91Ó°ÊÓ

The given expressions is$\mathrm{δ}\left(r-r_0\right)=\frac{\left.\mathrm{Δ}\left(r-r_0\right)(\cos \mathrm{θ}-\cos {\mathrm{θ}}_1\right)\mathrm{δ}(\mathrm{ϕ}-\mathrm{ϕ})}{r^2}$

Integration by partsor partial integration is a process that finds theintegralof aproductoffunctionsin terms of the integral of the product of theirderivativeandantiderivative.

Consider the below function.

$\mathrm{δ}\left(f\right(x\left)\right)=\underset{i}{∑}\frac{\mathrm{δ}\left(x-x_i\right)}{\left|f^{\text{'}}\left(x_i\right)\right|}$

Where, this formula can be used only if $f\left(x_i\right)=0$and $f^{\text{'}}\left(x_i\right)≠0$, to clarify this formula this simply states, that if the given delta function is variable in a function $f\left(x\right),$such that this function have i-th roots "ith number of values of x, which would make the function $f\left(x\right)=0$, then we have to find those roots "for example, the roots are $x_1,x_2,…,x_i$", finding the roots of the function $f\left(x\right),$we now differentiate the function $f\left(x\right),$to find $f(x{)}^{\text{'}}$, and then the delta function would be given by the following series

$f(x{)}^{\text{'}}\mathrm{δ}\left(f\right(x\left)\right)=\frac{\mathrm{δ}\left(x-x_1\right)}{\left|f^{\text{'}}\left(x_1\right)\right|}+\frac{\mathrm{δ}\left(x-x_2\right)}{\left|f^{\text{'}}\left(x_2\right)\right|}+…$

note: This divide the integral into ith-integrals, where not all of them would be non-zero, only those who have x_i which lies inside the limits integral would be evaluated.

Assume, $x_i={\mathrm{θ}}_0$

$$\begin{gathered} f\left(\mathrm{θ}\right)=\cos \mathrm{θ}-\cos {\mathrm{θ}}_0 \\ f\left(x_i\right)=0 \\ {f}^{\text{'}}\left(\mathrm{θ}\right)=-\sin \mathrm{θ} \\ {f}^{\text{'}}\left({\mathrm{θ}}_0\right)≠0 \end{gathered}$$

So, for a function of $\mathrm{θ},{\mathrm{θ}}_0$we can write,

$\mathrm{δ}\left(\cos \mathrm{θ}-\cos {\mathrm{θ}}_0\right)=\frac{\mathrm{β}\left({\mathrm{θ}}_0\right)}{\sin {\mathrm{θ}}_{\text{k}}}$

Thus,

$\mathrm{δ}\left(\cos \mathrm{θ}-\cos {\mathrm{θ}}_0\right)=\frac{\mathrm{β}\left({\mathrm{θ}}_0\right)}{\sin {\mathrm{θ}}_{\text{k}}}$