1. HW 5 (1) Exercise 8.3, 8.4
(2) Supplementary Exercise on Page 70, 71, all.
(3) Exercise 8.6, 8.7, 8.10, 8.12,
(4) Supplementary Exercise on Page 75, 76, all.
(5) Let ∆ be the Laplace operator onC ∼=R2 and f be a C2-function on an open set D⊂C.Show that
∆|f(z)|2 = 4|f0(z)|2.
(6) Letf, g, andh be holomorphic functions on a region1 Ω.
(a) Suppose |f(z)|=|g(z)|on Ω.Show that there is c∈Cso that f(z) =cg(z) on Ω.
(b) Suppose that |f(z)|2+|g(z)|2 = 1 on Ω.Show that both f and g are constant functions on Ω.
(c) Suppose that |f(z)|2 +|g(z)|2 = |h(z)|2 on Ω. Show that f, g, h are linearly dependent in Hol(Ω).(The space of holomorphic functions on Ω forms a vector space.)
(7) Letz1,· · · , zn, w1,· · ·, wm be distinct complex numbers.
(a) Let P(z) = (z−z1)· · ·(z−zn)∈C[z].Show that P0(z)
P(z) =
n
X
j=1
1 z−zj
.
Remark: You can not take lnP(z) at this moment yet because ln-function is not defined yet. Here is a hint: Consider
H(z) =P0(z)−
n
X
i=1
P(z) z−zj
. Show thatH(z) is the zero polynomial.
(b) LetDbe a closed disk centered at 0 of radiusRcontainingz1,· · · , znandC be its boundary. Compute
1 2πi
I
C
P0(z) P(z)dz.
(c) A rational function R(z) over C is a quotient of two complex polynomials P(z)/Q(z), i.e. R(z) = P(z)/Q(z) with P(z), Q(z) ∈ C[z]. Suppose that P(z) = (z−z1)· · ·(z−zn) and Q(z) = (z−w1)· · ·(z−wm). Let D be as above and also assume thatDcontainsw1,· · · , wm.Show that
R0(z)
R(z) = P0(z)
P(z) −Q0(z) Q(z) and compute
1 2πi
I
C
R0(z) R(z)dz.
Furthermore, letf(z) be a holomorphic function onD.Show that 1
2πi I
C
f(z)R0(z) R(z)dz =
n
X
i=1
f(zi)−
m
X
j=1
f(wj).
1An open connected set inCis called a region.
1
2
LetR(z) = (z−11)(z−12)· · ·(z−20)
(z−1)(z−2)· · ·(z−10) .Compute I
C
z2R0(z) R(z)dz.
(d) Let Ω be a region containing all w1,· · ·, wm whose boundary is denoted by
∂Ω = Γ.Show that I
Γ
R(z)dz = 2πi
m
X
i=1
P(zi)
(wi−w1)· · ·(wi−wi−1)(wi−wi+1)· · ·(wi−wn).
(e) Let R(z) = 5z2−z+ 2
z(z+ 1)(z−i).Use residues ofR(z) to find A, B, C such that R(z) = 5z2−z+ 2
z(z+ 1)(z−i) = A z + B
z+ 1+ C z−i. Also compute
I
Γ
R(z)dz and I
Γ
R0(z)
R(z)dz. Here Γ ={z:|z|= 100}.
(f) LetR(z) = 1
z4+ 1.Find all of its poles and their orders. Compute the residues of R at poles. Compute
Z
C
R(z)dz where C is the circle |z|= 2 with positive orientation.
(8) Using the Taylor expansion of 1−cosz atz= 0,we knowz= 0 is a zero of 1−cosz of order 2.Hence the function
f(z) = ez 1−cosz
has pole of order 2 atz= 0.In a deleted neighborhood ofz= 0,we write f(z) = a−2
z2 +a−1
z +a0+a1z+a2z2+· · · . (a) Use the Taylor expansion ofez atz= 0 and of 1−cosz and
ez = (1−cosz) a−2
z2 + a−1
z +a0+a1z+a2z2+· · ·+
to determinea−2, a−1, a0. (b) Compute
I
C
f(z)dz whereC is the unit circle|z|= 1 with positive orientation.
(9) Let P(z) be a complex polynomial of degree n and Q(x) = (z−a)m for m > n.In fact, we can write P(z) = a0+a1(z−a) +· · ·+an(z−a)n. It is very easy to see thatai =P(i)(a)/n!.Then we can write.
P(z)
(z−a)m = a0
(z−a)m + a1
(z−a)m−1 +· · ·+ an (z−a)n−m.
(a) Use this idea to find the expansion of the following rational functionz3+ 3z2+z+ 1 (z−1)5 in the power of (z−1)−1.
(b) Use similar idea (use the Taylor expansion of cosz at π) to find the series expansion of cosz
(z−π)5 in a neighborhood ofπ.
(c) Suppose f1(z), f2(z) are complex polynomials so that g.c.d(f1, f2) = 1, i.e. the greatest common divisor off1 andf2 is 1.Using division algorithm, we can find h1(z), h2(z) so that
f1(z)h1(z) +f2(z)h2(z) = 1.
3
SupposeF(z) =f1(z)f2(z) andG(z) are complex polynomials. Use this prop- erty and the division algorithm to show that there exist complex polynomials H(z) and g1(z), g2(z) so that
G(z)
F(z) =H(z) + g1(z)
f1(z) +g2(z) f2(z).
For example, ifa6=bthen (z−a)m and (z−b)nare relatively prime (g.c.d = 1).
Then
G(z)
(z−a)m(z−b)n =H(z) + g1(z)
(z−a)m + g2(z) (z−b)n.
Using this idea, we obtain the partial fraction expansion of G(z)/F(z) for a generalG(z)/F(z).An example is given below.
(d) Let g1(z), g2(z) be polynomials so that ((z−1)3 and (z2 + 1)2 are relatively prime.)
f(z) =2z6−4z5+ 5z4−3z3+z2+ 3z
(z−1)3(z2+ 1)2 = g1(z)
(z−1)3 + g2(z) (z2+ 1)2.
Furthermore,z2+ 1 = (z−i)(z+i).There are polynomialsh(z), k(z) so that g2(z)
(z2+ 1)2 = h(z)
(z−i)2 + k(z) (z+i)2. FindA1, A2, A3, B1, B2, C1, C2 so that
g1(z)
(z−1)3 = A1
z−1 + A2
(z−1)2 + A3 (z−1)3 h(z)
(z−i)2 = B1
z−i+ B2 (z−i)2 k(z)
(z+i)2 = C1
z+i+ C2 (z+i)2.
By definition, the residues of f(z) at 1, i,−iare A1, B1, C1 respectively. Com- pute
I
C
f(z)dz where C is the circle|z|= 2.
