In this problem we will find the image of the line x = 1 under the complex mapping w = 1/z...

In this problem we will find the image of the line x = 1 under the complex mapping w = 1/z.

(a) The line x = 1 consists of all points z = 1 + iy where −∞ < y < ∞. Find the real and imaginary parts u and v of f(z) = 1/z at a point z = 1 + iy on this line.

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(b) Show that  for the functions u and v from part (a).

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(c) Based on part (b), describe the image of the line x = 1 under the complex mapping w = 1/z.

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(d) Is there a point on the line x = 1 that maps onto 0? Do you want to alter your description of the image in part (c)?