Basic Special Functions

Basic Special Functions are Absolute Value Function, Signum Function, Floor and Ceiling Functions, Fractional Part Function and Integer Function.

$$ \begin{array}{|l|l|l|l|l|} \hline \text { notation } & \text { name } & \text { S\&O } & \text { Graham et al. } & \text { Wolfram Language } \\ \hline\lceil x\rceil & \text { ceiling function } & - & \text { ceiling, least integer } & \text { Ceiling[x] } \\ \hline \bmod (m, n) & \text { congruence } & - & -- & \operatorname{Mod}[\mathrm{m}, \mathrm{n}] \\ \hline\lfloor x\rfloor & \text { floor function } & \text { Int }(x) & \text { floor, greatest integer, integer part } & \text { Floor }[\mathrm{x}] \\ \hline x-\lfloor x\rfloor & \text { fractional value } & \text { frac }(x) & \text { fractional part or }\{x\} & \text { SawtoothWave }[\mathrm{x}] \\ \hline \operatorname{sgn}(x)(|x|-\lfloor|x|\rfloor) & \text { fractional part } & \mathrm{Fp}(x) & \text { no name } & \text { FractionalPart }[\mathrm{x}] \\ \hline \operatorname{sgn}(x)\lfloor|x|\rfloor & \text { integer part } & \operatorname{Ip}(x) & \text { no name } & \text { IntegerPart }[\mathrm{x}] \\ \hline \operatorname{nint}(x) & \text { nearest integer function } & - & - & \text { Round }[\mathrm{x}] \\ \hline m \backslash n & \text { quotient } & - & - & \text { Quotient }[\mathrm{m}, \mathrm{n}] \\ \hline \end{array} $$ *(S&O indicates Spanier and Oldham (1987).)

  1. Modulus Function or the Absolute Value Function. It is defined as $$ |x|= \begin{cases}x, & \text { if } x \geq 0 \\ -x, & \text { if } x<0\end{cases} $$ The absolute value of a complex number $z=x+iy$, where $x$ and $y$ are real numbers, is defined as $|z| = +\sqrt{x^2+y^2}$, also called the complex modulus.

The modulus function has the following properties: (i) For any real number $x$, ${\sqrt {x^{2}}}=|x|$

(ii) If $a, b$ are positive real numbers, then $x^{2} \leq a^{2} \Leftrightarrow|x| \leq a \Leftrightarrow-a \leq x \leq a$ $x^{2} \geq a^{2} \Leftrightarrow|x| \geq a \Leftrightarrow x \leq-a$ or, $x \geq a$ $x^{2}<a^{2} \Leftrightarrow|x|<a \Leftrightarrow-a<x<a>a^{2} \Leftrightarrow|x|>a \Leftrightarrow xa$ $a^{2} \leq x^{2} \leq b^{2} \Leftrightarrow a \leq|x| \leq b \Leftrightarrow x \in[-b,-a] \cup[a, b]$ $a^{2}<x^{2}<b^{2} \Leftrightarrow a<|x|<b \Leftrightarrow x \in(-b,-a) \cup(a, b)$

(iii) For real numbers $x$ and $y$, we have $|x+y|=|x|+|y| \Leftrightarrow(x \geq 0$ and $y \geq 0)$ or, $(x<0$ and $y<0)$ $|x-y|=|x|-|y| \Leftrightarrow(x \geq 0, y \geq 0$ and $|x| \geq|y|$ ) or, $(x \leq 0, y \leq 0$ and $|x| \geq|y|)$ $|x \pm y| \leq|x|+|y|$ $|x \pm y| \geq|| x|-| y||$

  1. Signum Function. It is defined as $$\operatorname{sgn}(x)=\left\{\begin{array}{cl}\frac{|x|}{x} & \text { or } \frac{x}{|x|} ; x \neq 0 \\ 0 & ; x=0\end{array}\right.$$ $$=\left\{\begin{array}{r}-1 ; x0\end{array}\right.$$

  2. Floor Function or the Greatest Integer Function is the Largest integer less than or equal to a given number. If $n$ is an integer and $x$ is a real number between $n$ and $n+1$, then

(i) $[-n]=-[n]$

(ii) $[x+k]=[x]+k$ for any integer $k$.

(iii) $[-x]=-[x]-1$

(iv) $[x]+[-x]=\left\{\begin{aligned}-1, & \text { if } x \notin Z \\ 0, & \text { if } x \in Z \end{aligned}\right.$

(v) $[x]-[-x]=\left\{\begin{array}{ll}2[x]+1, & \text { if } x \in Z \\ 2[x] & \text { if } x \in Z\end{array}\right.$

(vi) $[x] \geq k \Rightarrow x \geq k$, where $k \in Z$

(vii) $[x] \leq k \Rightarrow xk \Rightarrow x>k+1$, were $k \in Z$

(ix) $[x]<k \Rightarrow x<k$, where $k \in Z$

(x) $[x+y]=[x]+[y+x-[x]]$ for all $x, y \in R$

(xi) $[x]+\left[x+\frac{1}{n}\right]+\left[x+\frac{2}{n}\right]+\ldots+\left[x+\frac{n-1}{n}\right]=[n x], n \in N$

  1. Ceiling Function or the Smallest Integer Funtion is the Smallest integer larger than or equal to a given number. Following are some properties of smallest integer function: (i) $[-n\rceil=-\lceil n\rceil, \quad$ where $n \in Z$ (ii) $\lceil-x\rceil=-\lceil x\rceil+1$, where $x \in R-Z$ (iii) $\lceil x+n\rceil=\lceil x\rceil+n$, where $x \in R-Z$ and $n \in Z$ (iv) $\lceil x\rceil+\left\lceil-x\right\rceil=\left\{\begin{array}{ll}1 & , \text { if } x \notin Z \\ 0, & \text { if } x \in Z\end{array}\right.$ (v) $\lceil x\rceil-\lceil-x\rceil=\left\{\begin{array}{ll}2\lceil x\rceil-1, & \text { if } x \notin Z \\ 2\lceil x\rceil, & \text { if } x \in Z\end{array}\right.$

  2. Fractional Part Function. It is defined as $$ \operatorname{frac}(x)=\left\{\begin{array}{ll} x-\lfloor x\rfloor & x \geq 0 \\ x-\lceil x\rceil & x0 \\ 0 \text { if } x=0, \\ \lceil x\rceil \text { if } x<0\end{array}=\left\{\begin{array}{l}\lfloor x\rfloor \text { if } x \geq 0 \\ \lceil x\rceil \text { if } x<0\end{array}\right.\right.$$

The Fractional Part Function gives the fractional (non-integer) part of a real number and The Integer Part Function gives the integer part of a real number.