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