91影视

The Dirac Delta function which is represented as $\mathit{未}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$, is defined as $\mathit{胃}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{ll}\mathbf{1}& \mathbf{}\mathbf{x}\mathbf{鈮�}\mathbf{0}\\ \mathbf{0}\mathbf{,}& \mathbf{}\mathbf{x}\mathbf{=}\mathbf{0}\end{array}$, .the Dirac delta function has the property ${\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{鈭�}}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{未}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{d}\mathit{x}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$, where $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$is a continuous containing$\mathit{x}\mathbf{=}\mathbf{0}$.

(a)

Let function $u\left(x\right)=未\left(x\right)$and $v\left(x\right)=x$. Differentiate $u\left(x\right)=未\left(x\right)$and $v\left(x\right)=x$with respect to $x$.

$\begin{array}{l}u\text{'}\left(x\right)=\frac{d}{dx}未\left(x\right)\\ v^{\text{'}}\left(x\right)=1\end{array}$

Substitute $未\left(x\right)$for $u$, $x$for $v\left(x\right)$into $鈭�udv=uv-鈭�vdu$as,

localid="1657365897288" $$\begin{gathered} {鈭�}_{-鈭�}^{鈭�}未\left(x\right)dx={鈭�}_{-鈭�}^{鈭�}c\left(x\right)脳1dx \\ ={\overline{)x未\left(x\right)}}_{-鈭�}^{鈭�}-{鈭�}_{-鈭�}^{鈭�}x\frac{d}{dx}\left(未\left(x\right)\right)dx \end{gathered}$$ 鈥�.. (1)

Define $x未\left(x\right)$as,

$x未\left(\mathrm{x}\right)=\{\begin{array}{ll}0脳鈭�& \mathrm{x}=0\\ x=0& \mathrm{x}鈮�0\end{array}$

Thus the value of $x未\left(x\right)$is 0 for all x.

Hence, $x未{\overline{)\left(x\right)}}_{-鈭�}^{鈭�}=0$.

Now equation (1) can be rewritten as,

$$\begin{gathered} {鈭�}_{-鈭�}^{鈭�}未\left(x\right)dx={鈭�}_{-鈭�}^{鈭�}脳\frac{d}{dx}\left(未\left(x\right)\right)dx \\ ={鈭�}_{-鈭�}^{鈭�}x\frac{d}{dx}\left(未\left(x\right)\right)dx \end{gathered}$$ 鈥︹��. (2)

Equation (2) can be rewritten as $x\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{未}\left(\mathrm{x}\right)\right)=-未\left(\mathrm{x}\right)$

(b)

According to equation (1), ${鈭�}_{-鈭�}^{鈭�}未\left(x\right)dx={\overline{)x未\left(x\right)}}_{-鈭�}^{鈭�}-{鈭�}_{-鈭�}^{鈭�}脳\frac{d}{dx}\left(未\left(x\right)\right)dx$. The second property of Dirac Delta function can be written as follows

$$\begin{gathered} {鈭�}_{-鈭�}^{鈭�}f\left(x\right)\left(\frac{d胃}{dx}\right)dx=f\left(x\right)胃{\overline{)\left(x\right)}}_{-鈭�}^{鈭�}-{鈭�}_{-鈭�}^{鈭�}\frac{df}{dx}\left(胃\left(x\right)\right)dx \\ =f\left(x\right)胃{\overline{)\left(x\right)}}_{-鈭�}^{鈭�}-\left[{鈭�}_{-鈭�}^0\frac{df}{dx}\left(胃\left(x\right)\right)dx+{鈭�}_0^{鈭�}\frac{df}{dx}\left(胃\left(x\right)\right)dx\right] \\ =f\left(x\right)胃{\overline{)\left(x\right)}}_{-鈭�}^{鈭�}-\left[0+{鈭�}_0^{鈭�}\frac{df}{dx}dx\right] \\ =f\left(x\right)(鈭�)-{鈭�}_0^{鈭�}\frac{df}{dx}dx \end{gathered}$$

Solve further as,

$$\begin{gathered} {鈭�}_{-鈭�}^{鈭�}f\left(x\right)\left(\frac{d胃}{dx}\right)dx=(f\left(鈭�\right)-f\left(0\right)) \\ =f\left(0\right) \\ ={鈭�}_{-鈭�}^{鈭�}f\left(x\right)未\left(x\right)dx \end{gathered}$$

Thus, it can be written as $\left(\frac{d胃}{dx}\right)=未x$.