5021-hw1.pdf - Complex Analysis Fall 2017 Problem Set 1 Due...

This preview shows page 1 - 2 out of 2 pages.

Complex Analysis, Fall 2017 Problem Set 1 Due: September 12 in class 1. Find arg( i - 2 - 2 i ). 2. Find all the points at which f ( z ) = ¯ z 2 + z is complex differentiable. 3. Find the following roots. (i) All the fourth roots of - 1 - 3 i . Express the roots in rectangular coordinates and exhibit them as vertices of a certain square. (ii) The square roots of 8 i . 4. If z 1 , z 2 C , then show that z 1 ¯ z 2 = - 1 if and only if the pre-image of z 1 and z 2 under the stereographic projection correspond to diametrically opposite points on the Riemann sphere. 5. Let U 1 and U 2 be open subset of C and f : U 1 U 2 and g : U 2 C functions which are real differentiable. Prove the following chain rule: ( f g ) ∂z = ∂f ∂z ∂g ∂z + ∂f ¯ z ¯ g ∂z . (If f = u + iv , then ¯ f = u - iv ). Similarly ( f g ) ¯ z = ∂f ∂z ∂g ¯ z + ∂f ¯ z ¯ g ¯ z . (You don’t need to prove the second identity.) 6. Let U C be a connected open subset and
Image of page 1

Subscribe to view the full document.

Image of page 2
  • Fall '09

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern