91影视

Delta function under integral signs is given as:

${鈭�}_{-鈭�}^{+鈭�}f\left(x\right)D_1\left(x\right)dx={鈭�}_{-鈭�}^{+鈭�}f\left(x\right)D_2\left(x\right)dx$

The step function is given as:

$胃\left(x\right)=\left\{\begin{array}{l}1,x>0\\ 0,x<0\end{array}\right.$

The Dirac delta distribution is a distribution, where

• It is distributed over real numbers.
• Its value is zero, where x is not zero. Else value is infinity at all the other values of x .

Let $y=cx$

Then,

$dx=\frac{1}{c}dy.\left\{\begin{array}{l}Ifc>0,y:-鈭�鈫�鈭�\\ Ifc<0,y:鈭�鈫�-鈭�\end{array}\left.\begin{array}{r}\\ \end{array}\right\}\right.$

localid="1658205343693" ${鈭�}_{-鈭�}^{鈭�}f\left(x\right)未\left(cx\right)dx=\left\{\begin{array}{cc}\frac{1}{c}{鈭�}_{-鈭�}^{鈭�}f(y/c)未\left(y\right)dy=\frac{1}{c}f\left(0\right)(c>0);or& \\ \frac{1}{c}{鈭�}_{鈭�}^{-鈭�}f(y/c)未\left(y\right)dy=-\frac{1}{c}{鈭�}_{-鈭�}^{鈭�}f(y/c)未\left(y\right)dy=-\frac{1}{c}f\left(0\right)& (c<0)\end{array}\right.$

In either case,

$$\begin{gathered} {鈭�}_{-鈭�}^{鈭�}f\left(x\right)未\left(cx\right)dx=\frac{1}{\left|c\right|}f\left(0\right) \\ {鈭�}_{-鈭�}^{鈭�}f\left(x\right)\frac{1}{\left|c\right|}未\left(x\right)dx=\frac{1}{\left|c\right|}f\left(0\right) \end{gathered}$$

So,$未\left(cx\right)=\frac{1}{\left|c\right|}未\left(x\right)$

Thus, $未\left(cx\right)=\frac{1}{\left|c\right|}未\left(x\right)$.

Using method integration by parts to solve the integral as:

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

So,

$d胃/dx=未\left(x\right)$ .

[Makes sense: The 胃 function is constant (so the derivative is zero) except at x = 0, where the derivative is infinite.].

Thus, $d胃/dx=未\left(x\right)$.