Download Exercises involving analytic functions, harmonic func

c M. K. Warby
20161117171541 MA3614 Complex variable methods and applications
1
Exercises involving analytic functions, harmonic functions and harmonic conjugates
Some of the questions have been taken from past May exams of MA3614 and MA3914.
The format of the past exams was answer 3 from 4 in 3 hours with each question worth
20 marks. Hence if a question given here was worth 10 marks then as a percentage this
was worth 16.7%. The connection between analytic functions and harmonic functions is
unlikely to be covered in the lectures before week 5.
Ex. 1 — Let f (z) = |z|2 = z z. Show that f (z) has a complex derivative at z = 0 but
at no other point. Is f analytic at z = 0?
Ex. 2 — Use L’Hôpital’s rule to determine the following limits. (The first case was a
question from an earlier question sheet.)
1.
z 2 + iz + 2
.
z→i z 2 − 5iz − 4
lim
2.
z3 + i
lim
.
z→i z(z 2 + 1)
Ex. 3 — Let z1 , z2 , . . . , zn be points in the complex plane and let
pn (z) = (z − z1 )(z − z2 ) · · · (z − zn ).
Prove by induction on n that
p0n (z)
1
1
1
=
+
+ ··· +
.
pn (z)
z − z1 z − z2
z − zn
Ex. 4 — Verify that the following functions, denoted by u, of real variables x and y are
harmonic and find their harmonic conjugate. If f is given by f (z) = u(x, y) + iv(x, y) is
entire then find f (z) in each case.
1. u(x, y) = y 3 − 3x2 y.
2. u(x, y) = sinh x sin y.
c M. K. Warby
20161117171541 MA3614 Complex variable methods and applications
2
Ex. 5 — This was in the class test in November 2013 and was worth 30 marks.
Let z = x + iy, x, y ∈ R, and let f (z) = u(x, y) + iv(x, y) be a function which is defined
on C with u and v denoting respectively the real and imaginary parts.
State the Cauchy Riemann equations.
Determine if the following functions are analytic in C and if a function is analytic express
it in terms of z alone.
1.
f (x + iy) = y.
2.
f (x + iy) = 2 + y 3 − 3x2 y + 2x + i(x3 − 3xy 2 + 2y).
Ex. 6 — This was in the class test in November 2013 and was worth 15 marks.
Show that the following function u is a harmonic function and determine the harmonic
conjugate v which satisfies v(0, 0) = 2.
u(x, y) = x + sinh x cos y,
x, y ∈ R,
(Here sinh x = (ex − e−x )/2 denotes the usual hyperbolic function.)
Ex. 7 — This was question 1 of the 2016 MA3614 paper.
1. Let z = x + iy, with x, y ∈ R, and let f (z) = u(x, y) + iv(x, y) denote a function
defined in the complex plane C with u and v being real-valued functions which have
continuous partial derivatives of all orders.
State the Cauchy Riemann equations for the function f to be analytic. Give expressions for f 0 (z) in terms of only partial derivatives of u and also in terms of only
partial derivatives of v.
[3 marks]
2. For each of the following functions determine if they are analytic in the complex
plane, giving reasons for your answer in each case.
(a)
f1 (z) = (2x2 − 2y 2 + 6xy + 4x + 6y) + i(−3x2 + 3y 2 + 4xy + 4y − 6x).
(b)
f2 (z) = i
∂φ ∂φ
+
,
∂x ∂y
where φ(x, y) denotes any function which is harmonic at all points (x, y).
(c)
f3 (z) = ex ((x cos(y) − y sin(y) + cos(y)) + i(x sin(y) + y cos(y) + sin(y))) .
(d)
f4 (z) = ex (cos(y) − i sin(y)) .
c M. K. Warby
20161117171541 MA3614 Complex variable methods and applications
3
[11 marks]
3. Show that the function
u(x, y) = x3 y − xy 3
is harmonic and determine the harmonic conjugate v(x, y) satisfying v(0, 0) = 2.
Express u + iv in terms of z only.
[6 marks]
Ex. 8 — This was question 1 of the 2015 MA3614 paper.
1. Let z = x + iy, with x, y ∈ R, and let f (z) = u(x, y) + iv(x, y) denote a function
defined in the complex plane C with u and v being real-valued functions which have
continuous partial derivatives of all orders. In the following h is a real number. Given
that the function f (z) is such that the following limits exist, express each limit in
terms of the partial derivatives of u and v as appropriate.
(a)
f (z + h) − f (z)
.
h→0
h
lim
(b)
f (z + hi) − f (z)
.
h→0
hi
lim
[6 marks]
State the Cauchy Riemann equations for an analytic function.
[1 mark]
2. Let z = x+iy with x, y ∈ R. For each of the following functions determine whether
or not it is analytic in the domain specified giving reasons for your answers in each
case.
(a)
f1 : C → C,
f1 (z) = x.
(b)
f2 : C → C,
f2 (z) = x4 − 6x2 y 2 + y 4 + 4ixy(x2 − y 2 ).
f3 : C → C,
f3 (z) = cos(x) cosh(y) − i sin(x) sinh(y).
(c)
[7 marks]
c M. K. Warby
20161117171541 MA3614 Complex variable methods and applications
4
3. As in part (1) let z = x + iy, with x, y ∈ R, and let f (z) = u(x, y) + iv(x, y).
Explain why when f is analytic the function u is harmonic. In your answer you may
assume that partial derivatives of u and v of all orders exist and are continuous.
[2 marks]
Let
u(x, y) = −2x − y + x3 − 6x2 y − 3xy 2 + 2y 3 .
Given that the function u is harmonic and the function
g(z) =
∂u
∂u
−i
∂x
∂y
is analytic, express g(z) in terms of z only.
[4 marks]
c M. K. Warby
20161117171541 MA3614 Complex variable methods and applications
5
Ex. 9 — This was question 1 of the 2014 MA3614 paper.
1. Let z = x + iy, with x, y ∈ R and let f (z) = u(x, y) + iv(x, y) denote a function
defined in the complex plane C with u(x, y) and v(x, y) being real valued.
Define what it means for f (z) to be analytic at a point z0 and state the Cauchy
Riemann equations.
[3 marks]
2. Let z = x+iy with x, y ∈ R. For each of the following functions determine whether
or not it is analytic in the domain specified giving reasons for your answers in each
case.
(a)
f1 : C → C,
f1 (z) = |z|2 .
(b)
f2 : C → C,
f2 (z) = cos(3x) cosh(2y) − i sin(3x) sinh(2y).
(c)
f3 : C → C,
f3 (z) =
∂φ
∂φ
−i ,
∂x
∂y
where φ denotes any harmonic function.
(d)
f4 : C → C,
1
f4 (z) = (x2 − y 2 ) + ixy.
2
[9 marks]
3. Show that the function
u(x, y) = 2x2 − 2y 2 − 2xy + 3x − y + 1
is a harmonic function and determine the harmonic conjugate v(x, y) which satisfies
v(0, 0) = 0. For this function v(x, y) express u + iv in terms of z only.
[5 marks]
4. Let f : C → C be an analytic function with f (z) = u(x, y) + iv(x, y) with, as
usual, x, y, u, v being real. Show that the function
g(z) = u(x, −y) − iv(x, −y)
is also analytic in the complex plane.
[3 marks]
c M. K. Warby
20161117171541 MA3614 Complex variable methods and applications
6
Ex. 10 — This was question 1 of the 2013 MA3914 paper.
1. Let z = x + iy, with x, y ∈ R and let f (z) = u(x, y) + iv(x, y) denote a function
defined in the complex plane C with u(x, y) and v(x, y) being real valued.
Define what it means for f (z) to be analytic at a point z0 , state the Cauchy Riemann
equations and show that when f (z) is analytic the Cauchy Riemann equations are
satisfied.
[6 marks]
2. Let z = x+iy with x, y ∈ R. For each of the following functions determine whether
or not it is analytic in the domain specified giving reasons for your answers in each
case.
(a)
f1 : C → C, f1 (z) = z.
(b)
f2 : C → C,
f2 (z) = 3x + y + x2 − y 2 + 4xy + i(3y − x + 2xy − 2x2 + 2y 2 ).
(c)
f3 : C → C,
f3 (z) = cos x cosh y − i sin x sinh y.
(d)
f4 : D → C,
f4 (z) =
x − iy
,
x2 + y 2
where D = {z : |z| > 1}.
[9 marks]
3. Show that the function
u(x, y) = x3 − 3xy 2
is a harmonic function and determine the harmonic conjugate v(x, y) which satisfies
v(1, 0) = 1. For this function v(x, y) express u + iv in terms of z only.
[5 marks]
Ex. 11 — This was question 2b of the 2009 MA3914 paper and was worth 10 marks.
Show that the following function is harmonic in the whole complex plane except at the
origin
−cy
u(x, y) = 2
with c 6= 0 , c ∈ R.
x + y2
Write down the Cauchy-Riemann partial differential equations for the function f (z) =
u(x, y) + iv(x, y) and use them to determine the harmonic conjugate v(x, y) of u(x, y).
Can you write f (z) as a function of z alone? (hint: what is z z̄?) Is f (z) conformal on
the domain of its definition?
(Very little is done in MA3614 on conformal mapping at the moment but a function f has
this property if f (z) is analytic and f 0 (z) 6= 0.)