Practice Midterm Problems.pdf - Z u03b3 1 z 2 3 dz where...

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Math 185 Practice MidtermjacobOctober 20171Cool Math Problems!!!1.Letz=-2 + 3i.Find the modulus ofz, the argument ofz, logz, theargument ofz2, and the moduli and arguments ofz1/2.2. Find all complexzwhich satisfyz6= 8.3. Write the following inx+iyform:(2i-1)104. Compute:(-1 +i3)3/25. LetS={zC: 0≤ <(z)1 and=(z)0}.a. Sketch the image ofsunder the mapz7→(2 +i)z-3. Label ”corners” withthe corresponding complex numbers.b.Sketch the image ofSunder the mapz7→eiz.Label ”corners” with thecorresponding complex numbers.6. LetGbe an open subset ofC, letf:GC, and letz0G.a. Define what is meant byf0(z0).b. SupposeG=Candfis the functionf(z) ==(z) (=(z) denotes the imagi-nary part ofz). Show directly from the definition thatf0(z0) does not exist foranyz0. (If you don’t see how to do this, then for partial credit, get the sameconclusion by any method.)7. Supposef(x+iy) =u(x, y)+iv(x, y) is analytic, and supposeu(x, y) =x.What can we say aboutv(x, y)?8. Give an explicit formula for a pathf: [a, b]C(for real numbersa, bof your choice) that traces a circle of radius 2 centered at 7 + 3ionce around in1
theclockwisedirection.9. Evaluate

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