# sinc

Normalized sinc function

## Syntax

``sinc(x)``

## Description

example

````sinc(x)` returns `sin(pi*x)/(pi*x)`. The symbolic `sinc` function does not implement floating-point results, only symbolic results. Floating-point results are returned by the `sinc` function in Signal Processing Toolbox™.```

## Examples

collapse all

```syms x sinc(x)```
```ans = sin(pi*x)/(x*pi)```

Show that `sinc` returns `1` at `0`, `0` at other integer inputs, and exact symbolic values for other inputs.

```V = sym([-1 0 1 3/2]); S = sinc(V)```
```S = [ 0, 1, 0, -2/(3*pi)]```

Convert the exact symbolic output to high-precision floating point by using `vpa`.

`vpa(S)`
```ans = [ 0, 1.0, 0, -0.21220659078919378102517835116335]```

Although `sinc` appears in tables of Fourier transforms, `fourier` does not return `sinc` in output.

Show that `fourier` transforms a pulse in terms of `sin` and `cos`.

`fourier(rectangularPulse(x))`
```ans = (cos(w/2)*1i + sin(w/2))/w - (cos(w/2)*1i - sin(w/2))/w```

Show that `fourier` transforms `sinc` in terms of `heaviside`.

```syms x fourier(sinc(x))```
```ans = (pi*heaviside(pi - w) - pi*heaviside(- w - pi))/pi```

Plot the sinc function by using `fplot`.

```syms x fplot(sinc(x))```

Rewrite the `sinc` function to the exponential function `exp` by using `rewrite`.

```syms x rewrite(sinc(x),'exp')```
```ans = ((exp(-pi*x*1i)*1i)/2 - (exp(pi*x*1i)*1i)/2)/(x*pi)```

Differentiate, integrate, and expand `sinc` by using the `diff`, `int`, and `taylor` functions, respectively.

Differentiate `sinc`.

```syms x diff(sinc(x))```
```ans = cos(pi*x)/x - sin(pi*x)/(x^2*pi)```

Integrate `sinc` from `-Inf` to `Inf`.

`int(sinc(x),[-Inf Inf])`
```ans = 1```

Integrate `sinc` from `-Inf` to `x`.

`int(sinc(x),-Inf,x)`
```ans = sinint(pi*x)/pi + 1/2```

Find the Taylor expansion of `sinc`.

`taylor(sinc(x))`
```ans = (pi^4*x^4)/120 - (pi^2*x^2)/6 + 1```

Prove an identity by defining the identity as a condition and using the `isAlways` function to check the condition.

Prove this identity.

$\text{sinc}\left(x\right)=\frac{1}{\Gamma \left(1+x\right)\Gamma \left(1-x\right)}.$

```syms x cond = sinc(x) == 1/(gamma(1+x)*gamma(1-x)); isAlways(cond)```
```ans = logical 1```

## Input Arguments

collapse all

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.