basic
Basic notations
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Basic
| $ \displaystyle f:A \rightarrow B$ | This defines a function $f$ whose arguments take values from the set $A$ and which returns a unique value from the set $B$ for each corresponding argument. For example consider the function $f:\mathbb{R} \rightarrow \mathbb{R}_+$, given by $f(x) = 10x$. By evaluating this function at various values of the argument the following results are obtained, $f(10.2) = 102$, $f(-5.8) = -58$, $f(0) = 0$. | ||
| $ \displaystyle f \circ g $ | This defines the composite function $f \circ g$ defined by the relation $f \circ g (x) = f(g(x))$ for all values of $x$, where $f$ and $g$ are apropriately chosen functions. | ||
| $\displaystyle |x|$ | The absolute value function defined as: \( |x| = \left\{ \begin{array}{rl} x, & \text{for } x \geq 0 \\ -x, & \text{for } x < 0 \\ \end{array} \right. \) For example $|1.23| = 1.23$, $|0| = 0$, $|-3.14| = 3.14$. | ||
| $\displaystyle x \approx \alpha $ | This tells that $x$ has an approximate value of $\alpha$, in other words $|x - \alpha| < \varepsilon$ where $\varepsilon$ is a very small positive value. For example $\pi \approx 3.1415926535898$. | ||
| $\displaystyle x \gg \alpha $ | This tells that $x$ has a much greater value than $\alpha$, for example $10000 \gg 5.34$. | ||
| $ \displaystyle x \ll \alpha$ | This tells that $x$ has a much smaller value than $\alpha$, for example $-31000 \ll 0$. | ||
| $\displaystyle {\rm sign}(x) $ | The signum function defined as: \( \rm sign(x) = \left\{ \begin{array}{rl} 1, & \text{for } x > 0 \\ 0, & \text{for } x = 0 \\ -1, & \text{for } x < 0 \end{array} \right . \) For example ${\rm sign}(-1034.5) = -1$, ${\rm sign}(0) = 0$, ${\rm sign}(11.3) = 1$. | ||
| $ \displaystyle \lfloor x \rfloor $ | The floor function which gives the largest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$, $\lfloor -3.3 \rfloor = -4$. | ||
| $ \displaystyle \lceil x \rceil $ | The ceiling function which gives the smallest integer not less than $x$. For example $\lceil 5.4 \rceil = 6$, $\lceil -2.8 \rceil = -2$. | ||
| $\displaystyle \sqrt{x} $ | The square root of $x$ is a non-negative real number $y$ such that $y^2 = x$. Obviously $x$ always has to be non-negative. For example $\sqrt{144} = 12$, $\sqrt{256} = 16$ while $\sqrt{-16}$ is not a proper expression since $-16$ is negative. | ||
| $ \displaystyle \sqrt[n]{x} $ | The $n$-th root of $x$ is a real number $y$ such that $y^n = x$. If $n$ is even $x$ needs to be non-negative, as is the case for the square root with $n = 2$. For example $\sqrt[3]{-8} = -2$, $\sqrt[5]{1024} = 4$, while $\sqrt[4]{-256}$ is not a proper expression since the order of the root $n$ is even but $-256$ is negative. | ||
| $ \displaystyle \sum_{i=1}^n E(i) $ | If $E(i)$ is an expression that depends on the value of $i$, this evaluates the following sum \( \sum_{i=1}^n E(i) = E(1) + E(2) + E(3) + \ldots + E(n). \) For example if $E(i) = \sin(i\alpha)$ then the above sum becomes \( \sum_{i=1}^n \sin(i\alpha) = \sin(\alpha) + \sin(2\alpha) + \sin(3\alpha) + \ldots + \sin(n\alpha). \) | ||
| $ \displaystyle \sum_{P(i)} E(i) $ | If $E(i)$ is an expression that depends on the value of $i$, this evaluates the sum over those indices $i$ for which the predicate $P(i)$ becomes true. For example if $E(i) = \sin(i)$ and the predicate is $P(i)$: "$i$ is a prime and is less than $12$" then the sum becomes \( \sum_{\displaystyle i \mbox{ is a prime} \atop{\displaystyle i < 12}} \sin(i) = \sin(2) + \sin(3) + \sin(5) + \sin(7) + \sin(11). \) Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting $P(i)$: "$1 \leq i \leq n$" this generalised sum becomes the previous easier sum \( \sum_{1 \leq i \leq n} E(i) = \sum_{i=1}^n E(i). \) | ||
| $ \displaystyle \prod_{i=1}^n E(i) $ | If $E(i)$ is an expression that depends on the value of $i$, this evaluates the following product \( \prod_{i=1}^n E(i) = E(1) \cdot E(2) \cdot E(3) \cdot \ldots \cdot E(n). \) For example if $E(i) = i + \sqrt{2 \cdot i}$ then the above product becomes \( \prod_{i=1}^n i + \sqrt{2 \cdot i} = (1 + \sqrt{2 \cdot 1}) \cdot (2 + \sqrt{2 \cdot 2}) \cdot (3 + \sqrt{2 \cdot 3}) \cdot \ldots \cdot (n + \sqrt{2 \cdot n}). \) | ||
| $ \displaystyle \prod_{P(i)} E(i) $ | If $E(i)$ is an expression that depends on the value of $i$, this evaluates the product over those indices $i$ for which the predicate $P(i)$ becomes true. For example if $E(i) = i$ and the predicate is $P(i)$: "$i$ is a prime and is less than $16$" then the product becomes \[ \sum_{\displaystyle i \mbox{ is a prime} \atop{\displaystyle i < 16}} i = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13. \] Notice that if the predicate consists of several conditions, these are written one below the other as above. By letting $P(i)$: "$1 \leq i \leq n$" this generalised product becomes the previous easier product \[ \prod_{1 \leq i \leq n} E(i) = \prod_{i=1}^n E(i). \] | ||
| $ \displaystyle n! $ | The factorial function of argument $n \in \mathbb{N}$ defined through \( n! = \prod_{i=1}^n i \qquad 0! = 1. \) | ||
| $ \displaystyle \binom{n}{k} $ | The binomial coefficient of the natural numbers $n$ and $k$ defined through \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \qquad n \geq k \geq 0. \) |