math185f08-hw4

# math185f08-hw4 - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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

MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 4 For z C , recall that the argument of z , denoted arg( z ), is any θ R such that z = | z | e . Note that arg( z ) is only defined modulo 2 π . 1. This problem shows that the conditions in Theorem 2.8 in the lectures cannot be omitted. (a) Let f : C C be defined by f ( z ) = ( exp( - z - 4 ) z 6 = 0 , 0 z = 0 . Show that f satisfies the Cauchy-Riemann equation at all z C but f is not analytic on C . Why doesn’t this contradict the second part of Theorem 2.8 ? (b) Let g : C C be defined by g ( z ) = ( | z | - 4 z 5 z 6 = 0 , 0 z = 0 . Show that g satisfies the Cauchy-Riemann equation at z = 0 but g is not complex differen- tiable at z = 0. Why doesn’t this contradict the second part of Theorem 2.8 ? 2. Let Ω be a region and f : Ω C be analytic. (a) Let γ : [ a, b ] Ω be a smooth curve. If f 0 is non-zero on the image of γ , show that σ : [ a, b ] C defined by σ ( t ) = f ( γ ( t )) is also a smooth curve. (b) Let γ 1 : [ a, b ] Ω and γ 2 : [ a, b ] Ω be two smooth curves such that γ 1 ( t 1 ) = z = γ 2 ( t 2 ) for some t 1 , t 2 [ a, b ] and

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