View source for Cosc

[[File:CoscmapT100.png|800px|right|thumb|[[Complex map]] of [[cosc]], \(u+\mathrm i v = \mathrm{cosc}(x+\mathrm i y)\) ]]
'''Cosc'''\(=\)cosc is [[elementary function]] defined with
: \( \displaystyle \mathrm{cosc}(z)=\frac{\cos(z)}{z}\)

==History and ethimology==
First, the name [[Cip]] was suggested, as derivative of the [[IntegralCos]], named also [[Ci]], id est, "Ci-Prime";
and the name [[ArcCip]] of the inverse function had been suggested.
Then it happened that the similar formalism should be applied for function [[sinc]] defined with
: \( \displaystyle \mathrm{sinc}(z) = \frac{ \sin(z)}{z}\)
For the zero argument, it is defined that \(\mathrm{sinc}(0)=1\).
Function [[sinc]] already has etablishen name
<ref name="wolframsinc">
http://mathworld.wolfram.com/SincFunction.html
</ref>. Therefore, the natural choice for the similar function with [[cos]] instead of [[sin]] is [[cosc]], and the good name for the [[inverse function]] is [[ArcCosc]] or simplly [[acosc]].

==Complex map of [[cosc]]==
The [[complex map]] of function [[cosc]] is shown at the upper figure. It represents \(f\!=\!\mathrm{cosc}(x\!+\!\mathrm i y)\) in the \(~x\), \(y~\) plane. 
Levels  \(~u\!=\!\Re(f)\!=\!\mathrm{cost}~\) and
levels  \(~v\!=\!\Im(f)\!=\!\mathrm{cost}~\) are drawn.

Click the image to see the code used for its generation.

==Symmetry==
Cosc is real antisymmetric function:
: \(\mathrm{cosc}(-z)=- \mathrm{cosc}(z)\)
: \(\mathrm{cosc}(z^*)=-\mathrm{cosc}(z)^*\)

==Related functions==
[[File:SazaeconT.png|500px|right|thumb| cosc and [[cohc]] at real argument]]
For the case of argument with small real part (but, perhaps, large imaginary part), the modified function [[cosh]] is useful;
: \(\displaystyle \mathrm{cohc}(z)=\frac{\cosh(z)}{z}= \mathrm i ~ \frac{\cos(\mathrm{i}\, z)}{\mathrm{i} \, z}= \mathrm i ~ \mathrm{cosc}(\mathrm i z)\)

In the part of the [[complex plane]], the similar relation takes place for the [[inverse function]]s, id est, for 
[[ArcCosc]] and [[ArcCohc]].

The analysis of [[cohc]] and its inverse function [[ArcCohc]] is used to make the efficient implementation of the inverse function [[ArcCosc]]=cosc\(^{-1}\).

==Behavior along the real axis==
Due to the symmetry above, it is sufficient to consider behavior of cosc along the positive part of the real axis. Graphic of cosc is comparef to that for [[cohc]] in the figure at right. Values between zero and [[Sazae]] are used in the construction of function [[ArcCosc]].

The expansion at zero can be written as follows:
: \(\displaystyle \mathrm{cosc}(z) = \sum_{n=0}^{\infty}
\frac{(-1)^n z^{2n-1}}
{(2n)!}
\) \(\displaystyle = \frac{1}{z} - \frac{z}{2} +\frac{z^3}{24}-\frac{z^5}{720}+...\)

The series converges in the whole [[complex plane]] except \(z\!=\!0\).

Function [[cosc]] decreases from \(+\infty\) at zero, takes value zero at \(\pi\) and continues decreasing until its minimum called [[Sazae]];
: \( \mathrm{Sazae} \approx  2.798386045783887\)
: \( \mathrm{cosc}(\mathrm{Sazae})=\mathrm{Tarao} \approx −0.33650841691839534\)
: \( \mathrm{cosc}'(\mathrm{Sazae})=0\)

The expansion at Sazae has form
: \(\mathrm{cosc}(\mathrm{Sazae}+t)=
\mathrm{Tarao} \) \( \displaystyle
-\frac{
\mathrm{Tarao}}{2}t^2
-\frac{
\mathrm{Tarao}}{3 ~
\mathrm{Sazae}}  t^3
\) \( \displaystyle
+ 
\frac{\left(\mathrm
{Sazae}^2\!-\!8\right)
\mathrm{Tarao}}{24 ~
\text{Sazae}^2}   t^4 + .. \)
<!--
\( \) \displaystyle
+\frac{\left(\text{Sazae}^2-10\right) t^5
\text{Tarao}}{30 ~
\text{Sazae}^3}-\frac{t^6
\left(\left(\text{Sazae}^4 -24\, \text{Sazae}^2+240\right)
\text{Tarao}\right)}{720 ~
\text{Sazae}^4} \( \) \displaystyle
-\frac{t^7
\left(\left(\text{Sazae}^4-28\, \text{Sazae}^2+280\right)
\text{Tarao}\right)}{840 ~
\text{Sazae}^5}  \( \) \displaystyle
+O\left(t^8\right)
\(
!-->
[[Mathematica]] allows to calculate many terms of this expansion. The radius of convergence of the series is 
\) \mathrm{Sazae} \approx  2.798386045783887\(, which is just distance to the closest singularity.

At larger values of the argument, cosc oscillates, slowly decaying to zero.

==Invese function==
In [[TORI]], there is special article about function [[ArcCosc]]=cosc\)^{-1}$.
For the beginning of year 2012, the only [[principal branch]] of the inverse function is described and supplied with the efficient numerical implementation in [[C++]]. The other branches are expected be considered in the similar way.

==References==
<references/>

==Keywords==
[[Trigonometriv functions]], [[cos]], [[sin]]

[[Category:Cosc]]
[[Category:Elementary function]]
[[Category:Articles in English]]