# Thread: Complex analysis

1. ## Complex analysis

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.

2. Originally Posted by Messo
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.
First of all, you shouldn't post any more than 2 questions per thread. It helps for readability of the thread and also enables the questions to be answered quicker.

Second, what have you tried so far? Where exactly are you stuck?

3. Thanks you Prove It for your reply and explaination