Master’s Comprehensive Exam Complex Variables August 1995

1. (10 pts.) Compute Log(i), where Log represents the Principal branch of the logarithm.

2. (30 pts.) Let f(z) = z · \overline{z}.

(a) Where is f(z) continuous?

(b) Where is f(z) differentiable and what is its derivative at points where it is differentiable?

(c) Where is f(z) analytic?

3. (20 pts.) In each case, find the residue of the function at 0 and indicate whether 0 is a removable singular point, an essential point or a pole (in which case, give the order of the pole).

(a) \frac{e^{z} − 1}{z};

(b) \frac{(1 + z) sin (z)}{z^{2}}

4. (15 pts.) Using the Residue Theorem, compute the following integral:

∫_{0}^{+∞} \frac{dx}{1 + x^{2}} .

5. (10 pts.) Find the linear fractional transformation T which maps the points −1, 0, 1 onto −i, 1, i respectively.

6. (15 pts.) Find the image of the unit disk under the following linear fractional transformation

T:

T (z) = \frac{i − z}{i + z}