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Math Help - Complex analysis

  1. #1
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    Mar 2010
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    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.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by Messo View Post
    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?
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  3. #3
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    Mar 2010
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    Thanks you Prove It for your reply and explaination
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