Algebra and Geometry | Part IA, 2005

Give the real and imaginary parts of each of the following functions of z=x+iyz=x+i y, with x,yx, y real, (i) eze^{z}, (ii) cosz\cos z, (iii) logz\log z, (iv) 1z+1zˉ\frac{1}{z}+\frac{1}{\bar{z}}, (v) z3+3z2zˉ+3zzˉ2+zˉ3zˉz^{3}+3 z^{2} \bar{z}+3 z \bar{z}^{2}+\bar{z}^{3}-\bar{z},

where zˉ\bar{z} is the complex conjugate of zz.

An ant lives in the complex region RR given by z11|z-1| \leq 1. Food is found at zz such that

(logz)2=π216.(\log z)^{2}=-\frac{\pi^{2}}{16} .

Drink is found at zz such that

z+12zˉ(z12zˉ)2=3,z0\frac{z+\frac{1}{2} \bar{z}}{\left(z-\frac{1}{2} \bar{z}\right)^{2}}=3, \quad z \neq 0

Identify the places within RR where the ant will find the food or drink.

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