{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW11sol - P ROBLEM SET 11 Partial solution 6 Consider the...

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

P ROBLEM SET 11 Partial solution 6. Consider the function h = f - g . Since f and g are analytic, so is h . Since real parts of f and g are identical on the boundary B , the real part of h is equal to 0 on B . Let us denote by u the real part, and by v the imaginary part of h . By the consequence of the Maximum Modulus Principle, that was given in (4) on page 192 of the textbook, both the maximum and the minimum on D B of u are achieved on B . Therefore both the maximum and the minimum of u on B D are equal to 0 . Hence u is constant, and equal to 0 on D B . By the Cauchy-Riemann equations v y = u x = 0 and v x = - u y = 0 on D B . Hence v must be a constant on D B ; we denote it by α . Therefore f = g + h = g + on D B . 8. Let us consider first the case that f is equal to zero at some point w B . Then for all z D B , | f ( z ) | ≥ | f ( w ) | = 0 and hence | f | achieves its minimum on the boundary (which is what we wanted to prove).

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern