MATH3401: Complex Analysis (澳洲程序代写,数学Math代写)

Assignment Number 3

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本次CS代写的主要涉及如下领域: 澳洲程序代写,数学Math代写



MATH3401: Complex Analysis                    First Semester 2020



Assignment Number 3



Problem 1 (6 points) Show the following limits: 4z3


  1. lim

z→∞ z3

  1. lim

z→∞ z2




+ 1337z


= 4;


= ;



  1. lim

(az + b)2



2 =



2 if c ƒ= 0.


z→∞  (cz + d)       c


Problem 2 (3 points) Show that the following functions are defined on all of C, but are nowhere analytic (here z = x + iy):

a) z ›→ 2xy + i(x2 y2);         b) z ›→ sin z.


›→              −

Problem  3  (4  points) Show where the function z x4 + i(1      y)4 is:

  1. analytic;

|     |

differentiable (here z = x + iy).


Problem 4 (3 points) For z = x + iy, define f (z) =       xy .

  1. Show that f satisfies the Cauchy-Riemann equations at the origin. (Note: you will need to use the definition of the partial derivative to calculate ux(0, 0) and uy(0, 0)).
  2. Show that f is not differentiable at the origin. (Hint: approach on a suitable line).
  3. Explain why this doesn’t contradict any of the results from class.



Problem 5 (4 points)

  1. Define precisely what it means for a curve in C to be rectifiable.
  2. Give an example of a non-rectifiable curve in C. You don’t have to prove it is non- rectifiable.

Note: you will need to provide at least one reference, properly cited. This is not allowed

to be wikipedia.


Due: 11:50AM, Monday, 27/04/2020

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