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28) Find all the solutions of . (Hint: Rewrite the equations in terms of exponentials and manipulate it into a quadratic equation in the variable .)

29) Suppose is analytic on . Let . For which is analytic? (Hint: Use the Cauchy-Riemann equations.)

30) Suppose is analytic on . Let . For which is analytic?

31) Find a function such that and is analytic.

32) Sketch and describe in a few words the paths , , and , .

33) Evaluate , where is the union of the line segments joining to and then to .

34) Evaluate , where is the part of the unit circle joining to in a counterclockwise direction.

35) Evaluate , where is the unit circle traversed counterclockwise.

36) In class, we showed that , where , . Identify the flaw in the following argument: Since is an antiderivative of , the fundamental theorem of calculus implies that .

37) Evaluate where is the circle centered at with radius . (Hint: Deform the path into a more convenient one.)

38) Evaluate , where is the (a) circle or radius centered at ; (b) the circle of radius centered at ; (c) the circle of radius centered at .

39) Let

.

Show that

.

Conclude that is not continuous at . (Hint: Use the definition of the derivative to compute .) Challenge: Adapt this exercise to construct a function such that , , , exist but such that is discontinuous at .