MAT2410 – MANDATORY ASSIGNMENT 1
DUE 14.30, THURSDAY 18 SEPTEMBER 2014
LEVEL 7 NIELS HENRIK ABELS HUS
(1) Let f : C → C be given by f (z) = f (x, y) = e−y (cos(x) + i sin(x)).
(a) Show that f satisfies the Cauchy-Riemann equations everywhere in C, and
is therefore a holomorphic function.
(b) Using the fact that f 0 (z) = ∂f
= 2 ∂u
for holomorphic functions f = u + iv,
∂z
∂z
0
find a formula for f in terms of x and y.
(c) Now use the relations x = z+z
and y = z−z
, together with the definitions
2
2i
given in lectures for sin(z) and cos(z), to give a formula for f in terms of z.
You may also use the fact that ez ew = ez+w for all z, w ∈ C.
(d) Use your result from part (c) and the chain rule to find a formula for f 0 in
terms of z.
(e) Rewrite your expression for f 0 from part (d) as a function of x and y, and
check that it agrees with your answer from part (b).
(2) Let σ : C → C be the complex conjugation map, σ(z) = z.
(a) Let f : Ω → C be a real-differentiable function, where Ω ⊂ C is an open
)
set. Show that σ( ∂f
) = ∂(σ◦f
, or in other words, that
∂z
∂z
∂f
∂f
=
.
∂z
∂z
(b) Using the result of part (a) and the fact that σ ◦ σ(z) = z, that is, (z) = z,
)
carefully prove that σ( ∂f
) = ∂(σ◦f
, or in other words, that
∂z
∂z
∂f
∂f
=
.
∂z
∂z
(c) Let Ω = σ(Ω) = {z : z ∈ Ω} be the reflection of Ω in the real axis. Define
g : Ω → C by g = f ◦ σ, that is, g(z) = f (z). Prove that ∂g
(z) = ∂f
(z) for
∂z
∂z
z ∈ Ω.
(d) Suppose now that f : Ω → C is holomorphic. Using the previous results, or
otherwise, show that σ ◦ f ◦ σ : Ω → C is holomorphic, or in other words,
that f (z) is holomorphic.
1
(3) Recall from lectures that the holomorphic function (1−z)−1 , defined on the open
P
n
set C \ {1}, has the power series expansion ∞
n=0 z centred at 0, with radius of
convergence R = 1.
(a) Let m > 0 be a fixed positive integer. Using results from lectures, obtain
a power series expansion of (1 − z)−m centred at 0, and give the radius of
convergence of the series.
P
n
(b) Suppose that the power series for (1 − z)−m is ∞
n=0 an z . Show that, for
each fixed m > 0, the coefficients an satisfy the asymptotic relation
1
an ∼
nm−1 as n → ∞ .
(m − 1)!
an
n→∞ bn
(By definition, an ∼ bn as n → ∞ means that lim
= 1.)
R
(4) (a) Compute γ cos(z)dz where γ is the piecewise smooth curve from z1 = i + π2
to z2 = i − π2 as shown.
2
z2
−2
z1
1
−1
0
1
2
−1
1
(b) Let f (z) = z−1−i
R.
(i) Compute γ1 f (z)dz, where γ1 is the piecewise smooth curve consisting
of the vertical line segment from 2 − 2i to 2 + 2i followed by the
horizontal Rline segment from 2 + 2i to −2 + 2i.
(ii) Compute γ2 f (z)dz, where γ2 is the smooth curve consisting of the
diagonal line segment
from 2 − 2i to −2 + 2i.
R
(iii) Hence compute γ f (z)dz, where γ is the triangle with vertices
2 − 2i, 2 + 2i, and −2 + 2i, equipped with the counterclockwise orientation. Use the fact that arctan(1) + arctan(2) + arctan(3) = π to
simplify your answer as much as possible.
(iv) What, if anything, can we conclude about the existence of a primitive
1
on the set C \ {1 + i}? Give a reason for your answer.
for z−1−i
2