examples sheets - Tartarus

Lent 2013 COMPLEX METHODS G.Taylor
Conformal maps
Let U be an open subset of C. A map f : U → C is called conformal if it is analytic with non-zero
derivative at all points in U. Note that, by the chain rule, a composition of conformal maps is
conformal. We will want our conformal maps to be injective on U, so that we have a bijection
between U and f(U).
az +b
1. (i) Let f(z) = , with ad−bc ≠ 0. Where is f conformal?
cz +d
(ii) Let f(z) = z +1 . What are the images of the real axis, the imaginary axis, and the unit
z −1
circle? What are the images of the unit disc and the quadrant {x+iy : x,y > 0}?
(iii) Let C 1 be the unit circle, and C 2 the circle |z −(1+i)| = 1. Find a Möbius map which
simultaneously sends C 1 to the real axis and C 2 to the imaginary axis.
These circles partition the plane into four regions. Where does your map send each
region?
2. Find the images of the following maps, using the principal branches in (iii), (vi) and (vii).
If you haven’t met branches yet then have a go anyway, as though I hadn’t mentioned them.
(i) f(z) = z 2 on {x+iy : x,y > 0}
(ii) f(z) = z 2 on {z : |z| < 1, Re(z) > 0}.
(iii) f(z) = z 1/3 on C\{x+iy : x 0, y = 0}
(iv) f(z) = expz on {x+iy : 0 < y < π}
(v) f(z) = expz on {x+iy : x > 0, 0 < y < π/2}
(vi) f(z) = logz on {x+iy : x,y > 0}
(vii) f(z) = logz on {z : |z| < 1, Re(z) > 0}.
3. Use the identity
z
(z −1) 2 = ( 1
1−z − 1 2) 2
− 1 4
to show that f(z) = z/(z − 1) 2 is a conformal map from the disc |z| < 1 onto the domain
C\{x+iy : x −1/4,y = 0}.
4. For each of the following regions U, construct a function f which maps U conformally onto
the unit disc. If you construct f as a composition of several functions, it will be helpful (to
both you and me) if you sketch a picture for each step.
In each case, for the points on the boundary of U where f is analytic but f ′ (z) = 0, what
happens to angles at those points?
(i) U is the positive quadrant, {x+iy : x,y > 0}
(ii) U is the upper half-disc, {z : |z| < 1, Im(z) > 0}
(iii) U is the open region enclosed between the circles |z −1| = 1 and |z −2| = 2
(iv) U is the half-strip {x+iy : −1 < x < 1, y > 0}.
5. The Joukowski map. Consider the map f(z) = 1 2
(z + 1/z). Find the points where the map
is not conformal and determine how the angles between two vectors at those points change at
their image. Show that this map takes concentric circles with radius r > 1 centered at the
origin to cofocal ellipses. What is the image of the unit circle? Sketch the image of a circle
with center on the x-axis, passing through the point z = 1, with z = −1 as an interior point.
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Cauchy-Riemann equations ; harmonic functions
6. (i) Let u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations, and define
( z + ¯z
g(z,¯z) = u
2 , z − ¯z ) ( z + ¯z
+iv
2i 2 , z − ¯z )
.
2i
Use the chain rule to show that ∂g/∂¯z = 0. Thus any analytic function may be written
as a function of z only.
(ii) Where, if anywhere, in the complex plane are the following functions differentiable, and
where are they analytic?
Imz ; |z| 2 ; sechz .
(iii) Let f(z) = z 5 /|z| 4 (for z ≠ 0) and f(0) = 0. Show that at z = 0, the real and imaginary
parts of f satisfy the Cauchy-Riemann equations, but that f is not differentiable there.
When calculating, e.g., u x (0,0), set y = 0 before differentiating with respect to x.
7. Let f(z) be analytic on an open set D. Show that if |f(z)| is constant on D, then so is f(z).
8. Find complex analytic functions whose real parts are the following:
(i) x (ii) xy (iii) sinxcoshy
(iv) log(x 2 +y 2 )
(v)
y
(x+1) 2 +y 2
You should give functions of z, rather than of x and y. See 6(i).
(vi) tan −1 ( 2xy
x 2 −y 2 ) .
Deducethattheabovefunctionsareharmonicandgivethedomainonwhichtheyareharmonic.
9. By considering w(z) = i+z ( )
i−z , show that 2x
tan−1 x 2 +y 2 is harmonic. On what domain?
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10. Verify that the function φ(x,y) = e x (xcosy−ysiny) is harmonic. Find its harmonic conjugate
and determine the family of curves orthogonal to φ(x,y) = α, for an arbitrary constant α. Find
the analytic function f(z) such that Ref = φ. Can the expression f(z) = φ(z,0) be used to
determine f(z) in general?
11. The isothermal lines of a steady state temperature field are the family of curves x 2 +y 2 = α,
where α > 0. Find the general expression for the temperature, its harmonic conjugate and the
equation for the family of flux lines. Sketch the isothermal and heat flow lines.
12. Show that
g(z) = expz maps {x+iy : 0 < y < π} onto {x+iy : y > 0} ,
and that
h(z) = sinz maps {x+iy : −π/2 < x < π/2, y > 0} onto {x+iy : y > 0} .
Find a conformal map of {x+iy : −π/2 < x < π/2, y > 0} onto {x+iy : 0 < y < π}.
Find a function v which is harmonic on the strip {−π/2 < x < π/2, y > 0} with limiting
values on the boundaries given by: v = 0 on parts of the boundary in the left half-plane, and
v = 1 on parts of the boundary in the right half-plane. Is there only one such function?
If you can, give v as a function of x and y, rather than of z.
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Branches ; series ; singularities
13. Show how the principal branch of logz can be used to define a branch of z i which is singlevalued
on the set D = C\{x+iy : x 0,y = 0}. Evaluate i i for this branch.
Show, using polar coordinates, that the branch of z i defined above maps D onto an annulus
which is covered infinitely often.
14. How many branch points does f(z) = [z(z +1)] 1/3 have? Draw some possible branch cuts in
the complex plane.
If that went okay, repeat with f(z) = [z(z+1)(z+2)] 1/3 and f(z) = [z(z+1)(z+2)(z+3)] 1/2 .
15. Let f(z) = zlog(1+z) with arg(1+z) ∈ [−π,π), and g(z) = zlogz with argz ∈ [−π,π).
Explain why h(z) = f(z)−g(z) is analytic outside the unit disc, and find the Laurent series
for h(z) about the origin in this region. Hence evaluate
∮
h(z)dz .
|z|=2
16. Find the first two non-vanishing coefficients in the Taylor expansion about the origin of the
following functions, assuming principal branches for (i), (ii) and (iii). Where appropriate, you
may make use of standard series expansions for log(1+z), etc.
(i) z/log(1+z) ; (ii) √ cosz −1 ; (iii) log(1+e z ) ; (iv) e ez .
State the range of values of z for which each series converges.
In (i), (ii) and (iii), how would your series differ if you assumed branches different from the
principal branch?
17. Use partial fractions to find the Laurent expansions of 1/((z −a)(z −b)) about z = 0, where
|b| > |a| > 0, in each of the regions |z| < |a|, |a| < |z| < |b| and |z| > |b|.
18. Find and classify the singularities in the complex plane of the following functions:
(i)
(iv)
1
z 3 (z −1) 2 (ii) tanz (iii) zcothz
e z −e
(1−z) 3 (v) exp(tanz) (vi) tan(z −1 ) .
How does tanz behave as |z| → ∞ with Imz = 0?
19. Find the first three terms of the Laurent expansion of f(z) =
By considering the function
1
1−cosz
f(z) − 2 z 2 − 2
(z +2π) 2 − 2
(z −2π) 2 ,
valid for |z| < 2π.
find the three non-zero central terms of the Laurent expansion of f(z) valid for 2π < |z| < 4π.
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Integration
∮
20. Evaluate
C
∮
¯zdz and
C
z 1/2 dz (use the principal branch of z 1/2 ) in the two cases:
(i) C is the circle |z| = 1, and (ii) C is the circle |z −1| = 1.
21. Evaluate, using Cauchy’s theorem or the residue theorem:
∮
dz
(i)
C 1+z 2 where C is the ellipse x 2 +4y 2 = 1;
∮
dz
(ii)
C 1+z 2 where C is the circle x 2 +y 2 = 2;
∮
e z coszdz
(iii)
(1+z 2 )sinz where C is the circle |z −(2+i)| = √ 2.
C
∮
22. Evaluate
C
23. (i) Evaluate
z 3 e 1/z dz
1+z
∫ ∞
−∞
, where C is the circle |z| = 2.
dx
by closing the contour in the upper half-plane.
1+x2 How does the calculation differ if you close in the lower half-plane?
∫ R
xdx
(ii) Evaluate lim
R→∞ −R 1+x+x2. Why is the limit necessary?
(iii) Evaluate
∫ ∞
−∞
e ikx dx
for k > 0 and for k < 0.
1+x2 24. By integrating around a key-hole contour, show that
∫ ∞
0
x a−1 dx
1+x
=
π
sin(πa)
(0 < a < 1).
Explain why the given restriction on the value of a is necessary.
25. By integrating around a contour involving the real axis and the line z = rexp(2πi/n), evaluate
∫ ∞
0
dx
1+x n (n 2).
Check (by change of variable) that your answer agrees with that of the previous question.
∮
26. By evaluating
27. Establish the following:
(i)
∫ ∞
0
C
z −1 dz, where C is the ellipse x2
a 2 + y2
= 1, show that
b2 ∫ 2π
cosxdx
(1+x 2 ) 3 = 7π
16e ;
0
dθ
a 2 cos 2 θ +b 2 sin 2 θ = 2π
ab
(ii)
∫ ∞
0
(b > a > 0) .
sin 2 xdx
x 2 = π 2 ; (iii) ∫ ∞
0
logxdx
1+x 2 = 0 .
For (iii), you could integrate (logz)2
1+z 2 around a keyhole contour, or integrate logz
1+z 2 along the real
axis. (You could do both!) What goes wrong if you integrate logz
1+z 2 around a keyhole?
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28. Using a rectangular contour, show that for a real,
Find
∫ ∞
0
x
sinhx dx.
∫ ∞
0
sinax
sinhx dx = π πa
tanh
2 2 .
29. Let P(z) be a non-constant polynomial. Consider the contour integral
∮
P ′ (z)
I =
P(z) dz .
C
Show that, if C is a contour that encloses no zeros of P, then I = 0. Evaluate lim R→∞ I,
where C is the circle |z| = R, and deduce that P has at least one zero in the complex plane.
30. By considering the integral of f(z) = cotz
z 2 +π 2 around a suitable large contour, prove that,
a2 if ia is not an integer,
∞∑ 1
n 2 +a 2 = π a coth(πa) .
n=−∞
By considering a similar integral, prove also that, if a is not an integer,
∞∑ 1
(n+a) 2 = π 2
sin 2 (πa) .
n=−∞
Justify the limiting operation needed as a → 0 in order to deduce from each of these the value
of ∑ ∞
n=1 1/n2 .
*31. Evaluate the following contour integrals:
(i)
∫ ∞
0
log(x+1)
x a+1 dx (0 < a < 1) ; (ii)
∫ 1
−1
√ ∫ 1−x
2 2π
1+x 2 dx ; (iii) log(5+4cosθ)dθ .
0
Fourier Transforms
In all cases, the Fourier transform is denoted by a tilde and defined by ˜f(k) =
32. Let
f(x) =
{ 1 for |x| < a
0 for |x| > a
Show directly that ˜f(k) =
2sinak
k
and g(x) =
and ˜g(k) = 2(1−cosak)
k 2 .
∫ ∞
−∞
{ a−|x| for |x| < a
0 for |x| > a
f(x)e −ikx dx.
Find the convolution of f(x) with itself, and hence verify the formula for ˜g(k) using the
convolution theorem.
Verify by contour integration that the inversion formula holds for f(x).
33. Using the Fourier inversion formula, show that, for a > 0,
(i) e −a|t| = a π
∫ ∞
−∞
e iωt
ω 2 +a 2 dω, (ii) H(t)e−at sinbt = 1
2π
∫ ∞
−∞
be iωt
(iω +a) 2 +b 2 dω,
where H(t) is the Heaviside step function. What goes wrong with the integrals if a = 0? What
are the values of the integrals when a < 0?
.
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34. The displacement x(t) of a damped harmonic oscillator obeys the equation
and ˜x(ω) and ˜f(ω) exist. Show that
x(t) =
∫ ∞
−∞
ẍ+2γẋ+q 2 x = f(t) (γ > 0)
G(t−t ′ )f(t ′ )dt ′ , where G(t) = 1
2π
Verify formally, by differentiation under the integral sign, that
∫ ∞
−∞
d 2
dt 2G(t)+2γ d dt G(t)+q2 G(t) = δ(t) .
e iωt
q 2 +2iγω −ω 2 dω .
Assume that the system is underdamped, that is that 0 < γ < q, and let p = √ q 2 −γ 2 .
Deduce (see the previous question) that
G(t) = 1 p e−γt sinptH(t) .
35. Show that the convolution, h(x), of the function f(x) = e −|x| with itself is given by
{ (1−x)e
x
for x < 0
h(x) =
(1+x)e −x for x > 0 .
Use the convolution theorem to show that
h(x) = 2 π
Verify this result by contour integration.
∫ ∞
−∞
e ikx
(1+k 2 ) 2 dk .
Laplace Transforms
In all cases, the Laplace transform is denoted by L{f(t)} ≡ F(s) =
∫ ∞
0
f(t)e −st dt.
36. Use standard properties of the Laplace transform to find the Laplace transform of the following
functions: (i) t 3 e −3t , (ii) 2e 3t sin4t, (iii) e −4t cosh2t.
37. Use partial fractions and standard properties of the Laplace transform to find the inverse
s+3
Laplace transform of
(s−2)(s 2 . Verify this result using a contour integral.
+1)
38. Find the Laplace transforms of f(t) = t −1/2 and g(t) = t 1/2 . Note that ∫ ∞
0
e −x2 dx = 1 2√ π.
Verify (by integrating around a keyhole-ish contour) that the inversion formula holds for f(t).
39. The Gamma function is defined for n > 0 by Γ(n) =
∫ ∞
0
x n−1 e −x dx.
Show that Γ(n+1) = nΓ(n), and hence that Γ(n+1) = n! if n is a positive integer. Using the
previous question, write down the value of Γ( 1 2 ).
The Beta function is defined for m,n > 0 by B(m,n) =
Use the convolution theorem for Laplace transforms to establish that
∫ 1
B(m,n) = Γ(m)Γ(n)
Γ(m+n) .
0
x m−1 (1−x) n−1 dx.
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40. Use Laplace transforms to solve the differential equation:
y ′′′ −3y ′′ +3y ′ −y = t 2 e t ,
with initial conditions y(0) = 1, y ′ (0) = 0, y ′′ (0) = −2.
41. A linear system is described by the differential equation
with initial conditions y ′ (0) = y(0) = 0.
y ′′ −3y ′ +2y = u(t) ,
Use Laplace transforms to determine the response of the system to the signal u(t) = t. Determine
also the impulse response, i.e. the response y(t) to a signal u(t) = δ(t).
42. By taking Laplace transforms, solve
Verify your solution.
f(t)+4
∫ t
0
(t−τ)f(τ)dτ = t .
43. By taking Laplace transforms, solve
x ′ = 10y −5x
y ′ = y −x
with x(0) = 3, y(0) = 1.
44. The Initial/Final Value Theorems.
By considering the Laplace transform of f ′ (t), prove that
limf(t) = lim sF(s)
t→0 s→∞
and, assuming that lim
t→∞
f(t) exists, prove that
lim f(t) = lim sF(s).
t→∞ s→0
45. The zeroth order Bessel function J 0 (t) is defined by its series expansion to be
J 0 (t) = 1− t2
2 2 + t4
2 2 4 2 − t6
2 2 4 2 6 2 +···
Find its Laplace transform, given that it obeys the differential equation
tJ ′′
0 +J ′ 0 +tJ 0 = 0 .
Verify this by transforming the series expansion.
Show that
∫ ∞
0
J 0 (t)dt = 1. Find the convolution of J 0 with itself.
Please send any corrections or comments to me at glt1000@cam.ac.uk
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