$a=e^{\frac{Im(m)arg(\alpha ^{m+n})+Im(n)arg(\alpha ^{m+n})+iRe(m)arg(\alpha ^{m+n})+iRe(n)arg(\alpha ^{m+n})}{2Re(m)Re(n)+2Im(m)Im(n)+\left(Re(m)\right)^{2}+\left(Re(n)\right)^{2}+\left(Im(m)\right)^{2}+\left(Im(n)\right)^{2}}-\frac{2\pi n_{1}iRe(m)}{2Re(m)Re(n)+2Im(m)Im(n)+\left(Re(m)\right)^{2}+\left(Re(n)\right)^{2}+\left(Im(m)\right)^{2}+\left(Im(n)\right)^{2}}-\frac{2\pi n_{1}iRe(n)}{2Re(m)Re(n)+2Im(m)Im(n)+\left(Re(m)\right)^{2}+\left(Re(n)\right)^{2}+\left(Im(m)\right)^{2}+\left(Im(n)\right)^{2}}-\frac{2\pi n_{1}Im(m)}{2Re(m)Re(n)+2Im(m)Im(n)+\left(Re(m)\right)^{2}+\left(Re(n)\right)^{2}+\left(Im(m)\right)^{2}+\left(Im(n)\right)^{2}}-\frac{2\pi n_{1}Im(n)}{2Re(m)Re(n)+2Im(m)Im(n)+\left(Re(m)\right)^{2}+\left(Re(n)\right)^{2}+\left(Im(m)\right)^{2}+\left(Im(n)\right)^{2}}}\left(|\alpha ^{m+n}|\right)^{\frac{Re(m)+Re(n)-iIm(m)-iIm(n)}{2Re(m)Re(n)+2Im(m)Im(n)+\left(Re(m)\right)^{2}+\left(Re(n)\right)^{2}+\left(Im(m)\right)^{2}+\left(Im(n)\right)^{2}}}$
$n_{1}\in \mathrm{Z}$