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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 .
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 .