In each of Exercises 4.10-4.15, plot the z and its corresponding w under the res

Question

In each of Exercises 4.10-4.15, plot the z and its corresponding w under the respective function. In each to generalize as to how the z-plane or parts of it are transformed by the function. 4.10 W z 1 0, 1 i, 1, 3 2i, -i. 2i 0, 2i, 1, i, 2i, 1 i 4.11 w E 2 z 0, 1, 2, 3, 1 i, 2 2i, i, 2i, 3i. 4.12 12 4.13 Re (z) z 3i, i, 2i, 1 i -1 3i, 0, 2, z i, 1 -i, 1, 1 i -1 i, 2. 4i 4.14 12 4.15 w e cos y ie sin y z 0, 2Ti, 2ni, 1, 1 ni, 2, Ti, 2 Ti/2, 2 3TiV2. 16 Consider the function w z3 (a) Express the function in the form w w (r, o) iv (r, e) (b) Show that the angular region o arg z 3 is mapped under the function, onto the upper half-plane o arg w JT (c) Find the part of the z-plane that will be needed to cover the entire w- plane exactly once. (d) Argue that if we used every point of the z-plane exactly once, then except for w o, every point of the w-plane would be the image of three distinct points in the z-plane. How are the moduli and the arguments ofthese three points related? 4.17 Consider the function w exOcos y i sin y) (a) Find the image under this function of the points 1 T2, 2 Ti, -1 Ti, -2 Ti, and 3 Ti. Plot these w's. Then generalize your findings by locating the image in the w-plane of the line z Tri for all real x. (b) Find the image under the function, of the points 1 +Tri/4, 1 Tri/2, 1 Ti, 1 3Ti/2, and 1 2yti. Plot these w's. Then generalize your findings by locating the image of the line z 1 yi for all real y (c) Find the image of each of the lines Re (z) 2 and Im (z) -T/2 under the function (d) Attempt a generalization by formulating a rule that describes what happens to horizontal and vertical lines in general under the function.

Explanation / Answer

dude only 3 mins are left to answer tthe probkem so i cant solve bt here is a trick for many problems you can easily solve them using graphical methods.

it will help a lot.

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