Symbols:Real Analysis

Symbols used in Real Analysis

Convolution Integral

$\map f t * \map g t$

Let $f$ and $g$ be real functions which are integrable.

The convolution integral of $f$ and $g$ is defined as:

$\ds \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$

The $\LaTeX$ code for \(\map f t * \map g t\) is \map f t * \map g t .

Convolution of Real Sequences

$\sequence {f_i} * \sequence {g_i}$

Let $\sequence f$ and $\sequence g$ be real sequences.

The convolution of $f$ and $g$ is defined as:

$\ds \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop = 0}^i f_j g_{i - j}$

The $\LaTeX$ code for \(\sequence {f_i} * \sequence {g_i}\) is \sequence {f_i} * \sequence {g_i} .

Cross-Correlation Integral

$\map f t \star \map g t$

The cross-correlation of $f$ and $g$ is defined as:

$\ds \map f t \star \map g t := \int_{-\infty}^\infty \map f u \map g {t + u} \rd u$

The $\LaTeX$ code for \(\map f t \star \map g t\) is \map f t \star \map g t .

Limit

$\to$

$\map f x$ tends to the limit $L$ as $x$ tends to $c$, is denoted:

$\map f x \to L$ as $x \to c$

or

$\ds \lim_{x \mathop \to c} \map f x = L$

The latter is voiced:

the limit of $\map f x$ as $x$ tends to $c$.

The $\LaTeX$ code for \(\map f x \to L\) is \map f x \to L .

The $\LaTeX$ code for \(\ds \lim_{x \mathop \to c} \map f x\) is \ds \lim_{x \mathop \to c} \map f x .

Limit from the Left

Notations that may be encountered for the limit from the left:

$\ds \lim_{x \mathop \to b^-} \map f x$

$\map f {b^-}$ or $\map f {b -}$

$\map f {b - 0}$

$\ds \lim_{x \mathop \uparrow b} \map f x$

$\ds \lim_{x \mathop \nearrow b} \map f x$

The $\LaTeX$ code for \(\ds \lim_{x \mathop \to b^-} \map f x\) is \ds \lim_{x \mathop \to b^-} \map f x .

The $\LaTeX$ code for \(\map f {b^-}\) is \map f {b^-} .

The $\LaTeX$ code for \(\map f {b -}\) is \map f {b -} .

The $\LaTeX$ code for \($\map f {b - 0}\) is $\map f {b - 0} .

The $\LaTeX$ code for \(\ds \lim_{x \mathop \uparrow b} \map f x\) is \ds \lim_{x \mathop \uparrow b} \map f x .

The $\LaTeX$ code for \(\ds \lim_{x \mathop \nearrow b} \map f x\) is \ds \lim_{x \mathop \nearrow b} \map f x .

Limit from the Right

Notations that may be encountered for the limit from the right:

$\ds \lim_{x \mathop \to a^+} \map f x$

$\map f {a^+}$ or $\map f {a +}$

$\map f {a + 0}$

$\ds \lim_{x \mathop \downarrow a} \map f x$

$\ds \lim_{x \mathop \searrow a} \map f x$

The $\LaTeX$ code for \(\ds \lim_{x \mathop \to a^+} \map f x\) is \ds \lim_{x \mathop \to a^+} \map f x .

The $\LaTeX$ code for \(\map f {a^+}\) is \map f {a^+} .

The $\LaTeX$ code for \(\map f {a +}\) is \map f {a +} .

The $\LaTeX$ code for \(\map f {a + 0}\) is \map f {a + 0} .

The $\LaTeX$ code for \(\ds \lim_{x \mathop \downarrow a} \map f x\) is \ds \lim_{x \mathop \downarrow a} \map f x .

The $\LaTeX$ code for \(\ds \lim_{x \mathop \searrow a} \map f x\) is \ds \lim_{x \mathop \searrow a} \map f x .

Symbols commonly used in both Real Analysis and Number Theory

Ceiling

$\ceiling x$

The ceiling function of $x$: the smallest integer greater than or equal to $x$.

The $\LaTeX$ code for \(\ceiling x\) is \ceiling x .

Floor

$\floor x$

The floor function of $x$: for $x \in \R$, the greatest integer less than or equal to $x$.

The $\LaTeX$ code for \(\floor x\) is \floor x .

Nearest Integer

$\nint x$

The nearest integer function is defined as:

$\forall x \in \R: \nint x = \begin {cases}

\floor {x + \dfrac 1 2} & : x \notin 2 \Z + \dfrac 1 2 \\ x - \dfrac 1 2 & : x \in 2 \Z + \dfrac 1 2 \end{cases}$ where $\floor x$ is the floor function.

The $\LaTeX$ code for \(\nint x\) is \nint x .

Fractional Part

$\fractpart x$

Let $x \in \R$ be a real number.

Let $\floor x$ be the floor function of $x$.

The fractional part of $x$ is the difference:

$\fractpart x := x - \floor x$

The $\LaTeX$ code for \(\fractpart x\) is \fractpart x .