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

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;
2.  

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

Current assignments will be available at

http://www.maths.uq.edu.au/courses/MATH3401/Tutorials.html