Verification of the solution in the three regions
Region 1 \(\rho < \alpha ,\,\,\rho < \beta\), \(0 \le \phi \le \pi\).
So far, we have gotten the displacement field which is the basis for determining other elastic fields such as stress field and stress intensity factor. To avoid having a misleading result in the end, we subject this result to thorough verifications at the three regions. Does it satisfy the governing equation
$$\frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} = 0$$
(39)
and the boundary condition
$$W\left( {\rho ,0} \right) = 0,\quad 0 < \rho < \alpha$$
(40)
$$\frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) = 0,\quad 0 < \rho < \alpha $$
(41)
Substituting Eqs. (26–28) and (32–34) in to Eq. (25), we have
$$\begin{aligned} W\left( {\rho ,\phi } \right) & = \frac{bQ}{{c_{44} }}\left\{ {I_{\beta }^{(1)} - I_{\beta }^{(2)} + I_{\beta }^{(3)} } \right\} - \frac{bQ}{{c_{44} }}\left\{ {I_{\alpha }^{(1)} - I_{\alpha }^{(2)} + I_{\alpha }^{(3)} } \right\} = \frac{bQ}{{c_{44} }}\left\{ {\frac{2\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. \\ & \quad + \;\left( {c_{1} 2^{1} - c_{2} 2^{2} } \right)\frac{1}{\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \left( {c_{2} 2^{2} - c_{3} 2^{3} } \right)\frac{1}{{\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad + \;\left( {c_{3} 2^{3} - c_{4} 2^{4} } \right)\frac{1}{{\pi \beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \left( {c_{4} 2^{4} - c_{5} 2^{5} } \right)\frac{1}{{\pi \beta^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad + \;\left( {c_{5} 2^{5} - c_{6} 2^{6} } \right)\frac{1}{{\pi \beta^{5} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)\left( {n - \frac{1}{2}} \right)}} + \left( {c_{6} 2^{6} - c_{7} 2^{7} } \right)\frac{1}{{\pi \beta^{6} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{11}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} } \\ & \quad - \;\frac{bQ}{{c_{44} }}\left\{ {\frac{2\alpha }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. + \left( {c_{1} 2^{1} - c_{2} 2^{2} } \right)\frac{1}{\pi \alpha }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad + \;\left( {c_{2} 2^{2} - c_{3} 2^{3} } \right)\frac{1}{{\pi \alpha^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \left( {c_{3} 2^{3} - c_{4} 2^{4} } \right)\frac{1}{{\pi \alpha^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad + \;\left( {c_{4} 2^{4} - c_{5} 2^{5} } \right)\frac{1}{{\pi \alpha^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \left( {c_{5} 2^{5} - c_{6} 2^{6} } \right)\frac{1}{{\pi \alpha^{5} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(42)
Therefore,
$$W\left( {\rho ,0} \right) = 0$$
(43)
Differentiating Eq. (42), we have
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \phi } & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} + \left( {c_{1} 2^{1} - c_{2} 2^{2} } \right)\frac{1}{\pi \tau }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } } \right. \\ & \quad + \;\left( {c_{2} 2^{2} - c_{3} 2^{3} } \right)\frac{1}{{\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \left( {c_{3} 2^{3} - c_{4} 2^{4} } \right)\frac{1}{{\pi \beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad + \;\left( {c_{4} 2^{4} - c_{5} 2^{5} } \right)\frac{1}{{\pi \beta^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} + \left( {c_{5} 2^{5} - c_{6} 2^{6} } \right)\frac{1}{{\pi \beta^{5} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}}} \\ & \quad - \;\frac{bQ}{{c_{44} }}\left\{ {\frac{2\alpha }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. + \left( {c_{1} 2^{1} - c_{2} 2^{2} } \right)\frac{1}{\pi \alpha }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad + \;\left( {c_{2} 2^{2} - c_{3} 2^{3} } \right)\frac{1}{{\pi \alpha^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \left( {c_{3} 2^{3} - c_{4} 2^{4} } \right)\frac{1}{{\pi \alpha^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad + \;\left( {c_{4} 2^{4} - c_{5} 2^{5} } \right)\frac{1}{{\pi \alpha^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} + \left( {c_{5} 2^{5} - c_{6} 2^{6} } \right)\frac{1}{{\pi \alpha^{5} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}}} \\ \end{aligned}$$
(44)
Therefore,
$$\frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) = 0,\quad 0 < \rho < \alpha$$
(45)
Solving to derive other terms of Eq. (39), we have
$$\begin{aligned} W\left( {\rho ,\phi } \right) & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)\left( {n - \frac{1}{2}} \right)}} - \frac{1}{2\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}} - \frac{1}{{\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}} + \cdots } } } } \right\} \\ & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)\left( {n - \frac{1}{2}} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}} + \cdots } } } } \right\} \\ \end{aligned}$$
$$\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\}$$
$$\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\}$$
$$\frac{1}{{\rho^{2} }}\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\}$$
$$\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{3}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\}$$
$$\frac{1}{\rho }\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\}$$
$$\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \rho^{2} }} = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\}$$
Hence, for \(\frac{bQ}{{c_{44} }}\left( {I_{\beta }^{\left( 1 \right)} - I_{\beta }^{\left( 2 \right)} + I_{\beta }^{\left( 3 \right)} } \right)\)
$$\begin{aligned} & \frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} \\ & \quad = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\} \\ & \quad \quad + \;\frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\} \\ & \quad \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\} = 0 \\ \end{aligned}$$
Similarly, for \(\frac{bQ}{{c_{44} }}\left( {I_{\alpha }^{\left( 1 \right)} - I_{\alpha }^{\left( 2 \right)} + I_{\alpha }^{\left( 3 \right)} } \right)\)
$$\begin{aligned} & \frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} \\ & \quad = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\} \\ & \quad \quad + \;\frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\} \\ & \quad \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}} + \cdots } } } } \right\} = 0 \\ \end{aligned}$$
(46)
Therefore, for \(W\left( {\rho ,\phi } \right) = \frac{bQ}{{c_{44} }}\left\{ {I_{\beta }^{(1)} - I_{\beta }^{(2)} + I_{\beta }^{(3)} } \right\} - \frac{bQ}{{c_{44} }}\left\{ {I_{\alpha }^{(1)} - I_{\alpha }^{(2)} + I_{\alpha }^{(3)} } \right\}\)
$$\frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} = 0 + 0 = 0$$
(47)
satisfying the governing equation.
Satisfaction of the governing equation for \(\alpha < \rho < \beta\), \(0 \le \phi \le \pi\) (region II)
We now prove that Eq. (39) satisfies both the governing Laplace equation
$$\frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} = 0$$
(48)
and the boundary conditions
$$W\left( {\rho ,0} \right) = 0\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha < \rho < \beta$$
(49)
$$\frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) = \frac{bQ\rho }{{c_{44} }}\left[ {\frac{{\left( {\rho - 1} \right)}}{{\sqrt {\rho \left( {\rho - 2} \right)} }} + 1} \right],\quad \alpha < \rho < \beta ,\;\rho > 2$$
(50)
Now using Eqs. (35–37) for \(\rho > \alpha\) (region II)
$$\begin{aligned} W\left( {\rho ,\phi } \right) & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2\alpha }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. + \frac{1}{2}\sin \phi \rho^{ - 1} + \frac{1}{2\pi \alpha }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad + \;\frac{\sin 2\phi }{2}\rho^{ - 2} + \frac{1}{{\pi \alpha^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{5}{8}\frac{\sin 3\phi }{3}\rho^{ - 3} + \frac{15}{{8\pi \alpha^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\alpha }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(51)
Therefore,
$$W\left( {\rho ,0} \right) = 0$$
(52)
Rewriting Eq. (51), we have
$$\begin{aligned} W\left( {\rho ,\phi } \right) & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. + \frac{1}{2}\sin \phi \rho^{ - 1} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{\sin 2\phi }{2}\rho^{ - 2} \\ & \quad + \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{15}{8}\frac{\sin 3\phi }{3}\rho^{ - 3} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(53)
Differentiating Eq. (53), we have
$$\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. + \frac{1}{2}\cos \phi \rho^{ - 1} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} + \cos 2\phi \rho^{ - 2}$$
(54)
Therefore,
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ { - \frac{1}{2}\rho^{ - 1} + \rho^{ - 2} - \frac{15}{8}\rho^{ - 3} + \cdots } \right\}$$
(55)
$$\begin{aligned} \frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{1}{2}\sin \phi \rho^{ - 1} } } \right. - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - 2\sin 2\phi \rho^{ - 2} \\ & \quad - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{45}{8}\sin 3\phi \rho^{ - 3} - \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(56)
$$\begin{aligned} \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} \\ & \quad - \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} - \sin 2\phi \rho^{ - 4} - \frac{1}{8}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{45}{8}\sin 3\phi \rho^{ - 5} \\ & \quad - \;\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}} + \cdots } \\ \end{aligned}$$
(57)
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 2} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad - \;\sin 2\phi \rho^{ - 3} \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} - \frac{15}{8}\sin 3\phi \rho^{ - 4} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(58)
$$\begin{aligned} \frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \rho^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} } } \right.\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} + \sin \phi \rho^{ - 3} \\ & \quad + \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} + 3\sin 2\phi \rho^{ - 4} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}} + \frac{15}{2}\sin 3\phi \rho^{ - 5} } \\ & \quad + \;\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}} + \cdots } \\ \end{aligned}$$
(59)
$$\begin{aligned} \frac{1}{\rho }\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} - \sin 2\phi \rho^{ - 4} \\ & \quad - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{15}{8}\sin 3\phi \rho^{ - 5} + \frac{5}{8\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(60)
$$\begin{aligned} \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} \\ & \quad - \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} - \sin 2\phi \rho^{ - 4} - \frac{1}{8}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{45}{8}\sin 3\phi \rho^{ - 5} \\ & \quad - \;\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}} + \cdots } \\ \end{aligned}$$
(61)
Therefore,
$$\begin{aligned} \frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} } } \right.\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} + \sin \phi \rho^{ - 3} \\ & \quad + \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} + 3\sin 2\phi \rho^{ - 4} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} \\ & \quad + \;\frac{15}{2}\sin 3\phi \rho^{ - 5} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}} + \cdots } \\ & \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} - \sin 2\phi \rho^{ - 4} \\ & \quad - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{15}{8}\sin 3\phi \rho^{ - 5} + - \frac{5}{8\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ & \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \alpha^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} - \sin 2\phi \rho^{ - 4} \\ & \quad - \;\frac{1}{8}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{45}{8}\sin 3\phi \rho^{ - 5} - \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\alpha }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}} = 0} \\ \end{aligned}$$
(62)
Now for \(\rho < \beta\) (region II)
$$\begin{aligned} W\left( {\rho ,\phi } \right) & = \frac{bQ}{{c_{44} }}\left\{ { - 2\sin \phi \rho + \frac{2\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. - \frac{1}{2\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad - \;\frac{1}{{\pi \tau^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{15}{{8\pi \tau^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad - \;\frac{7}{{2\pi \beta^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{63}{{8\pi \beta^{5} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(63)
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \phi } & = \frac{bQ}{{c_{44} }}\left\{ { - 2\cos \phi \rho + \frac{2\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. - \frac{1}{2\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad - \;\frac{1}{{\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \frac{15}{{8\pi \beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad - \;\frac{7}{{2\pi \beta^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} + \frac{63}{{8\pi \beta^{5} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(64)
Therefore,
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ {2\rho } \right\}$$
(65)
$$\begin{aligned} \frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ {2\sin \phi \rho - \frac{2\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. + \frac{1}{2\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad + \;\frac{1}{{\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} - \frac{15}{{8\pi \beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad + \;\frac{7}{{2\pi \beta^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} - \frac{63}{{8\pi \beta^{4} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{n - \frac{1}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(66)
$$\begin{aligned} \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ {2\sin \phi \rho^{ - 1} - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad + \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad + \;\frac{7}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{7}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} - \frac{63}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{9}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(67)
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } & = \frac{bQ}{{c_{44} }}\left\{ { - 2\sin \phi \rho + \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{\frac{3}{2} - n}} \rho^{{n - \frac{3}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad - \;\frac{7}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{7}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} + \frac{63}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{9}{2}}} \rho^{{n - \frac{3}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}} + \cdots } \\ \end{aligned}$$
(68)
$$\begin{aligned} \frac{1}{\rho }\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } & = \frac{bQ}{{c_{44} }}\left\{ { - 2\sin \phi \rho^{ - 1} + \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad \left. { - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} + \cdots } \right\} \\ \end{aligned}$$
(69)
$$\begin{aligned} \frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \rho^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \,\beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} } } \right. \\ & \quad - \;\frac{{\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{{\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{3}{2}} \right)}} \\ & \quad + \;\left. {\frac{{\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{5}{2}} \right)}} - \frac{{\frac{7}{2\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{7}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{7}{2}} \right)}} - \cdots } \right\} \\ \end{aligned}$$
(70)
Therefore,
$$\begin{aligned} \frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \,\beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} } } \right. \\ & \quad - \;\frac{{\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{1}{2}} \right)}} - \frac{{\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{3}{2}} \right)}} \\ & \quad + \;\frac{{\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{5}{2}} \right)}} - \frac{{\frac{7}{2\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{3}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \,\left( {\frac{1}{\beta }} \right)^{{n + \frac{7}{2}}} \rho^{{n - \frac{5}{2}}} } }}{{\left( {n + \frac{7}{2}} \right)}} \\ & \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - 2\sin \phi \rho^{ - 1} + \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad \left. { - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} + \cdots } \right\} \\ & \quad + \;\frac{bQ}{{c_{44} }}\left\{ {2\sin \phi \rho^{ - 1} - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{\frac{3}{2} - n}} \rho^{{n - \frac{5}{2}}} }}{{\left( {\frac{3}{2} - n} \right)}}} } \right. + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{1}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \\ & \quad + \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{3}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{3}{2}} \right)}}} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{5}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{5}{2}} \right)}}} \\ & \quad \left. { + \;\frac{7}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{7}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{7}{2}} \right)}}} - \frac{63}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{n + \frac{9}{2}}} \rho^{{n - \frac{5}{2}}} }}{{\left( {n + \frac{9}{2}} \right)}}} } \right\} = 0 \\ \end{aligned}$$
(71)
Conversion of the boundary condition for region II to series form
Recall the boundary condition for region II
$$\frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) = \frac{bQ}{{c_{44} }}\left[ {\frac{{\rho \left( {\rho - 1} \right)}}{{\sqrt {\rho \left( {\left( {\rho - 2} \right)} \right)} }} + \rho } \right]\,\,,\,\,\,\,\,\alpha < \rho < \beta ,\,\,\,\,\rho > 2$$
(72)
We convert the above equation to series form using the formula
$$\left( {1 - t} \right)^{{ - \frac{1}{2}}} = \sum\limits_{k = 0}^{\infty } {c_{k} t^{k} }$$
(73)
Now,
$$\begin{aligned} \frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) & = \frac{bQ}{{c_{44} }}\left[ {\frac{{\rho \left( {\rho - 1} \right)}}{{\sqrt {\rho \left( {\rho - 2} \right)} }} + \rho } \right] = \frac{bQ}{{c_{44} }}\left[ {\frac{{\rho \left( {\rho - 1} \right)}}{{\rho^{\frac{1}{2}} \left( {\rho - 2} \right)^{\frac{1}{2}} }} + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\frac{{\rho \left( {\rho - 1} \right)}}{{\rho^{\frac{1}{2}} \left( {\rho - 2} \right)^{\frac{1}{2}} }} + \rho } \right] = \frac{bQ}{{c_{44} }}\left[ {\rho^{\frac{1}{2}} \left( {\rho - 1} \right)\left( {\rho - 2} \right)^{{ - \frac{1}{2}}} + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\rho^{\frac{1}{2}} \rho \left( {\rho - 2} \right)^{{ - \frac{1}{2}}} - \rho^{\frac{1}{2}} \left( {\rho - 2} \right)^{{ - \frac{1}{2}}} + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\rho^{\frac{1}{2}} \rho \rho^{{ - \frac{1}{2}}} \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} - \rho^{\frac{1}{2}} \rho^{{ - \frac{1}{2}}} \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\rho \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} - \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} + \rho } \right] \\ \end{aligned}$$
(74)
But
$$\left( {1 - t} \right)^{{ - \frac{1}{2}}} = \sum\limits_{k = 0}^{\infty } {c_{k} t^{k} } \Rightarrow \, \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} = \sum\limits_{k = 0}^{\infty } {c_{k} \left( {\frac{2}{\rho }} \right)^{k} = } \sum\limits_{k = 0}^{\infty } {c_{k} 2^{k} \rho^{ - k} }$$
Hence,
$$\rho \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} = \sum\limits_{k = 0}^{\infty } {c_{k} 2^{k} \rho^{1 - k} }$$
(75)
Therefore,
$$\begin{aligned} \frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) & = \frac{bQ}{{c_{44} }}\left[ {\rho \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} - \left( {1 - \frac{2}{\rho }} \right)^{{ - \frac{1}{2}}} + \rho } \right] = \frac{bQ}{{c_{44} }}\left[ {\sum\limits_{k = 0}^{\infty } {c_{k} 2^{k} \rho^{1 - k} } - \sum\limits_{k = 0}^{\infty } {c_{k} 2^{k} \rho^{ - k} } + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\left( {c_{0} 2^{0} \rho - c_{0} 2^{0} \rho^{0} } \right) + \left( {c_{1} 2^{1} \rho^{0} - c_{1} 2^{1} \rho^{ - 1} } \right) + \left( {c_{2} 2^{2} \rho^{ - 1} - c_{2} 2^{2} \rho^{ - 2} } \right) + \left( {c_{3} 2^{3} \rho^{ - 2} - c_{3} 2^{3} \rho^{ - 3} } \right) + \left( {c_{4} 2^{4} \rho^{ - 3} - c_{4} 2^{4} \rho^{ - 4} } \right) + \cdots + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\left( {\rho - 1} \right) + \left( {1 - \rho^{ - 1} } \right) + \left( {\frac{3}{8} \times 4\rho^{ - 1} - \frac{3}{8} \times 4\rho^{ - 2} } \right) + \left( {\frac{5}{16} \times 8\rho^{ - 2} - \frac{5}{16} \times 8\rho^{ - 3} } \right) + \left( {\frac{35}{{128}} \times 16\rho^{ - 3} - \frac{35}{{128}} \times 16\rho^{ - 4} } \right) + \cdots + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {\left( {\rho - 1} \right) + \left( {1 - \rho^{ - 1} } \right) + \left( {\frac{3}{2}\rho^{ - 1} - \frac{3}{2}\rho^{ - 2} } \right) + \left( {\frac{5}{2}\rho^{ - 2} - \frac{5}{2}\rho^{ - 3} } \right) + \left( {\frac{35}{8}\rho^{ - 3} - \frac{35}{8}\rho^{ - 4} } \right) + \cdots + \rho } \right] \\ & = \frac{bQ}{{c_{44} }}\left[ {2\rho + \frac{1}{2}\rho^{ - 1} + \rho^{ - 2} + \frac{15}{8}\rho^{ - 3} + \cdots } \right] \\ \end{aligned}$$
(76)
From region II \(\left( {\rho > \alpha } \right)\)
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ { - \frac{1}{2}\rho^{ - 1} + \rho^{ - 2} - \frac{15}{8}\rho^{ - 3} + ...} \right\}\,\,\,,\rho > \alpha \,\,\,\,\,\,$$
(77)
From region II \(\left( {\rho < \beta } \right)\)
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ {2\rho } \right\},\;\rho < \beta$$
(78)
Hence by superposition principle
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } & = \frac{bQ}{{c_{44} }}\left\{ {2\rho } \right\} - \frac{bQ}{{c_{44} }}\left\{ { - \frac{1}{2}\rho^{ - 1} + \rho^{ - 2} - \frac{15}{8}\rho^{ - 3} + \cdots } \right\} \\ & = \frac{bQ}{{c_{44} }}\left\{ {2\rho + \frac{1}{2}\rho^{ - 1} - \rho^{ - 2} + \frac{15}{8}\rho^{ - 3} + \cdots } \right\} \\ \end{aligned}$$
(79)
which satisfies the boundary condition in series form.
Satisfaction of the governing equation for \(\rho > \beta \,,\,\rho \, > \alpha\), \(0 \le \phi \le \pi\) (region III)
For region III, it is not a difficult algebra to show that
$$W\left( {\rho ,\phi } \right) = \frac{bQ}{{c_{44} }}\left\{ {I_{\beta }^{(1)} - I_{\beta }^{(2)} + I_{\beta }^{(3)} } \right\} - \frac{bQ}{{c_{44} }}\left\{ {I_{\alpha }^{(1)} - I_{\alpha }^{(2)} + I_{\alpha }^{(3)} } \right\}$$
(80)
satisfies both the governing Laplace equation
$$\frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} = 0$$
(81)
and the boundary condition
$$W\left( {\rho ,0} \right) = 0,\quad \rho > \beta$$
(82)
$$\frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) = 0,\quad \rho > \beta.$$
(83)
For \(\frac{bQ}{{c_{44} }}\left( {I_{\beta }^{(1)} - I_{\beta }^{(2)} + I_{\beta }^{(3)} } \right)\)
$$\begin{aligned} W\left( {\rho ,\phi } \right) & = \frac{bQ}{{c_{44} }}\left( {I_{\beta }^{(1)} - I_{\beta }^{(2)} + I_{\beta }^{(3)} } \right) = \frac{bQ}{{c_{44} }}\left\{ {\frac{\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} } \right. - \phi + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{1}{2}} \right)^{2} }}} + \frac{3}{2}\sin \phi \rho^{ - 1} \\ & \quad + \;\frac{3}{2\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{5}{2}\frac{\sin 2\phi }{2}\rho^{ - 2} + \frac{5}{{2\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{35}{8}\frac{\sin 3\phi }{3}\rho^{ - 3} \\ & \quad + \;\frac{35}{{8\pi \beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots + \phi - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{1}{2}} \right)^{2} }}} - \sin \phi \rho^{ - 1} \\ & \quad - \;\frac{1}{\pi \beta }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} - \frac{3}{2}\frac{\sin 2\phi }{2}\rho^{ - 2} - \frac{3}{{2\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} - \frac{5}{2}\frac{\sin 3\phi }{3}\rho^{ - 3} \\ & \quad - \;\frac{5}{{2\beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots + \frac{\beta }{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} \\ & \quad + \;\frac{1}{{\pi \beta^{2} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \frac{15}{8}\frac{\sin 3\phi }{3}\rho^{ - 3} + \frac{15}{{8\pi \beta^{3} }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{\rho }{\beta }} \right)^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(84)
Therefore,
$$w\left( {\rho ,0} \right) = 0$$
(85)
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \phi } & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{{n + \frac{1}{2}}} \cos \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. + \frac{1}{2}\cos \phi \rho^{ - 1} + \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{{n + \frac{1}{2}}} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} \\ & \quad + \;\cos 2\phi \rho^{ - 2} + \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{{n + \frac{1}{2}}} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} + \frac{15}{8}\cos 3\phi \rho^{ - 3} \\ & \quad + \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{{n + \frac{1}{2}}} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} + \frac{15}{8}\cos 3\phi \rho^{ - 3} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{{n + \frac{1}{2}}} \cos \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(86)
Therefore,
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ {\frac{5}{2}\rho^{ - 1} + \rho^{ - 2} - \frac{15}{8}\rho^{ - 3} + \cdots } \right\}$$
(87)
$$\begin{aligned} \frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)}^{n + 1} \frac{{\left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} \right. - \frac{1}{2}\sin \phi \rho^{ - 1} \\ & \quad - \;\frac{{\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)}^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}} - 2\sin 2\phi \rho^{ - 2} \\ & \quad - \;\frac{{\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)}^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}} - \frac{45}{8}\sin 3\phi \rho^{ - 3} \\ & \quad - \;\frac{{\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)}^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n + \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}} + \cdots \\ \end{aligned}$$
(88)
$$\begin{aligned} \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \frac{{\left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} \\ & \quad - \;\frac{1}{2\pi }\frac{{\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} } }}{{\left( {n - \frac{3}{2}} \right)}} - 2\sin 2\phi \rho^{ - 4} \\ & \quad - \;\frac{1}{\pi }\frac{{\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} } }}{{\left( {n - \frac{5}{2}} \right)}} - \frac{45}{8}\sin 3\phi \rho^{ - 5} \\ & \quad - \;\frac{15}{{8\pi }}\frac{{\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} } }}{{\left( {n - \frac{7}{2}} \right)}} + \cdots \\ \end{aligned}$$
(89)
$$\begin{aligned} \frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n + \frac{1}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} - \frac{1}{2}\sin \phi \rho^{ - 2} } \right. \\ & \quad + \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n - \frac{3}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} - \sin 2\phi \rho^{ - 3} \\ & \quad - \;\frac{3}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n - \frac{5}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} - \frac{15}{8}\sin 3\phi \rho^{ - 4} \\ \end{aligned}$$
$$\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( { - n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{1}{2}}} }}{{\left( {n - \frac{7}{2}} \right)\left( {n - \frac{1}{2}} \right)}}} { + } \cdots$$
$$\begin{aligned} \frac{{\partial^{2} W\left( {\rho ,\phi } \right)}}{{\partial \rho^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} } \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} + \sin \phi \rho^{ - 3} } \right. \\ & \quad + \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} + 3\sin 2\phi \rho^{ - 4} \\ & \quad + \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} + \frac{15}{2}\sin 3\phi \rho^{ - 5} + \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ \end{aligned}$$
$$\begin{aligned} \frac{1}{\rho }\frac{{\partial W\left( {\rho ,\phi } \right)}}{\partial \rho } & = \frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} - \frac{1}{2}\sin \phi \rho^{ - 3} } \right. - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} \\ & \quad - \;\sin 2\phi \rho^{ - 4} - \frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{15}{8}\sin 3\phi \rho^{ - 5} - \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ \end{aligned}$$
(90)
Therefore,
$$\begin{aligned} \frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} & = \frac{bQ}{{c_{44} }}\left\{ {\frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} } + \sin \phi \rho^{ - 3} } \right. \\ & \quad + \;\frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} + 3\sin 2\phi \rho^{ - 4} \\ & \quad + \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} + \frac{15}{2}\sin 3\phi \rho^{ - 5} \\ & \quad + \;\frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \left( {n + \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + \cdots \\ & \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} - \frac{1}{2}\sin \phi \rho^{ - 3} } \right. - \frac{1}{2\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{3}{2}} \right)}}} - \sin 2\phi \rho^{ - 4} \\ & \quad - \;\frac{1}{\pi }\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{5}{2}} \right)}}} - \frac{15}{8}\sin 3\phi \rho^{ - 5} - \frac{15}{{8\pi }}\sum\limits_{n = 1}^{\infty } {\frac{{\left( { - 1} \right)^{n + 1} \sin \phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n - \frac{7}{2}} \right)}}} + ... \\ & \quad + \;\frac{bQ}{{c_{44} }}\left\{ { - \frac{2}{\pi \tau }\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \frac{{\left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \beta^{{n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} }}{{\left( {n + \frac{1}{2}} \right)}}} } \right. - \frac{1}{2}\sin \phi \rho^{ - 3} \\ & \quad - \;\frac{1}{2\pi }\frac{{\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{3}{2}}} \rho^{{ - n - \frac{3}{2}}} } }}{{\left( {n - \frac{3}{2}} \right)}} - 2\sin 2\phi \rho^{ - 4} \\ & \quad - \;\frac{1}{\pi }\frac{{\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{5}{2}}} \rho^{{ - n - \frac{3}{2}}} } }}{{\left( {n - \frac{5}{2}} \right)}} - \frac{45}{8}\sin 3\phi \rho^{ - 5} \\ & \quad - \;\frac{15}{{8\pi }}\frac{{\sum\limits_{n = 1}^{\infty } {\left( { - 1} \right)^{n + 1} \left( {n - \frac{1}{2}} \right)\sin \left( {n - \frac{1}{2}} \right)\phi \left( {\frac{1}{\beta }} \right)^{{ - n + \frac{7}{2}}} \rho^{{ - n - \frac{3}{2}}} } }}{{\left( {n - \frac{7}{2}} \right)}} = 0 \\ \end{aligned}$$
(91)
Similarly, for \(W\left( {\rho ,\phi } \right) = \frac{bQ}{{c_{44} }}\left( {I_{\alpha }^{(1)} + I_{\alpha }^{(2)} + I_{\alpha }^{(3)} } \right)\)
$$\frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} = 0$$
Therefore,
$$\frac{{\partial^{2} W}}{{\partial \rho^{2} }} + \frac{1}{\rho }\frac{\partial W}{{\partial \rho }} + \frac{1}{{\rho^{2} }}\frac{{\partial^{2} W}}{{\partial \phi^{2} }} = 0 + 0 = 0$$
(92)
satisfying the governing equation.
Now for \(\rho > \beta\)
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ { - \frac{1}{2}\rho^{ - 1} + \rho^{ - 2} - \frac{15}{8}\rho^{ - 3} + \cdots } \right\}$$
Similarly, for \(\rho > \alpha\)
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \frac{bQ}{{c_{44} }}\left\{ { - \frac{1}{2}\rho^{ - 1} + \rho^{ - 2} - \frac{15}{8}\rho^{ - 3} + \cdots } \right\}$$
Therefore.
$$\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi } = \left. {\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi }} \right|_{\beta } - \left. {\frac{{\partial W\left( {\rho ,\pi } \right)}}{\partial \phi }} \right|_{\alpha } = 0$$
(93)
From the results gotten at the three regions, we can see that our displacement equation satisfies the governing equation in the regions. The relevance of this satisfaction is that suppose we intend to derive the stress field or the stress intensity factor say, we would be fully convinced that our result is not misleading when a numerical analyst decides to use our analytic result to compare his numerical findings. Also in region 2, notice that we use the formula \(\left( {1 - t} \right)^{{ - \frac{1}{2}}} = \sum\nolimits_{k = 0}^{\infty } {c_{k} t^{k} }\) to convert the boundary condition
$$\frac{\partial W}{{\partial \phi }}\left( {\rho ,\pi } \right) = \frac{bQ}{{c_{44} }}\left[ {\frac{{\rho \left( {\rho - 1} \right)}}{{\sqrt {\rho \left( {\left( {\rho - 2} \right)} \right)} }} + \rho } \right],\quad \alpha < \rho < \beta ,\;\rho > 2$$
to series form to obtain our desired result. This is also novel.