Real analysis代写

数学代写|Real analysis实变分析代考,Complex analysis复变分析代考,数学代考

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下面是复分析中Comformal Mappings部分的作业代写实战

  1. Let D  C be the intersection of the half planes y < 2x and y > 􀀀2x, where z = x +
    iy 2 C. Find a conformal mapping of D onto the following domains:
    a) Find a conformal mapping of D onto the right half plane.
    b) Find a conformal mapping of D onto the unit disk D = fjzj < 1g.
    c) Determine the automorphism group for D.
  2. Given a 2 (􀀀1; 1) we wish to find a conformal map  of the slit disk D n (􀀀1; a]
    onto the unit disk D such that (i=2) = 0. In order to do this proceed as follows:
    a) Find a conformal map of D n (􀀀1; a] onto D n (􀀀1; 0].
    b) Find a conformal map of D n (􀀀1; 0] onto D.
  3. Let D = D n fjz + 1=2j  1=2g. Find a conformal map from D to D.
  4. Consider the unit disk
    a) Show that D is not conformally equivalent to C.
    b) Find an analytic mapping f : D ! C such that f(D) = C.
  5. Prove that the only entire functions f : C ! C which are injective (one to one) are the
    linear functions f(z) = az + b.
  6. Let Dj ; j = 1; 2; : : : ; be a sequence of simply connected domains such that Dj+1 
    Dj  C for all j = 1; 2; : : : , and such that the interior D of \jDj is nonempty and
    a) Prove that D is simply connected.
    b) Pick any point z0 2 \jDj . Let gj : D ! Dj be a conformal map, normalized so that
    gj (0) = z0 and g0
    j (0) 2 (0; 1). Prove that the sequence gj ; j = 1; 2; : : : ; converges
    locally uniformly on D to an analytic function g with the properties: if z0 2= D then g
     z0; if z0 2 D then g : D ! D is a conformal map.

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