list of problems

Problems for MATH-4300
Complex Variables
Gregor Kovaˇciˇc
February 10, 2015
This list will change as the semester goes on. Please make sure you
always have the newest version of it.
1. Reduce the following numbers to the form a + ib:
(i)
5i
,
(1 − i)(2 − i)(3 − i)
(ii)
2+i
3 − 2i
2
,
(iii)
z−1
, where z = x + iy, with x and y real,
z+1
(iii)
1
, where z = x + iy, with x and y real.
z2
2. (i) If z is a complex number such that |z| = 1, compute |1 + z|2 + |1 − z|2 .
(ii) For complex a and b, calculate that
|a + b|2 + |a − b|2 = 2(|a|2 + |b|2 ),
and interpret this result geometrically.
3. If z, a, b, and c are complex numbers, show that, geometrically, the equation az+b¯
z +c = 0
may represent a straight line or a single point, and give the conditions on the coefficients a,
b, and c for each case to occur. Are there any other possibilities?
1
4. Find all the solutions of the equation z 3 = 1. Write the real and imaginary parts of these
roots in terms of fractions involving integers and square roots of integers.
5. If a and b are real numbers, b 6= 0, show that
s

s
√
√
2
2
2
2
√
a+ a +b
b −a + a + b 
a + ib = ± 
+i
,
2
|b|
2
and explain why the expressions under the square root signs are non-negative.
HINT: Assume (x + iy)2 = a + ib, and find two equations for x and y. From these equations,
deduce (x2 + y 2 )2 = a2 + b2 , and then deduce the expressions for x2 and y 2 . Finally, be
careful about choosing the relative signs of x and y.
6. Use the result of problem 5 (even if you did not derive it), to compute
√
(i) 1 + i,
s
√
1 − 3i
.
(ii)
2
7. If n is a positive integer, and
ω = cos
2π
2π
+ i sin ,
n
n
compute that
1 + ω h + ω 2h + · · · + ω (n−1)h = 0
for any integer h which is not a multiple of n. What is the geometric interpretation of this
equality? What happens if h is a multiple of n?
HINT: For the geometric representation, dividing the expression by n may be helpful.
8. Let z = x + iy, with x and y real.
(i) Write the function z 3 − 2z 2 + z − 1 in the form u(x, y) + iv(x, y), and verify that u and
v satisfy the Cauchy-Riemann equations ux = vy , vx = −uy and also Laplace’s equation
uxx + uyy = 0 = vxx + vyy .
(ii) Write the function z z¯ + z¯ − 1 in the form u(x, y) + iv(x, y), and verify that u and v do
not satisfy the Cauchy-Riemann equations ux = vy , vx = −uy . Does either u or v satisfy the
Laplace equation uxx + uyy = 0 or vxx + vyy = 0? What can you conclude from this?
2
9. Verify that the function u(x, y) = ex cos y is harmonic, i.e., satisfies Laplace’s equation
uxx + uyy = 0. Then, find its harmonic conjugate, i.e., a function v(x, y) such that ux = vy ,
vx = −uy ? Verify that v also satisfies Laplace’s equation, vxx + vyy = 0?
10. Let z = x + iy and w = u + iv, with x, y, u, and v real. Consider the mapping by the
analytic function w = z 2 .
(i) What regions in the z-plane are mapped into the strips 0 < u < a and −a < u < 0, with
a > 0, in the w-plane? What about the strips b < u < c, where b and c are real?
(ii) What regions in the z-plane are mapped into the strips 0 < v < a and −a < v < 0, in
the w-plane? What about the strips b < v < c?
(iii) What region in the w-plane is the wedge z = reθ , r > 0, 0 < θ < α mapped onto? What
about the wedge −β < θ < α? Here α, β > 0.
Make appropriate sketches of the regions involved in all these cases.
11. Solve the
equation w0 = w with the initial condition w(0) = 1 using a power
Pdifferential
∞
series w = n=0 cn z n ? What values of the coefficients do you get? What is the function
w = f (z), represented by this series?
12. Let z = x + iy and w = u + iv, with x, y, u, and v real. Consider the mapping by
the analytic function w = ez . What regions in the w-plane are the rectangles a < x < b,
c < y < d, with a, b, c, and d real, mapped into? Make appropriate sketches of the regions
involved in all these cases. Pay specific attention to limiting cases of strips when one or both
of the sides drift off to infinity. Make appropriate sketches of the regions involved.
13. Show that log ez = z + 2nπi, where n runs through all the integers.
14. Compute all the values of log(−ei).
15. Show that, when restricting to values on the principal branch of the logarithm, defined
by −π < arg z < π,
(i) log(1 + i)2 = 2 log(1 + i),
(ii) log(−1 + i)2 6= 2 log(−1 + i).
16. Find all the values for (1 + i)i .
3
17. Define
cosh z =
ez + e−z
,
2
sinh z =
ez − e−z
.
2
(i) Sketch cosh x and sinh x for real x.
(ii) Derive the addition formulas for cosh(z + w) and sinh(z + w), and also the formula
cosh2 z − sinh2 z = 1.
(iii) What kind of a curve does the parametrization ξ = cosh t, η = sinh t, −∞ < t < ∞
represent in the (ξ, η)-plane?
(iv) Find the derivatives of cosh z and sinh z.
(v) Find the power series representations for cosh z and sinh z.
(vi) Derive the formulas cos z = cosh iz and sin z = −i sinh iz.
(vii) Derive the formulas for the inverse functions cosh−1 z and sinh−1 z in terms of logarithms, and then derive the formulas for their derivatives.
HINT: Ignore any question of branches and take all the square roots with the plus sign.
18. Using the polar representation, show that
(i) (−1 + i)7 = −8(1 + i).
√
√
(ii) (1 + 3i)−10 = 2−11 (−1 + 3i).
19. (i) If w0 is one of the cube roots of a nonzero complex number z0 , show that the other
two cube roots are w0 and w0 2 , where = e2πi/3 .
(ii) Express in the rectangular coordinates.
√
√
(iii) Let z0 = 4 2(−1 + i). Verify that one of its roots is w0 = 2(1 + i), and find the other
two roots.
20. (i) Given complex α, show that the function
∞ X
α n
f (z) =
z
n
n=0
where
α
α
α(α − 1) · · · (α − n + 1)
, n = 1, 2, . . . ,
= 1 and
=
0
n!
n
4
where n! = n(n − 1) · · · 1, is analytic for |z| < 1.
HINT: Compute the radius of convergence of the power series representing it.
(ii) Compute that its derivative f 0 (z) equals αf (z)/(1 + z).
(iii) Deduce that the derivative of (1 + z)−α f (z) vanishes, and therefore
f (z) = (1 + z)α .
(iv) Which branch of (1 + z)α does f (z) represent?
HINT: What is f (0)?
21. (i) Show that, if n is a positive integer and |z| ≤ 21 n,
z
= z + fn (z),
n log 1 +
n
where
|z|2
|fn (z)| ≤
.
n
(ii) Deduce that
z n
1+
→e
n
as n → ∞ for any complex z.
22. Let C be the line segment x = t, y = t, 0 < t < 1 in the complex plane. Let
f (z) = u(x, y) + iv(x, y) = z 2 . Compute the following integrals:
Z
(i)
f (z)dz,
C
Z
(ii)
u(x, y)dx + v(x, y)dy,
C
Z
(iii)
f (z)|dz|, where |dz|2 = dx2 + dy 2 .
C
I
23. Compute
xdz over the circle traversed counter-clockwise in two ways. First, using
1
1
r2
a suitable parametrization, and second, by observing that x = (z + z¯) =
z+
on
2
2
z
the circle.
|z|=r
5
I
24. Let f (z) be analytic in a region in which the closed curve C lies. Show that
f (z)f 0 (z)dz
C
is purely imaginary.
HINT: Show that the real part of the integrand is an exact differential.
25. Let C be the circle |z − z0 | = r traversed counter-clockwise, and let α be any nonzero
real number. Parametrize C by z = z0 + reiθ , with −π < θ < π, and compute that
I
sin(πα)
(z − z0 )α−1 dz = 2iRα
α
C
on the principal branch of the integrand. What does this show for α = n, integer? In
particular, can you deduce the value for α = 0?
26. Show that if C is a positively oriented simple closed contour, then the area of the region
enclosed by C can be calculated as
I
1
z¯ dz.
2i C
27. Use the method described below to derive the integration formula
√
Z ∞
π −b2
−x2
e
cos 2bx dx =
e
(b > 0).
2
0
(i) Let C be the circumference of the rectangle with the vertices −a, a, a + ib, and −a + ib,
2
a > 0, traversed counter-clockwise. Show that the sum of the integrals of e−z along the
lower and upper horizontal legs of C can be written as
Z a
Z a
2
−x2
b2
e
2
dx − 2e
e−x cos 2bx dx,
0
0
and that the sum of the integrals along the vertical legs on the right and left can be written
as
Z b
Z b
2
−a2
y 2 −2iay
−a2
ie
e
dy − ie
ey +2iay dy.
0
0
Thus, with the aid of the Cauchy’s theorem, show that
Z a
Z a
Z b
2
−x2
−b2
−x2
−(a2 +b2 )
e
cos 2bx dx = e
e
dx + e
ey sin 2ay dy
0
0
0
(ii) Rewrite
Z
∞
e
−x2
2
dx
ZZ
∞
e−(x
=
−∞
−∞
6
2 +y 2 )
dx dy
in polar coordinates, and thus show that
√
Z ∞
π
−x2
e
dx =
.
2
0
(iii) Use the estimate
Z b
Z b
2
2
y
<
e
sin
2ay
dy
ey dy,
0
0
and let a → ∞ in the last formula in part (i) to deduce the desired integration formula.
28. Let C be the boundary of the square whose sides lie along the lines x = ±2 and y = ±2,
described in the positive sense. What is the value of
I
cos z
dz ?
2
C z(z + 8)
29. Let
Pn (z) =
n
1 dn
z2 − 1 .
n
n
2 n! dz
(i) Show that Pn (z) is a polynomial of order n. These polynomials are called Legendre’s
polynomials.
(ii) Show that
Pn (z) =
1
2n+1 πi
I
C
(s2 − 1)n
ds,
(s − z)n+1
n = 0, 1, 2, . . . ,
where C is any positively-oriented simple closed contour surrounding the point z.
(iii) When z = 1, show that the integrand in (ii) can be written as (s + 1)n /(s − 1), and
deduce that Pn (1) = 1. Likewise, calculate that Pn (−1) = (−1)n , n = 0, 1, 2, . . ..
30. Find an example showing that an function analytic in a region Ω can have its minimum
modulus inside Ω, provided the value of this minimum modulus is zero.
7