6.5: Impulse functions

Unit impulse function = Dirac delta function is a generalized function with the properties:

1. $\delta(t) =0, t \not=0$
2. $\int_{\infty}^{-\infty} \delta(t) dt =1$
3. $\mathcal{L}(\delta(t-t_0))=e^{-s t_0}$

Since the first item is just a definition we will prove 2. and 3.

Proof of 2.

We let $d_k(t) = \begin{cases} \frac{1}{2k} \quad -k <t<k \\ 0 \quad t \leq -k \\ 0 \quad t \geq -k  \end{cases}$

We note that $\lim_{k \to 0} d_k(t)=0$ if $t \not=0$ and $\lim_{k \to 0} \int_{-\int}^{\int} d_k(t) = \lim_{k \to 0} 1=1 = \int_{-\infty}^{\infty}\delta(t) dt.$

Proof of 3.

$$\notag \begin{align} \mathcal{L}(\delta(t-t_0)) &= \lim_{k \to 0^+}\mathcal{L}\left(d_k(t-t_0)\right) \notag\\ &= \lim_{k \to 0} \int_0^{\infty}e^{-st}\left(d_k(t-t_0)\right)dt \notag \\ &= \lim _{k \to 0} \frac{1}{2 k} \int_{t_0-k}^{t_0+k} e^{-s t} d t \notag\\ &= \lim _{k \to 0} \frac{-1}{2 s k} e^{-s t} \Big|_{t_0-k} ^{t_0+k} \notag\\ &= \lim _{k \to 0} \frac{1}{2 s k} e^{-s t_0}\left(e^{s k}-e^{-s k}\right) \notag\\ & =\lim _{k \to 0} \frac{\sinh (s k)}{s k} e^{-s t_0}\notag\\ &=\lim _{k \to 0} \frac{s \cosh (s k)}{s} e^{-s t_0}=e^{-s t_0} \notag \end{align}$$

Note

The following are some useful identities:

1. $\sin (t)=\frac{e^{i t}-e^{-i t}}{2 i}$
2. $\cos (t)=\frac{e^{i t}+e^{-i t}}{2}$
3. $\sinh (t)=\frac{e^t-e^{-t}}{2}$
4. $\cosh (t)=\frac{e^t+e^{-t}}{2}$
5. $[\sinh(t)]' = \cosh(t)$
6. $[\cosh(t)]' = \sinh(t)$
7. $\sinh (0)=\frac{e^0-e^0}{2}=0$
8. $\cosh (0)=\frac{e^0+e^0}{2}=1$

Intro to Group Theory
Define the $\cdot$ product on $R^2$ by
$$\notag \left(x_1, y_1\right) \cdot\left(x_2, y_2\right)=\left(x_1 x_2-y_1 y_2, x_1 y_2-y_1 x_2\right)$$

Note $\cdot$ is
1.) commutative:$\left(x_1, y_1\right) \cdot\left(x_2, y_2\right)=\left(x_1 x_2-y_1 y_2, x_1 y_2-y_1 x_2\right) =\left(x_2 x_1-y_2 y_1, x_2 y_1-y_2 x_1\right)=\left(x_2, y_2\right) \cdot\left(x_1, y_1\right)$
2.) associative: $(f \cdot g) \cdot h=f \cdot(g \cdot h)$
3.) distributive w.r.t $+: f \cdot\left(g_1+g_2\right)=f \cdot g_1+f \cdot g_2$
4.) $\left(x_1, y_1\right) \cdot(0,0)=(0,0)$

Note: $(0,1) \cdot(0,1)=(-1,0)$