Elementary Analytic Functions and their Mapping Properties
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    8.Find for each of the following functions the values of $z$ which make the function (a) purely real and (b) purely imaginary.
    (i) $\sin(\overline{z})$ $\qquad$ (ii) MATH $\qquad$ (iii) MATH $\qquad$ (iv) MATH
    9.Prove that MATH where $\Im(z)$ denotes the imaginary part of $z$. Deduce that MATH tends to $\infty$ as MATH.
    10.Find all the values of $z$ for which the equation $\cos z = \cosh 2$ holds.
    11.Sketch the families of level curves of the component functions $u$ and $v$ of $f=u+iv$, when $f(z) = z^{2}$.
    12.Assume that MATH is analytic in a domain $D$ and that MATH in $D$. Consider the families of level curves MATH and MATH. Prove that the two families of level curves are orthogonal.
    13.Find the image of the vertical line $x = c_{1}$ and the horizontal line $y = c_{2}$ where $c_{1}$ and $c_{2}$ are real constants under the following mappings:
    (i) $w = \exp(z)$ $\qquad$ (ii) $w = (1/z)$ (Assume, $c_{1}$ and $c_{2}$ are non-zero) $\qquad$ (iii) $w = z^{2}$ $\qquad$ (iv) $w = \tan(z)$.
    14.Evaluate the following:
    (i) $\log(1 + i)$ $\qquad$ (ii) $(-3)^{\sqrt{2}}$ $\qquad$ (iii) $(3 - 2i)^{(1+i)}$ $\qquad$ (iv) $(3 + 4i)^{(1+i)}$
    15.Find the image of the sector bounded by the rays MATH and MATH in the $z$-plane under the mapping MATH.

     

 
   
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