Give the real and imaginary parts of each of the following functions of $z=x+i y$, with $x, y$ real, (i) $e^{z}$, (ii) $\cos z$, (iii) $\log z$, (iv) $\frac{1}{z}+\frac{1}{\bar{z}}$, (v) $z^{3}+3 z^{2} \bar{z}+3 z \bar{z}^{2}+\bar{z}^{3}-\bar{z}$,

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

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

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

Drink is found at $z$ such that

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

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