5. (10 points) Section 3.4 Solve the following equations. a. Log(z2 - 1) = 2 7T 2 b. e27 + ez + 1 = 0. 6. (20 points) Sections 3.1-3.5, 3.7 Differentiate the following functions, state the regions where the functions are analytic. a. cos(ez) b. ez + 1 c. Log (z2 + 1) (Hint: To find where it is analytic, you may let z = x +iy.) d. tanh (z), use z = 0 and the negative real axis as branch cut for the logarithmic function involved. For calculating the derivative, you need to show intermediate steps, since the answer is already given in the textbook. 7. (10 points) Sections 3.6-3.7 Evaluate the following and write your answer in such a way that the real and imaginary part can be identified clearly (polar form or exponential form will be fine). a. 2" , note that this is not (2?). You can use this result: 2 = e-2-2ku, k E Z (the negative sign in front of k is just a personal preference, you can use plus sign if you wish, since k can be negative). b. tan-1(1), using the formula tan 1 z = , log (? ?). 8. (12 points) Section 3.8 Find a conformal map that maps the horizontal strip Imz E (0, 7) onto the unit disk | z|