Tanc function

The tanc function in the range from −11 to 11

The tanc function is a math function made by

${\displaystyle \operatorname {tanc} (x):={\dfrac {\tan(x)}{x}}}$

is defined. Here denotes the ordinary tangent . ${\displaystyle \tan(x)}$

Analogous to the more common sinc function is the function of the liftable definition gap at by its limit continued. Despite its structural similarity, it does not count among the cardinal functions . ${\displaystyle x=0}$ ${\displaystyle \operatorname {tanc} (0)=1}$

properties

General

At the liftable singularity at , the functions are continued continuously by the limit value or , which results from de l'Hospital's rule ; sometimes the definition equation is also written with case distinction: ${\displaystyle x=0}$ ${\displaystyle \operatorname {tanc} (0)=1}$ ${\displaystyle \operatorname {tanc} (0)=1}$

${\displaystyle \operatorname {tanc} (x)={\begin{cases}{\frac {\tan x}{x}}&x\neq 0\vee x\neq \pi n\\1&x=0\end{cases}}}$.

zeropoint

The tanc function has its zeros at integer multiples of : ${\displaystyle \pi }$

${\displaystyle \operatorname {tanc} (x)={\frac {\tan(x)}{x}}=0}$ applies to ${\displaystyle \ x\in \{n\pi \ \mid \ n\in \mathbb {Z} \setminus \{0\}\}}$

Asymptotic borderline behavior

For coordinates of the form with an integer , the function has an asymptotic limit behavior , since it diverges. ${\displaystyle x}$ ${\displaystyle x_{n}={\frac {1}{2}}+\pi n}$ ${\displaystyle n}$ ${\displaystyle \operatorname {tanc} (x_{n})}$ ${\displaystyle \tan(x_{n})}$

Derivatives

The first derivative of is given by: ${\displaystyle \operatorname {tanc} (x)}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\;\operatorname {tanc} (x)={\frac {\sec ^{2}(x)}{x}}-{\frac {\tan(x)}{x^{2}}}}$

Demarcation

This is structurally very similar to the function, although it is not a cardinal function, but has gaps in its definition . Therefore, in physics, for example, the use of is more common. ${\displaystyle \operatorname {tanc} (x)}$ ${\displaystyle \operatorname {sinc} (x)}$ ${\displaystyle {\frac {1}{2}}+\pi n}$ ${\displaystyle \operatorname {sinc} (x)}$

Web links

• Information on the tanc function at Wolfram Research

Individual evidence

1. ↑ Eric W. Weisstein: Tanc Function. Retrieved January 23, 2020 (English). 
2. ^ Cardinal Function , Eric W. Weisstein, Wolfram Web Resource.