The first equation should be
a m , n = r ^ α z m , n ∫ 0 r ^ ∫ 0 2 π r f ( r , θ ) cos m θ J m ( z m , n r / r ^ ) d θ d r ( ∫ 0 2 π cos 2 m θ d θ ) ( ∫ 0 r ^ r J m 2 ( z m , n r / r ^ ) ) {\displaystyle a_{m,n}={\frac {\hat {r}}{\alpha z_{m,n}}}{\frac {\int _{0}^{\hat {r}}\int _{0}^{2\pi }rf(r,\theta )\cos m\theta J_{m}(z_{m,n}r/{\hat {r}})d\theta dr}{\left(\int _{0}^{2\pi }\cos ^{2}m\theta d\theta \right)\left(\int _{0}^{\hat {r}}rJ_{m}^{2}(z_{m,n}r/{\hat {r}})\right)}}}
The second equation should be
b m , n = r ^ α z m , n ∫ 0 r ^ ∫ 0 2 π r f ( r , θ ) sin m θ J m ( z m , n r / r ^ ) d θ d r ( ∫ 0 2 π cos 2 m θ d θ ) ( ∫ 0 r ^ r J m 2 ( z m , n r / r ^ ) ) {\displaystyle b_{m,n}={\frac {\hat {r}}{\alpha z_{m,n}}}{\frac {\int _{0}^{\hat {r}}\int _{0}^{2\pi }rf(r,\theta )\sin m\theta J_{m}(z_{m,n}r/{\hat {r}})d\theta dr}{\left(\int _{0}^{2\pi }\cos ^{2}m\theta d\theta \right)\left(\int _{0}^{\hat {r}}rJ_{m}^{2}(z_{m,n}r/{\hat {r}})\right)}}}
The third equation should be
c m , n = r ^ α z m , n ∫ 0 r ^ ∫ 0 2 π r g ( r , θ ) cos m θ J m ( z m , n r / r ^ ) d θ d r ( ∫ 0 2 π cos 2 m θ d θ ) ( ∫ 0 r ^ r J m 2 ( z m , n r / r ^ ) ) {\displaystyle c_{m,n}={\frac {\hat {r}}{\alpha z_{m,n}}}{\frac {\int _{0}^{\hat {r}}\int _{0}^{2\pi }rg(r,\theta )\cos m\theta J_{m}(z_{m,n}r/{\hat {r}})d\theta dr}{\left(\int _{0}^{2\pi }\cos ^{2}m\theta d\theta \right)\left(\int _{0}^{\hat {r}}rJ_{m}^{2}(z_{m,n}r/{\hat {r}})\right)}}}
and the fourth equation should be
d m , n = r ^ α z m , n ∫ 0 r ^ ∫ 0 2 π r g ( r , θ ) sin m θ J m ( z m , n r / r ^ ) d θ d r ( ∫ 0 2 π cos 2 m θ d θ ) ( ∫ 0 r ^ r J m 2 ( z m , n r / r ^ ) ) . {\displaystyle d_{m,n}={\frac {\hat {r}}{\alpha z_{m,n}}}{\frac {\int _{0}^{\hat {r}}\int _{0}^{2\pi }rg(r,\theta )\sin m\theta J_{m}(z_{m,n}r/{\hat {r}})d\theta dr}{\left(\int _{0}^{2\pi }\cos ^{2}m\theta d\theta \right)\left(\int _{0}^{\hat {r}}rJ_{m}^{2}(z_{m,n}r/{\hat {r}})\right)}}.}
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