Question Number 655 by 123456 last updated on 19/Feb/15
$${f}\left({z}\right)=\frac{\mathrm{1}−\frac{\mathrm{1}−{z}^{\mathrm{2}} }{\mathrm{2}+{z}^{\mathrm{2}} }}{\mathrm{2}+\frac{\mathrm{1}−{z}^{\mathrm{2}} }{\mathrm{2}+{z}^{\mathrm{2}} }} \\ $$$$\frac{{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)}{\mathrm{2}}−{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=? \\ $$
Answered by prakash jain last updated on 19/Feb/15
$${f}\left(\mathrm{0}\right)=\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{5}} \\ $$$${f}\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${f}\left(\mathrm{0}\right)=\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{3}}}=\frac{\mathrm{2}}{\mathrm{7}} \\ $$$$\frac{{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)}{\mathrm{2}}\:−\:{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{2}}}{\mathrm{2}}−\frac{\mathrm{2}}{\mathrm{7}}=\frac{\mathrm{7}}{\mathrm{20}}−\frac{\mathrm{2}}{\mathrm{7}}=\frac{\mathrm{9}}{\mathrm{140}} \\ $$