Practice Final Exam
.pdf
School
Columbia University**We aren't endorsed by this school
Course
MATH 3007
Subject
Mathematics
Date
Jan 10, 2025
Pages
11
Uploaded by BaronAntelope2230
1Math 3007 Practice Final ExamName:Problem12345678910Total:Max10101010101010101010100ScoresInstructions•Time for test:3 hours.•Do not use any notes or any textbooks.
21. True False questions (no explanation needed):(a) T FThe functionu(x, y) =x5-10x3y2+ 15xy4is harmonic.(b) T FThere exists an analytic branch of(z3-1)12defined outside the unit diskB1.(c) T FThe functionf(z) =1zn-1hasnsimple poles.(d) T FThe minimum value of|f(z)|for an analytic functionfover adomainDoccurs on the boundary of the domain.(e) T FIffis an entire function andlimz→∞f(z)zn= 0thenfis a polynomial.
32.(a) Write the definition of an analytic function in a domainD.(b) Give an example of an analytic function inB1\ {0}which has apole of order 4 at 0.(c) State the hypotheses and write the bounds forf(n), wherefis ananalytic function defined inD.(d) Write the Fundamental theorem of Calculus for analytic functions.(e) Write the definition forz0to be a zero of ordermforf.
43.(a) State and prove (using complex variables) the fundamental theo-rem of algebra.(b) Show that iffis analytic inDandBr(z0)⊂Dthenf(z0) =12πZ2π0f(z0+reiθ)dθ.
54. ComputeZ2π01(2 + cosθ)2dθ.
65. Computep.v.Z∞-∞e2xcosh(πx)dxby integrating over a rectangle with verices at±R,±R+i.
76. Computep.v.Z∞0x-sinxx3dx.
87.(a) Letα∈C. Compute the Taylor series expansion around 0 for theprincipal branch of (1 +z)αand find its radius of convergence.(b) Find the points where the functionf(z) =|z|2z-1-¯z22is complex differentiable.
98.(a) ComputeZΓLogz dzwhere Γ is the half circle parametrized byeiθwithθ∈[-π/2, π/2].(b) ComputeZ{|z|=2π}tanz dz
109.(a) Assume thatfandghave zeros of ordermatz0. Show thatlimz→0f(z)g(z)=f(m)(z0)g(m)(z0).(b) Assume that atz0,fhas a pole of ordermandga pole of ordern, withm > n. Show thatf+ghas a pole of ordermatz0.
1110. Assume thatfis analytic inC\ {z1, .., zm}, has poles atz1, .., zm, andthat limz→∞f(z) = 0. Show thatfis a rational functionf(z) =p(z)q(z)withp, qpolynomials and degp <degq.