75443978076916 \,t^2$$ $$- 0.4192454853893647 \,t^3$$ $$+ 0.4123348061601586 \,t^4$$ $$- 0.3332232452918583 \,t^5$$ $$+ 0.2798560879822419 \,t^6 +..$$

With Mathematica software one can easy calculate a dozen coefficients of this expansion and evaluate them.

The InverseSeries service allows to express the expansion of the inverse function ArcCoshc:

$$\displaystyle \mathrm{ArcCoshc}(J+t)=H+\left(\frac{2 t}{J}\right)^{1/2} + \frac{1}{2H} \left(\frac{2 t}{J}\right) -\frac{4+3H^2}{72\,H^2} \left(\frac{2 t}{J}\right)^{3/2} + O(t^2)$$

The replacement of $$J$$ and $$H$$ to thier approximations gives the approximation
$$\mathrm{ArcCoshc}(J+t)= 1.199678640257734$$ $$+1.1512978931181814\, t^{1/2}$$ $$+0.3682894163539221\, t$$ $$+0.12249075985507865\, t^{3/2}$$ $$-0.11091147932942629\, t^2$$ $$+0.02623011940391186\, t^{5/2}$$ $$+O(t^3)$$