1. HW3
(1) Find the radius of convergence of the power series P∞
n=0anzn, where {an} are se- quence of complex numbers defined by
(a) an= (ln n)2 (b) an= (n!)3/(3n)!
(2) Denote (a)0= 1 and (a)n= a(a + 1) · · · (a + n − 1). Find the radius of convergence of the hypergeometric series:
F (α, β, γ, x) =
∞
X
n=0
(α)n(β)n
n!(γ)n
xn.
(3) Find the radius of convergence of the Bessel function of order r : Jr(z) =z
2
r ∞
X
n=0
(−1)n n!(n + r)!
z 2
2n
.
(4) Let z(t) = x(t) + iy(t), a ≤ t ≤ b be a parametrization of a smooth curve C on C.
From calculus, we know the arc length of C is given by the formula L(C) =
Z b a
p(x0(t))2+ (y0(t))2dt.
We use the notation dz = (x0(t) + iy0(t))dt and |dz| =p(x0(t))2+ (y0(t))2dt. Com- pute the arc length of the following smooth curves C.
(a) z(t) = cos3t + i sin3t, 0 ≤ t ≤ 2π.
(b) z(t) = t + i ln cos t, 0 ≤ t ≤ π/4.
(5) Compute the line integral Z
C
−ydx + xdy x2+ dy2 ,
where C is the unit circle x2+ y2= 1 with positive orientation.
(6) Show that if |a| < r < |b|, then Z
Cr
dz
(z − a)(z − b) = 2πi a − b. (7) Compute
Z
C
f (z)dz.
(a) f (z) = 1/z with z ∈ C \ {0} and C is the unit circle |z| = 1 with positive orientation.
(b) f (z) = 1/z with z ∈ C \ {0} and C is the circle |z − 3| = 1 with positive orientation.
(c) f (z) = z2 with z ∈ C and C is given by the parametrization z(t) = t + it2 with
−1 ≤ t ≤ 1.
(d) f (z) = z2 with z ∈ C and C is the unit circle |z| = 1.
(8) Which of the following regions are simply connected?
(a) Ω = {z ∈ C : |z| < 1}.
(b) Ω = {z ∈ C : 1 < |z| < 2}.
(c) Ω = {z = x + iy ∈ C : |x| < 1, |y| < 1}.
(d) Ω = C \ {−1, 0, 1}.
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