# 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 -} .