MAT389 Fall 2014, Problem Set 6 (due Oct 30) Conformal transformations

MAT389 Fall 2014, Problem Set 6 (due Oct 30)
Conformal transformations
Suppose f (z) is conformal at z0 —that is, it is holomorphic at z0 and f 0 (z0 ) 6= 0. Recall the setup:
we take a path C : [a, b] → C in the z-plane, and its image f ◦ C : [a, b] → C in the w-plane; a point
z0 in the z-plane that C goes through at time λ ∈ (a, b) (i.e., C(λ) = z0 ), and its image w0 = f (z0 ).
Then the tangent vectors to C and f ◦ C at the points z0 and w0 , respectively, are related by the
equation
(f ◦ C)0 (λ) = f 0 (z0 ) C 0 (λ).
This is a rotation by an angle arg f 0 (z0 )) —which we call the angle of rotation of f at z0 — and a
dilation by a factor |f 0 (z0 )| —the scale factor of f at z0 .
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6.1 Determine the angle of rotation at the point z = 2 + i of the transformation f (z) = z 2 and
illustrate it for some particular curve. Show that the scale factor of the transformation at
√
that point is 2 5.
6.2 Show that the angle of rotation at a nonzero point z = r0 eiθ0 under the transformation
f (z) = z n (n ≥ 1) is (n − 1)θ0 . Determine the scale factor of the transformation at that
point.
6.3 Fix some x0 ∈ R. Consider the M¨
obius transformation
w = T (z) =
z−i
z+i
It takes the real line to the circle of radius 1 (see Problem 2.11), and H to D. Let Lx0 be
the half-line x = x0 , y > 0. Since it is contained in H, its image is contained in D. And
because Lx0 is contained in a line in the z-plane, its image T (Lx0 ) is contained in a line or
a circle in the w-plane). We do know two points in (the closure of) T (Lx0 ): (x0 − i)/(x0 + i)
and 1 —the images of x0 and ∞. Explain why the fact that Lx0 is perpendicular to the
real axis in the z-plane is enough to determine T (Lx0 ), without having to compute the
image of some third point in Lx0 .
Critical points
6.4 Find the critical points of the following functions. Can you draw how they behave near
those points?
(i) f (z) = z − i,
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(ii) f (z) = 1 − z 2 ,
(iii) f (z) = z 4 − 2z 2 ,
(iv) f (z) = esin z .
6.5 Let k be an positive integer, and α ∈ C. Find all the critical points of
f (z) =
1
z k+1 − αz,
k+1
as well as their order.
Hint: the cases α = 0 and α 6= 0 are radically different!
Challenge: to see how f (z) looks like some z m around a critical point z0 , use a computer
algebra system to plot u = u0 and v = v0 (where u0 + iv0 = w0 = f (z0 )) for some concrete
choices of k and α. The case k = 2, α = −1 is the one I showed in class.
Double dog dare: true, hardcore engineers don’t use computer algebra systems: they
code their stuff! Take on the above challenge using the python library matplotlib
(http://matplotlib.org) —and let me know if you do.
6.6 Find all the critical points of the function
f (z) = cos(z 2 ),
together with their order.
The exponential
6.7 Suppose that a function f (z) = u(x, y) + iv(x, y) satisfies the following two conditions:
(1) f (x + i0) = ex , and
(2) f is entire, with derivative f 0 (z) = f (z).
Follow the steps below to show that f (z) must, in fact, be the exponential function.
(i) Obtain the equations ux = u and vx = v and then use them to show that there exist
real-valued functions φ and ψ of the real variable y such that
u(x, y) = ex φ(y),
and v(x, y) = ex ψ(y).
(ii) Use the fact that u is harmonic to obtain the differential equation φ00 (y) + φ(y) = 0
and thus show that φ(y) = A cos y + B sin y, where A and B are complex numbers.
(iii) After pointing out why ψ(y) = A sin y − B cos y and noting that
u(x, 0) + iv(x, 0) = ex ,
find A and B. Conclude that
u(x, y) = ex cos y,
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and v(x, y) = ex sin y.
6.8 If w = ez is purely imaginary, what restriction is placed on z? In other words, what is the
inverse image by the exponential function of the imaginary axis u = 0?
6.9 Describe the behavior of ez as
(i) x → −∞, with y fixed; and
(ii) y → +∞, with x fixed.
Trigonometric and hyperbolic functions
6.10 Show that eiz = cos z + i sin z for every complex number z.
Hint: start from the right-hand side and work your way towards the left-hand side.
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6.11 Show that sin z = sin x cosh y + i cos x sinh y. Deduce from it that the formula
| sin z |2 = sin2 x + sinh2 y
Note: in class I tried to prove the latter formula by brute force because I wanted to reserve
this method of proof for a homework problem.
6.12 Use the identities
sin(z1 ± z2 ) = sin z1 cos z2 ± cos z1 sin z2 ,
cos(z1 ± z2 ) = cos z1 cos z2 ∓ sin z1 sin z2 ,
and the relationship between trigonometric and hyperbolic functions,
sinh z = −i sin(iz),
cosh z = cos(iz)
to deduce expressions for sinh(z1 ± z2 ) and cosh(z1 ± z2 ).
6.13 Let f (z) = tanh z = sinh z/ cosh z. Find the domain of holomorphicity of f (z), as well as
all of its zeroes.
6.14 Find all roots of the equations
(i) cosh z = 1/2,
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(ii) sinh z = i,
(iii) cosh z = −2.
Hint: notice that these functions are linear combinations of ez and e−z . Letting w = ez ,
we have e−z = w−1 , which yields quadratic equations for w —and you know how to solve
those. Once you have solved for w, solve for z.
The transformation w = sin z
The next three problems work out the images of vertical and horizontal lines in the z-plane under the
transformation w = sin z. Once you know how those work, you can find the image of any (possibly
infinite) rectangle in the z-plane with sides parallel to the real and imaginary axes —for example,
that in the last problem.
A hint for Problems 6.15–6.17: use the formula in Problem 6.11.
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6.15
(i) Show that the inverse image under the transformation w = sin z of the imaginary
axis v = 0 consists of the following collection of lines in the z-plane:
− the real axis y = 0, and
− the lines x = π/2 + kπ, where k ∈ Z.
(ii) Show that the inverse image under the transformation w = sin z of the real axis u = 0
consists of the lines x = kπ, where k ∈ Z.
6.16 Show that the image of the line given by x = c1 for some fixed 0 < c1 < π/2 under the
transformation w = sin z is a branch of hyperbola given by the equation
u
sin c2
2
−
v
cos c2
2
= 1.
Show that, if π/2 < c1 < 0, the image of x = c1 is the other branch of the same hyperbola.
6.17 Show that the image of the line segment given by
−π ≤ x ≤ π,
and y = c2
for some fixed c2 > 0 under the transformation w = sin z is given by the ellipse with
equation
2 2
u
v
+
= 1.
cosh c2
sinh c2
What happens if c2 < 0?
6.18 Find a conformal transformation w = f (z) that takes the semi-infinite strip 0 < x < π/2,
y > 0 onto the upper half-plane, H = {z ∈ C | Im z > 0}.
Hint: start by considering the image of the domain given under Z = sin z. Do you know of
a conformal transformation w = g(Z) that takes the resulting domain to the entire upper
half-plane?