Taylor polynomials of eᶻ converge uniformly on the closed unit disk, proving **C**. The contour integral of every polynomial around the unit circle is zero, and uniform convergence preserves it. The other choices have nonzero integrals because they contain respectively z⁻¹,(z+z⁻¹)/2, and 2+z+z⁻¹ on the circle.
More Analytic and Harmonic Functions questions
GATE 2023 MA Q42– On the unit disk f(z)=Σₙ₌₀∞a nzⁿ, where a₀=a₁=1 and a n=1/2(2n) for n≥2. P:f is one-to-one…
GATE 2022 MA Q13– Let u=e2x[3x2y3y-3x2y3y]. Its harmonic conjugate v satisfies v(0,0)=2. For z=x+iy and…
GATE 2021 MA Q12– A nonconstant entire f(z)=u(x,y)+iv(x,y),z=x+iy, defines g₁=u x−iu y and g₂=v x+iv y.…
GATE 2026 MA Q17– f: C C is f(z)= z 2-5 z+2. At which listed point is f complex differentiable?
GATE 2026 MA Q18– Let D = z ∈ ℂ: z < 1. An analytic map f:D→D has f(0) = 0. Which is a possible f′(0)?
GATE 2026 MA Q39– On which domain does f(z)= n=15-n(nz) represent an analytic function?