*3.2. Case 2 (When* μ = *r)*

Using Equations (16), (18), and (22), the following is implied for μ = *r*:

$$I(r) = \int \frac{r}{2.r} \mathrm{d}r = \frac{r}{2!} \tag{38}$$

$$\left(L\_r^{-1}\right)^n I(r) = \sum\_{n=0}^{\infty} \frac{r^{n+1}}{(n+2)!(1.2.3.4\ldots, (n+1))}\tag{39}$$

$$I\_1(r) = \int \frac{r^2}{3.r} \mathrm{d}r = \frac{r^2}{3!} \tag{40}$$

$$\left(L\_r^{-1}\right)^n I\_1(r) = \frac{r^2}{3!} + \sum\_{n=1}^{\infty} \frac{r^{n+2}}{(n+3)!(3.4.5\dots(n+2))}\tag{41}$$

With the help of these values, and using boundary conditions, the closed form of *w*(*r*, *z*) can be written as

$$w(r,z) = b\_{14}\chi\_4(r) + b\_{18}\chi\_5(r) + \frac{\mathrm{d}p}{\mathrm{d}z}(\chi\_6(r) + b\_{19}\chi\_5(r) - b\_{15}\chi\_4(r))\tag{42}$$

The constants appearing in the above equations are defined in Appendix A.
