sinc

Create a sinc function of a symbolic variable x.

syms x
sinc(x)

ans = 
$$\frac{\sin \left(\pi \hspace{0.17em}x\right)}{x\hspace{0.17em}\pi }$$

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 = 
$$\left(\begin{array}{cccc}0& 1& 0& -\frac{2}{3\hspace{0.17em}\pi }\end{array}\right)$$

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

vpa(S)

ans = $$\left(\begin{array}{cccc}0& 1.0& 0& -0.21220659078919378102517835116335\end{array}\right)$$

Although sinc can appear in tables of Fourier transforms, fourier does not return sinc in output.

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

syms x
fourier(rectangularPulse(x))

ans = 
$$\frac{\sin \left(\frac{w}{2}\right)+\cos \left(\frac{w}{2}\right)\hspace{0.17em}\mathrm{i}}{w}-\frac{-\sin \left(\frac{w}{2}\right)+\cos \left(\frac{w}{2}\right)\hspace{0.17em}\mathrm{i}}{w}$$

Show that fourier transforms sinc in terms of heaviside.

fourier(sinc(x))

ans = 
$$\frac{\pi \hspace{0.17em}\mathrm{heaviside}\left(\pi -w\right)-\pi \hspace{0.17em}\mathrm{heaviside}\left(-w-\pi \right)}{\pi }$$

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 = 
$$\frac{\frac{{\mathrm{e}}^{-\pi \hspace{0.17em}x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{2}-\frac{{\mathrm{e}}^{\pi \hspace{0.17em}x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{2}}{x\hspace{0.17em}\pi }$$

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

Differentiate sinc.

syms x
diff(sinc(x))

ans = 
$$\frac{\cos \left(\pi \hspace{0.17em}x\right)}{x}-\frac{\sin \left(\pi \hspace{0.17em}x\right)}{x^2\hspace{0.17em}\pi }$$

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 = 
$$\frac{\mathrm{sinint}\left(\pi \hspace{0.17em}x\right)}{\pi }+\frac{1}{2}$$

Find the Taylor expansion of sinc.

taylor(sinc(x))

ans = 
$$\frac{{\pi }^4\hspace{0.17em}{x}^4}{120}-\frac{{\pi }^2\hspace{0.17em}{x}^2}{6}+1$$

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

Prove the identity $\mathrm{sinc}\left(\mathit{x}\right)=\frac{1}{\Gamma \left(1+\mathit{x}\right)\Gamma \left(1-\mathit{x}\right)}$.

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

ans = logical
   1