MATH502_HW1 - Serge Ballif MATH 502 Homework 1(1 Let f = u...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Serge Ballif MATH 502 Homework 1 January 25, 2008 (1) Let f = u + iv be a holomorphic function of z = x + iy . Show that the function g = log | f | 2 = log( u 2 + v 2 ) satisfies Laplaces equation 2 g ∂x 2 + 2 g ∂y 2 = 0 in a neighborhood of any point where f is a nonzero. (You may assume that u and v are twice continuously differen- tiable. (Try to do this in two different ways: once by a brute force calculation using the Cauchy-Riemann equations and once by using the properties of the locally defined holomorphic logarithm function, discussed in lecture 5) Using the “brute force” method, we calculate a first partial derivative of g with respect to x ∂g ∂x = ∂x [log( u 2 + v 2 )] = ∂x [ u 2 + v 2 ] u 2 + v 2 = 2 u ∂u ∂x + 2 v ∂v ∂x u 2 + v 2 . Then the second partial with respect to x is 2 g ∂x 2 = ∂x " 2 u ∂u ∂x + 2 v ∂v ∂x u 2 + v 2 # = 2 ( ∂u ∂x ) 2 + 2 u 2 u ∂x 2 + 2 ( ∂v ∂x ) 2 + 2 v 2 v ∂x 2 ( u 2 + v 2 ) - ( 2 u ∂u ∂x + 2 v ∂v ∂x ) 2 ( u 2 + v 2 ) 2 = 2 u 3 2 u ∂x 2 + 2 u 2 ( ∂v ∂x ) 2 + 2 u 2 v 2 v ∂x 2 + 2 v 2 ( ∂u ∂x ) 2 + 2 uv 2 2 u ∂x 2 + 2 v 3 2 v ∂x 2 - 4 uv ∂u ∂x ∂v ∂x ( u 2 + v 2 ) 2 . By symmetry 2 g ∂y 2 = 2 u 3 2 u ∂y 2 + 2 u 2 ∂v ∂y 2 + 2 u 2 v 2 v ∂y 2 + 2 v 2 ∂u ∂y 2 + 2 uv 2 2 u ∂y 2 + 2 v 3 2 v ∂y 2 - 4 uv ∂u ∂y ∂v ∂y ( u 2 + v 2 ) 2 .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

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