Midterm 2 Practice exam
.pdf
School
Columbia University**We aren't endorsed by this school
Course
MATH 3007
Subject
Mathematics
Date
Dec 18, 2024
Pages
6
Uploaded by SuperHumanBearMaster1162
1Math 3007 Practice Midterm IIName:Problem12345678910Total:Max10101010101010101010100ScoresInstructions•Time for test:75 minutes.•Do not use any other notes or calculators or any textbooks.
21. ComputeLog(π2+Log i).2. Show that sin(2z) = 2 sinz·coszfor allz∈C.
33. Define an analytic branch of1-z1 +z13that is analytic inC\[-1,1].(for allzexcept the ones on the segment [-1,1].)4. ComputeZΓ¯z2+z dzwhere Γ is the upper part of the circle of radius 2 centered at 1 (whichconnects 3 to-1).
45. ComputeZΓz2-2z(z-1)2dzwhere Γ is the square with side length 6 around the origin.6. Show that ifais a real number with|a|<1 thenZ2π01-acosθ1-2acosθ+a2dθ= 2πby relating it to the contour integralRC11z-adz.
57. True or false (explain your reasoning):Iffis continuous inD=B1(0)\ {0}(the unit disk without the origin)andZγf dz= 0for any closed loopγ⊂D,thenfmust be analytic inD.8. Use the maximum modulus principle to show that iffis analytic inthe annulus 1<|z|<5 and|f(z)| ≤2 when|z|= 2 and|f(z)| ≤12when|z|= 4 then|f(z)| ≤89when|z|= 3.(Hint: Find an explicit functiongsuch that|f(z)| ≤ |g(z)|)
69. Assumefis analytic in the unit diskB1(0) and|f(z)| ≤11-|z|2. Showthat|f00(0)| ≤8.10. Assume thatfis analytic inD=B1(0)\ {0}(the disk of radius 1without the origin) and|f| ≤MinDfor some constantM.Show thatfcan be defined at 0 so that is analytic in the wholeB1(0).(Hint: Use Cauchy integral formula.)