Reciprocal of T 1 z T_{\frac{1}{z}}

Algebra Level 2

What is the reciprocal of T 1 z T_{\frac{1}{z}} where T z = 2 z − 1 2 z + 1 T_z=\frac{2z-1}{2z+1} ?

z − 2 z + 2 \frac{z-2}{z+2} 2 − z 2 + z \frac{2-z}{2+z} 2 + z 2 − z \frac{2+z}{2-z} z + 2 z − 2 \frac{z+2}{z-2}

Zeeshan Ali by Zeeshan Ali , 25, Pakistan

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1 solution

Munem Shahriar
Aug 31, 2018

T 1 z = 2 z − 1 2 z + 1 = 2 − z z × z 2 + z = 2 − z 2 + z \large T_{\frac 1z} = \dfrac{\frac 2z - 1}{\frac 2z + 1} = \dfrac{2 - z}z \times \dfrac z{2 +z} = \dfrac{2 - z}{2+z}

The reciprocal of T 1 z , \large T_{\frac 1z}, 1 T 1 z = 1 2 − z 2 + z = 2 + z 2 − z \dfrac 1{T_{\frac 1z}} = \dfrac 1{\frac{2 - z}{2+z}} = \dfrac{2 + z}{2-z}

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