MATH2007 / G12COF COMPLEX FUNCTIONS 2018/19
GENERAL INFORMATION
Module lecturer: Dr Matthias Kurzke (B10 Mathematics Building)
Email: matthias .kurzke@nottingham .ac .uk Office hours: Monday 15-16, Friday 13-14
AIMS AND LEARNING OUTCOMES
Aims of the module are: to teach the introductory theory of functions of a com- plex variable; to teach the computational techniques of complex analysis, in particular residue calculus, with a view to potential applications in subsequent modules.
Learning outcomes: a successful student will be able to
(1) identify analytic functions and singularities
(2) apply the main results of the module to functions of a complex variable
(3) evaluate certain classes of integrals
(4) compute Taylor and Laurent series expansions
MODULE CONTENT
In this module we concentrate on functions which can be regarded as functions of a complex variable, and are differentiable with respect to that complex variable. These “good” functions include most common functions like polynomials, rational functions, exponentials, sine, cosine etc. (logarithms and roots require some special care). These are important in applied maths, and they turn out to satisfy some very useful and quite surprising and interesting formulas. For example, one technique we learn in this mod- ule is how to calculate integrals like
Z− dx
without finding an antiderivative.
LECTURE AND PROBLEM CLASS SCHEDULE
Lectures take place on Wednesdays 11-12 in Pope C14 and Fridays 12-13 in Pope C16. Problem classes or review sessions will be on Wednesdays 12-13 in Pope C14.
There will be problem classes (where you will solve problems aided by me and post- graduate helpers), on 6/2, 20/2, 6/3, 20/3, 3/4.
There will be review/revision sessions (where I present sample questions and dis- cuss how to solve them) on 30/1, 13/3 (class test revision), 10/4 (exam revision). There maybe additional sessions on the remaining Wednesdays if necessary.
TEACHING AND LEARNING METHODS
To learn the material, we will have parts where I present material (lectures and worked example classes) and parts where you work on the material with or without assistance (problem classes and practice coursework). As the only way to learn math- ematics is to do mathematics, the problem classes and practice coursework are just as important as the lectures, if not more.
LECTURE NOTES
I will distribute printed copies of the slides used in the lectures. These are available on Moodle along with more complete book-style lecture notes on Moodle that also cover optional proofs.
OFFICE HOURS
Please feel free to come to my office hours with any questions about the content. PRACTICE COURSEWORK
A major part in learning mathematics is to solve problems on your own. To allow you to get better feedback on your problem-solving techniques and your mathematical writing, three (unassessed) practice courseworks will be issued, to be submitted on 20 February, 13 March, 3 April (after the lectures).
They will be returned (along with feedback available from markers) in the problem classes.
Additionally, practice questions for the problem classes will be issued. All solutions will be made available on Moodle.
ASSESSMENT
There will be one 2-hour written exam (90%) plus an “in-class” test (10%). The ex- amination will mainly be based on using the facts and theorems of the module to solve problems of a computational nature, or to derive facts about functions. Single- or dual- line calculators will be permitted. You will not be expected to memorise the proofs of the theorems from the notes or slides. However, you may be asked to state theorems, and/or to use them to deduce some fact. The problem class and practice coursework questions, as well as past exam papers, provide good practice in the essential tech- niques required for passing the module. Doing the practice coursework is the easiest way to get feedback on your performance and writing style, so it is strongly recom- mended.
In the examination there will be 4 questions, from which the best 3 answers will count. There will be an “in-class” test on Thursday 21 March (17-18, T&L Building A03). It will not be multiple choice. Calculators will NOT be allowed.
RE-ASSESSMENT
If a re-assessment is required due to failing the May/June examination then the re- assessment will be by resit examination only. (The in-class test will not count). This is inline with the School policy in the undergraduate student handbook.
REVISION HANDOUT: COMPLEX NUMBERS AND USEFUL FORMULAS
This handout contains some formulas and a short review of material from G11LMA. For more details, see Chapter 1 of the lecture notes on Moodle. We assume in this sheet x, y ∈ R and set z = x + iy so Re z = x, Im z = y. (Note that Im z is always a real number).
The complex conjugate of z is z = Re z - iIm z = x - iy. It obeys the rules (z) = z, z + w = z + w, zw = z w
and we can write Re z = z = . The absolute value or modulus of z = x + iy is
|z| = √ (Re z)2 +(Im z)2 = √x2 + y2 = √zz.
Geometrically, |z| is the distance of z from 0 in the complex plane. We have the estimates |x| ≤ |z| , |y| ≤ |z| , |z| ≤ |x| + |y| and the multiplication rule | zw | = |z|| w | . For z 0, we can write = complex number.
Warning: For z ∈ C \ R we have z2 |z|2 and |z| ±z.
For any z, w ∈ C we have the triangle inequality |z + w | ≤ |z| + | w | and as a consequence the reverse triangle inequality |z - w| ≥ |z| - | w | . We also have |z - w| ≥ | w | - |z|, so we can write |z - w| ≥ '' |z| - | w | '' . Note that |z - w| is the distance from z to w.
Polar and exponential form. Let z ∈ C. We associate z with the point (Re z,Im z) ∈ R2 . If z 0, then at least one of x = Re z and y = Im z are nonzero, and r = |z| > 0. We let θ denote the angle between the positive real axis and the line from 0 to z, measured counterclockwise in radians. Then we have the polar form.
x = Re z = r cos θ, y = Im z = r sin θ, z = r cos θ + irsin θ .
The real number θ is called an argument of z, and we write θ = arg z. Note that (i) arg0 does not exist; (ii) if θ is an argument of z, then so is θ +2kπ for any k ∈ Z; (iii) arg z ± π is an argument of -z.
We can always choose an arg z with -π < arg z ≤ π; this is called the principal argument Arg z. For z on the negative real axis, Arg z = π; however, we have Arg z → -π if z approaches the negative real axis from below.
For x > 0 we have θ = Arg z = arctan . For x < 0, y ≥ 0 we have Arg z = arctan π, and for x < 0, y < 0 we have Arg z = arctan - π . If x = 0 andy > 0, then Arg z = , while for x = 0 andy < 0 we have Arg z = - .
For t ∈ R, we define eit := cos t + isin t. For s, t ∈ R, we can use the usual trigono- metric formulas to see
eis eit = cos s cos t − sin s sint + i(cos s sint + sin s cos t) = ei(s+t) .
It follows that e −iteit = ei0 = 1 so = e −it . We also have eit = e −it . Furthermore, if z, w ∈ C \{0} then
zw = |z|ei arg z | w |ei arg w = |z|| w |ei(arg z+arg w), z = |z|e−i arg z, = e −i arg z
(we multipliy complex numbers by multiplying their absolute values and adding their arguments), so arg z + arg w is an argument of zw, and − arg z of z.
Warning: It is NOT always true that Arg(zw) = Arg z + Arg w.
Question: for which s, t ∈ R is eis = eit? We have eit = 1 if cos t + isin t = 1,which is true if and only if cos t = 1, so if and only if t is an integer multiple of 2π . It follows that eis = eit if and only if ei(s−t) = 1, so when s − t is an integer multiple of 2π .
De Moivre’s theorem: For all t ∈ R and n ∈ Z we have (eit)n = eint .
Roots of unity: If n is a positive integer, then the n solutions z of zn = 1 are ζk = ek2πi/n, k = 0, . . . , n − 1.
In the complex plane, the ζk are equally spaced around the unit circle (forming a regular n-gon). .
If w ∈ C \{0}, we can solve zn = w as follows. Write w = | w |ei Arg w and set
z0 = | w | 1/nei Arg w /n then z0(n) = | w |ei Arg w = w.
Here | w | 1/n is the positive nth root of the positive real number | w | . We call z0 the principal root. For any other solution z of zn = w, we have (z/z0)n = zn /z0(n) = 1 and so
z/z0 is an nth root of unity. All of the solutions of zn = w are thus
z k = z0ζk = | w | 1/nei(Arg w /n+k2π/n), k = 0, . . . , n − 1.
As an example, we solve z3 = −2 + 2i = w. We have w = √8e3πi/4 and z0 = √2e πi/4 . The other roots are z1 = √2e πi/4+2πi/3 = √2eiπ11/12 and z2 = √2e πi/4+4πi/3 = √2e πi19/12 = √2e −πi5/12 .
Quadratic equations: For w1, w2 ∈ C we can solve the quadratic equation z2 + w1z +
w2 = 0 by completing the square. We write z2 + w1z + w2 = (z 2 + w2 − so we just need to solve (z 2 = − w2 by the method above.
As an example, consider z2 + (2 + 2i)z + 6i = 0. We obtain (z + 1 + i)2 = −6i + (1 + i)2 = −4i = 4e−iπ/2 . So the solutions are z + 1 + i = 2e−iπ/4 = √2 (1 − i) so z = √2 − 1 + i(−√2 − 1) and z + 1 + i = 2e−iπ/4+iπ = 2eiπ3/4 = √2 (−1 + i) so z = −(√2 + 1)+ i(√2 − 1).
MATH2007 / G12COF COMPLEX FUNCTIONS: PRACTICE PROBLEMS 2018/19
In the problem classes on Wednesdays, you will solve some of these problems. Other questions are included here for your own practice and revision, or for revision sessions. Solutions will appear on Moodle.
1. BASIC PROPERTIES OF COMPLEX NUMBERS AND SETS IN C
See the handout from the first lecture for the basic properties you need.
1.1. Solve z6 + 2iz3 + 1 = 0. Do this by first solving the quadratic u2 + 2iu + 1 = 0 to get two roots u1, u2, and then solving z3 = u1, z3 = u2 by the method in the handout on complex numbers. Express the roots in exponential (reiθ) form.
1.2. Find all complex numbers z with z5 + z = 0.
1.3. Determine the region given by Re (z) > 0 , |Im (1 + z + z2)| < 3 (hint: write z = x + iy).
1.4. Let θ ∈ R and n ∈ N. Show that
provided cos θ 0 and cos nθ 0. Hint: Use de Moivre’s formula.
1.5. Let a, b ∈ R. By writing
e (a+ib)t = eat (cos bt + isin bt),
verify that
for t real, and hence determine I = e (4+i)t dt.
1.6. Let a, b ∈ R with ab < 1. Show that arctan a + arctan b = arctan - . What happens if ab ≥ 1?
Hint: If x > 0, we have Arg (x + iy) = arctan . Consider the numbers
Arg (1 + ia), Arg (1 + ib) and Arg (1 + ia)(1 + ib). 1.7. Let w, z ∈ C. Show that
Hint: Calculate the real and imaginary parts of both sides.
2. BASIC COMPLEX INTEGRATION
2.1. Let gbe the contour which describes twice counter-clockwise the cir- cle with centre 2i and radius 1. Write down a parametrization of g and
z2 dz, dz.
2.2. Let gbe the piecewise smooth contour which consists of, in order, the line segments from −3 − 3i to 3 − 3i, and from 3 − 3i to 3 + 3i.
2z + dz.
2.3. Let g(t) = exp(it),0 ≤ t ≤ 2π . Determine the contour integral
; (b) x2 + y2 + ixy.
2.4. Let gbe the contour consisting of the straight line from 0 to 2 followed by the semi-circular arc of radius 1, centre 1, counter-clockwise from 2 back to 0.
dz.
2.5. Let gbe the straight line segment from 1 + i to 3 + 2i, and let
I = |z|2 dz, J =
Determine the value of I. Without evaluating J, show that |J| 5 (hint: use the fundamental estimate).
2.6. Determine the integral of along the parabolic arc Im z = (Re z)2 from
0 to 1 + i.
2.7. Determine the integral I of the function ()3 once counter-clockwise around the circle of centre i and radius 2.
2.8. Let σ denote the straight line segment from 0 to i and consider
(i) Show that |J| ≤ 1 by the fundamental estimate (hint: |1 − z2| ≥ 1 on σ; why?) and (ii) compute J.
2.9. For a ∈ C arbitrary and r > 0, let g(t) = a + r exp(it), 0 ≤ t ≤ 2π (a parametrisation of the circle with radius r around a).
Determine dz) and compare it to the area enclosed by g.
OPTIONAL: What happens if g is not a circle, but a triangle or some other shape?
2.10. Let gbe the circle of radius 2 around 0 (once counterclockwise). Let
f (z) = z and determine (i) A = (z) dzI, (ii) B = (z)|dz. Compare
the results. What does the fundamental estimate tell you about A?
3. OPEN SETS AND DOMAINS
3.1. Sketch the following sets, and determine which are open. Of those which are open, which are domains? Justify answers brieflybut clearly:
(i) {z ∈ C : |z| < 1 and |z − 1| < 1} ; (ii) {z ∈ C : |z| < 1 or |z − 4i| < 1} ; (iii) {z ∈ C : Re (z) ≥ 0 and |z + 1| < 2} ; (iv) {z : z = x + iy, x, y ∈ R, |x| > |y|} ; (v) {z ∈ C : Re (z2) > 0} ; (vi) {z ∈ C : |z + i| > |z − i|}; (vii) {z ∈ C : |ez | < 1}; (viii) {z ∈ C : |Argz| < π/2}.
A widely used convention is a “dotted” curve for a part of the bound- ary/frontier not in the set, “full” curve for parts which are in the set.
4. FUNCTIONS, LIMITS AND THE CAUCHY-RIEMANN EQUATIONS
4.1. Define the function u(x, y) by u (0,0) = 0 and, if x, y are not both zero,
Show that u(x, y) tends to 0 as (x, y) tends to (0,0) along any straight line through the origin (hint: consider the lines x = 0 and y = cx ). Next, find a curve tending to (0,0) along which u (x, y) does not tend to 0 (this shows that lim(x,y)→ (0,0) u (x, y) does not exist).
4.2. Use the Cauchy-Riemann equations to determine all points where the function f = u + iv is complex differentiable, when u, v are as follows (x, y real):
(i) u (x, y) = x4 + y2 , v(x, y) = xy2 ; (ii) u (x, y) = exp(x2) cos y , v(x, y) = exp(x2)siny; (iii) u (x, y) = x2 + cos y, v(x, y) = y2; (iv) u (x, y) = x siny, v(x, y) = x2 + y; (v) u (x, y) = x/(x2 + y2 ), v(x, y) = y/(x2 + y2 ).
4.3. Where in C is the function
g(x + iy) = sin(x2 + y2 ) − i2xy (x, y ∈ R)
(i) complex differentiable (ii) analytic?
4.4. Suppose that the function f is complex differentiable at every point in a domain D, and that f (z) is real for every z in D. Explain why f must be constant on D.
4.5. Is there a function f, complex differentiable in all of C, such that the real part off(x + iy), when x, y are real, is: (i) x3 − 3xy2 + 4y ;(ii) ex+y ?
Hint: assume such an f , write f = u + iv , and see what Cauchy-Riemann tells you about v.
4.6. Is there a function f, complex differentiable in all of C, such that the real part off (x + iy), when x, y are real, is u (x, y) = e2y cos x?
4.7. Determine all analytic functions (if any exist) f on C such that Ref (x + iy) = x2 − y2 .
4.8. Determine those z ∈ C at which the following functions are not de- fined:
(i) cos (ii) (iii) z4 exp(exp(1/z2)).
4.9. Let f and gbe analytic on a domain D , such that g(z) = if (z) for all z in D. Prove that f is constant.
4.10. Statement: “The function f (z) = is analytic on the circle |z| = 1 because for |z| = 1 we have = 1/z and g(z) = 1/z is analytic except at 0” . Is this correct?
4.11. Complex analysis is a very geometric subject, and one of the many uses of complex functions is to provide mappings between regions.
(i) The complex numbers w and z are related by
with z 1. Show that Re (w) > 0 if and only if |z| < 1.
(ii) Determine the image of the upper right quadrant V = {z ∈ C : Re (z) > 0,Im z > 0} under the mappings w1 = z2 and w2 = z4 . Hint: what happens to the argument?
(iii) Find a function mapping the disc {z ∈ C : |z| < 1} one-one onto (i.e. a bijection to) the sector {u ∈ C : |arg u| < π/4}.
4.12. Let a ∈ C. The function za is defined on the domain D0 obtained by deleting from C the non-positive real axis, by za = exp(aLog z).
(i) Use the method in the notes to find all z in D0 such that zi = i.
(ii) Determine the limits of zi as z tends to −1 (a) from above (b) from below the negative real axis.
(iii) Is the following statement correct?
“As z → 0 in D0 , we have zi → 0, because 0 to any non-zero power is 0” .
(iv) Taking z = e2π, a = b = i , show that (za)b need not equal zab .
4.13. Determine all z in D0 = C \{x ∈ R : x ≤ 0} which satisfy z5+i = −1. Do this using the method given in the notes, by first determining Log z.
4.14. Let gbe the semi-circular arc from −i to i via 1. Determine
(a) by direct calculation, and (b) by using a suitable antiderivative.
Can the same antiderivative be used to calculate the integral of 1/z along the semi-circular arc σ from −i to i via −1?
4.15. Let gbe the simple piecewise smooth contour consisting of the straight line segment from 0 to 1, followed by that from 1 to 1 + i, followed by that from 1 + i to 2 + i. Determine the integral ) dz.
4.16. Find all solutions of the equations (i) sin z = 0 (ii) cos z = 2 (iii) sin z = i. Hint: Write z = x + iy and determine real and imaginary part of sin z and cos z.