91Ó°ÊÓ

The given condition is

(a) $m>n$.

(b) $mâ\copyright ½n$.

(c)$mâ\copyright ½n$

(d)$\left(x^2+y^2+z^2\right){∇}^2\left[\mathrm{δ}\right(x\left)\mathrm{δ}\right(y\left)\mathrm{δ}\right(z\left)\right]=6\mathrm{δ}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)$

A function operator is a function that takes one (or more) functions as input and returns a function as output. In some ways, function operators are similar to functional

Consider the integral.

$∫\left[x^m\left(x\right)\mathrm{ϕ}\left(x\right)\right]{\mathrm{δ}}^{\left(n\right)}\left(x\right)dx(m>n)$

Solve above integral.

$∫x^m{\mathrm{δ}}^{\left(n\right)}\left(x\right)\mathrm{ϕ}\left(x\right)dx=(-1{)}^n\frac{d^{*}}{d{x}^{*}}{\left[x^m\mathrm{ϕ}\left(x\right)\right]}_{m+x=0}$

In Above integral all the terms are multiplied by x,0 at$x=0$the terms a will be zero.

Hence, $x^m{\mathrm{δ}}^n\left(x\right)=0$.

Consider the integral.

$∫x^m{\mathrm{δ}}^{\left(n\right)}\left(x\right)\mathrm{Ï•}\left(x\right)dx(mâ\copyright ½n)$

Solve above integral.

$∫x^n{\mathrm{δ}}^{\left(n\right)}\left(x\right)\mathrm{ϕ}\left(x\right)dx=(-1{)}^n\frac{d^n}{d{x}^n}\left|x^m\mathrm{ϕ}\left(x\right)\right|$

$=(-1{)}^n{\left[{}^{n}C_0\mathrm{ϕ}\left(x\right)\frac{d^{*}}{d{s}^n}\left(x^s\right)+{}^{n}C_1{\mathrm{ϕ}}^1\left(x\right)\frac{d^{-1}}{d^{n-1}}\left(x^n\right)+……+{}^{n}C_m{x}^m{\mathrm{ϕ}}^{\left(x\right)}\left(x\right)x^n\right]}_{afs=0}$

$=(-1{)}^n[n!\mathrm{ϕ}(0)+0+…..+0]$

$=(-1{)}^{n}n!\mathrm{δ}\left(x\right)$

Hence, $x^{\mathrm{Î\mu }}{\mathrm{δ}}^{\left(n\right)}\left(x\right)=(-1{)}^{c}n!\mathrm{δ}\left(x\right)$.

Consider the integral.

${∫}_{-}^7{x}^n{\mathrm{δ}}^{\text{(e)}}\left(x\right)\mathrm{Ï•}\left(x\right)dx(mâ\copyright ½n)$

Solve above integral.

$$\begin{gathered} ∫x^m{\mathrm{δ}}^{\left(\mathrm{Î\mu }\right)}\left(x\right)\mathrm{Ï•}\left(x\right)dx=(-1{)}^{\mathrm{Î\mu }}\frac{d^n}{d{x}^{*}}\left|x^m\mathrm{Ï•}\left(x\right)\right| \\ =(-1{)}^n\left[n_{c_0}\mathrm{Ï•}\left(x\right)\frac{d^n}{d_{x^{*}}}\left(x^m\right)+n_{c_1}{\mathrm{Ï•}}^1\left(x\right)\frac{d^{n-1}}{d{x}^{x-1}}\left(x^m\right)+....+n_{c-m}m!{\mathrm{Ï•}}^{(n-m)}\left(x\right)+n_{c_c}{x}^{m}m!{\mathrm{Ï•}}^{\left(n\right)}\left(x\right)\right] \\ =\frac{(-1)}{{(-1)}^{-\mathrm{Î\pm }}}{n}_{e-m}m!∫\mathrm{Ï•}\left(x\right){\mathrm{δ}}^{(\mathrm{Ï€}-\mathrm{Ï€})}dx \\ =(-1{)}^{\mathrm{Ï€}}{n}_{c-∞}m!{\mathrm{δ}}^{(\mathrm{Ï€}-\mathrm{Ï€})}\left(x\right) \end{gathered}$$

Hence,$x^m{\mathrm{δ}}^{(*)}\left(x\right)=(-1{)}^n{n}_c-m!{\mathrm{δ}}^{(n-m)}\left(x\right)$ .

(d)

Solve

$$\begin{gathered} \left(x^2+y^2+z^2\right){∇}^2\left[\mathrm{δ}\right(x\left)\mathrm{δ}\right(y\left)\mathrm{δ}\right(z\left)\right] \\ \left(x^2+y^2+z^2\right)\left[{\mathrm{δ}}^{*}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)+\mathrm{δ}\left(x\right){\mathrm{δ}}^{*}\left(y\right)\mathrm{δ}\left(z\right)+\mathrm{δ}\left(x\right)\mathrm{δ}\left(y\right){\mathrm{δ}}^{*}\left(z\right)\right] \end{gathered}$$

Use above result further.

$$\begin{gathered} x^2{j}^4\left(x\right)=21\mathrm{δ}\left(x\right) \\ {x}^2\mathrm{δ}\left(x\right)=2s\left(x\right) \end{gathered}$$

Solve$\left(x^2+y^2+z^2\right){∇}^2\left[\mathrm{δ}\right(x\left)\mathrm{δ}\right(y\left)\mathrm{δ}\right(z)âˆ\pounds$by use of above results.

$$\begin{gathered} \left(x^2+y^2+z^2\right)\left[{\mathrm{δ}}^{*}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)\right]=\left(x^2{\mathrm{δ}}^{*}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)+y^2{\mathrm{δ}}^{*}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)+z^2{\mathrm{δ}}^{*}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)\right) \\ =2\mathrm{δ}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)+0+0 \end{gathered}$$

So, it is clear that $\left(x^2+y^2+z^2\right){∇}^2\left[\mathrm{δ}\right(x\left)\mathrm{δ}\right(y\left)\mathrm{δ}\right(z\left)\right]=6\mathrm{δ}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)$.

Hence,$\left(x^2+y^2+z^2\right){∇}^2\left[\mathrm{δ}\right(x\left)\mathrm{δ}\right(y\left)\mathrm{δ}\right(z\left)\right]=6\mathrm{δ}\left(x\right)\mathrm{δ}\left(y\right)\mathrm{δ}\left(z\right)$.