# Delta functions and derivatives

Nick Trefethen, 1st August 2012

(Chebfun example calc/DeltaDerivs.m)

[Tags: #DIRAC, #delta, #impulse, #deltafunction]

Here is a sine wave on the interval $[0,20]$ to which have been added a sequence of Dirac delta functions of random amplitudes, with a constant function then subtracted to make the mean zero:

x = chebfun('x',[0 20]); f = 0.5*sin(x); randn('seed',3) for j = 1:19 f = f + randn*dirac(x-j); end f = f - mean(f); LW = 'linewidth'; lw = 1.6; FS = 'fontsize'; fs = 12; plot(f,LW,1.6) title('f: a sine wave plus a sequence of delta impulses',FS,fs)

Can you explain each of these numbers?

max(f)

ans = Inf

min(f)

ans = -Inf

sum(f)

ans = -8.881784197001252e-16

norm(f,1)

ans = 24.404776060337156

norm(f,2)

ans = Inf

norm(f,inf)

ans = Inf

If we integrate $f$ with CUMSUM, each delta function becomes a jump:. The value at the left is $0$ because CUMSUM always does that, and the value at the right is $0$ because $f$ has zero mean.

g = cumsum(f); plot(g,'r',LW,1.6) title('The integral of f',FS,fs)

If we integrate a second time, we get a continuous function, that is, a function of class $C^0$:

h = cumsum(g); plot(h,LW,1.6,'color',[0 .7 0]) title('The second integral of f',FS,fs)

Our eye is good at detecting this degree of non-smoothness. One final integration gives a $C^1$ function whose lack of smoothness is not so obvious:

q = cumsum(h); plot(q,LW,1.6,'color',[1 .5 0]) title('The third integral of f',FS,fs)

Taking the third derivative of this last function brings us back where we started:

```
f2 = diff(q,3);
plot(f2,LW,1.6)
title('f again, obtained via a third derivative')
```