Dear There,

Please if there any one can help me with these question

.

Problem 1 Graph the following regions in the complex plane:

a) {z: Re z > 2Im z };

b) {z: π/8 < Arg z ≤ π/4};

c) {z: |z− 2i + 2| > 2}.

Problem 2 Express the following in the form x + iy

i

a) (1 − i) + 1 − i;

b) all the 6th roots of unity;

c) (1 + i)177.

Problem 3 Find the image under the exponential function of the sets:

a) {z: Re z < 0, |Imz | < π };

b) {z: π/4 < |Imz| < π/2}.

Problem 4 Let T be a mapping from C to C. A ﬁxed point of T is a point z

satisfying T (z) = z .

a) Show: any M¨obius transformation, apart from the identity, can have at most 2 ﬁxed

points in C. (The identity is the transformation z 7→ z ).

b)Give examples of M¨obius transformations having (i) 2; (ii) 1 and (iii) no ﬁxed points

in C.

Problem 5 Determine the M¨obius transformation mapping 0 to 2, −2i to 0,

and i to 3/2.

Problem 6 Write a few lines, and draw a picture, on the Joukowski

transformation. Neither your lines nor your picture should be from Wikipedia.

I will be thankful for any one helps me.