MA3614

Complex variable methods and applications (2024/5)

Lecture notes

Chapter plan

OHPs in Lectures

Exercises sheets and solutions

A summary of what was done

Miscellaneous

The Brightspace pages

This is the 3rd year that Brunel University has been using the virtual learning environment (VLE) called Brightspace.

Click here for Brightspace,
Click here for MA3614 pages on Brightspace,
Click here for post record panopto MA3614 pages,
Click here for handwriting in the post record panopto MA3614 videos.

In all of the Brightspace links the Brunel login process is likely to be needed, if not already done.

Brightspace will have a link to these web pages and most of the material for the module can be directly accessed from here. A brief text summary (just a few lines) of what has been covered in the previous sessions will be put in the ``summary of what was done part'' which is later in this file.

Welcome to the module slides

If you are browsing and wish to see welcome to the module type slides then Click here. These slides are also identified as week 00 in the table below.

Main lecture notes

A PDF of main lecture notes so far (chapters 1-7) is available if you click here.

An article about complex variables from SIAM News in 2012. click here.

Exercises sheets (and solutions when ready)

Exercises on the introductory material (mainly chapter 1 related) (without answers) click here, (with all the solutions) click here,

Exercises involving analytic functions and harmonic functions etc (chapter 3 material) (without answers) click here, (with all the solutions) click here,

Exercises involving elementary functions (chapter 4 material) (without answers) click here, (with all the solutions) click here,

Extra exercises before the class test for the week 15 revision sessions (without answers) click here, (with all the solutions) click here,

Exercises involving contour integrals (chapter 5 material) (without answers) click here, (with all the solutions) click here,

Exercises involving the Cauchy integral formula (chapter 6 material) (without answers) click here, (with solution to Qu 1,5) click here,

Exercises involving series. (without answers) click here,

Plan of the chapters

Chap 1: An introduction to the module and revision of previous study of complex numbers.
This was covered in week 1 and part of week 2.
Chap 2: Functions of a complex variable -- the domain and continuity.
This was covered quickly in week 2. There are no exercise sheets on chapter 2 material.
Chap 3: The complex derivative and analytic functions.
This started in week 3 and completed at the end of week 5.
Chap 4: The elementary functions of a complex variable.
This started in week 7 and completed at the end of week 9.
Chap 5: Contour integrals, the Cauchy integral theorem and examples involving trigonometric integrals.
This started in week 10 and will most likely finish after about the first teaching week of semester 2, i.e. in about week 18.
Chap 6: Cauchy's integral formula, consequences and bounds. This started in week 18 and finished in week 19.
Chap 7: Taylor series and Laurent series This started in week 20 and may finish in about week 22.
Chap 8: Residue theory This is likely to start in about week 24 and complete in week 26.

Week 01: OHPs used, 2x2 version to print.
A post-record version of the week 1 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: An introduction to the topics in the module and an indication as to the order in which things will be covered and in particular comments about chapter 3--4 at the start of the session.
The start of chapter 1 material. Revision of complex numbers, the cartesian and polar forms. arg z and the principal argument Arg z. The complex conjugate. Multiplication and division in polar form.
The exp(i \theta) notation and justification by considering the series.
The n roots of unity. The n roots of any number as equally spaced points on a circle. The triangle inequality.
The convergence of a sequence of complex numbers and some comments about series which is a topic in chapter 7. 1/(1-z): The geometric series when |z| is less than 1 and a Laurent series when |z| is greater than 1.
There was no exercises hour but in the latter part of the last hour Qu 3 of the introductory exercises was done.
Week 02: OHPs used, 2x2 version to print.
A post-record version of the week 2 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: The presentation had many more slides than is usually the case. There were examples of what the complex case helps to explain such where a series converges 1/(1+z^2) and that an infinitely differentiable function in the real sense can be unbounded in the complex sense, i.e. the function involving exp(-1/(x^2)).
The gain in interest in complex is usually considered to be associated with work to find the roots of cubics. There were comments about Cardano's method for finding roots of a cubic where non-real computations can be needed to find real roots.
The fundamental theorem of algebra about polynomials having roots in the complex plane and the roots of real polynomials occur in complex conjugate pairs was mentioned. Some of the steps in showing that p(\conj{z})=\conj(p(z)) where the coefficients are real were given
The start of chapter 2 about functions of a complex variable. The definition of many terms: an open disk, the unit disk, a neighbourhood, an interior point, a boundary point, the boundary, a polygonal path, connected, a domain and a region. A domain being an open connected set. There was mention of a simply-connected domain being a domain without holes. Bounded and unbounded domains and a number of examples of sets which are domains or are not domains was given.
Graphically representing w=f(z) by showing a uniform grid in the z-plane and the image grid in the w-plane. Explaining the polar grid of z^2 near the point 1 by a finite Taylor expansion about 1.
An example of a ``bilinear map'' f(z) which maps the unit disk onto itself. It was proved that when |z|=1 we have |f(z)|=1.
The definition of a limit in the complex sense and the definition of continuity in the complex sense with mention that this is stronger requirement than in the corresponding real case as there are more possibilities of how z tends to z_0.
Results about combining continuous functions, e.g. f(z)+g(z), f(z)g(z) and f(z)/g(z). f(z)=u(x,y)+iv(x,y) is continuous at z_0=x_0+iy_0 if and only if u and v are continuous at (x_0, y_0).
Some examples of continuous functions and some examples of discontinuous functions were given.
Exercises session: Qu 4 and 7 from the introduction exercises were done.
Week 03: OHPs used, 2x2 version to print.
A post-record version of the week 2 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: The start of chapter 3 starting with the definition of the complex derivative, and analytic at a point. The derivative of the monomials using the definition. The derivative of 1/z using the definition.
Combining analytic functions, e.g. adding, scalar multiplication, product rule, quotient rule and chain rule.
The derivative of 1/z^n when n is a natural number.
L'Hopital's rule statement and one example.
The complex differentiable property for f=u+iv implies the Cauchy Riemann equations linking the partial derivatives of u and v.
Using the Cauchy Riemann equations to show that the conjugate of z is not analytic.
Exercises session: Q5, Q9, Q10 and Q11 from the first exercise sheet were done.
Week 04: OHPs used, 2x2 version to print.
A post-record version of the week 4 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: Comments again of how the complex differentiable property for f=u+iv implies the Cauchy Riemann equations. Showing the reverse that Cauchy Riemann equations implies complex differentiability using a Taylor expansion of g(t)=u(x_0+t*h_1, y_0+t*h_2) to get the first terms in g(1)-g(0). With the first terms of u(x_0+h_1, y_0+h_2)-u(x_0, y_0) and v(x_0+h_1, y_0+h_2)-v(x_0, y_0) we get an expression for f(z_0+h)-f(z_0) which has a factor of h when the Cauchy Riemann equations hold.
Examples of (1+2i)z^2 and exp(z) being analytic using the CR equations. The derivative of exp(z) is exp(z).
Example of g(z)=f(\conj{z}) not being analytic when f(z) is analytic and not constant.
Example of Log(z) using cartesian form to show that the Cauchy Riemann equations hold. The Cauchy Riemann equations in polar form. The derivative of Log(z) is 1/z.
Exercises session: Q3 (from Dec 2023) and Q2 were done from the exercise sheet on analytic functions etc. More than one way was given to show how to express the analytic function f_3(z) in terms of z only.
Week 05: OHPs used, 2x2 version to print.
A post-record version of the week 5 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: When f=u+iv is analyic it is shown that u and v are harmonic functions. Examples were done of starting with a harmonic function u to construct a harmonic conjugate v so that u+iv is analytic. In the examples some cases with u such that u+iv is a polynomial were considered. There was also an example involving a trig function and a hyperbolic function.
By using the finite Taylor series representation of a polynomial the analytic function was expressed in terms of z alone in the polynomial cases.
It was shown that if f=u(x, y)+iv(x, y) is written as g(z, \conj{z} ) then the partial derivative of g wrt \conj{z} is 0 if and only if the function is analytic. It is in this sense that an analytic function cannot depend on \conj{z}.
On the slides that when w=f(z) is considered as a mapping angles are preserved when f(z) is analytic and we are at points where f'(z) is not zero.

Exercises session: Q5 and Q12(a)(iv) of the exercises on chapter 3 material. Both of these where concerned with whether or not a function of x and y is analytic and writing in z alone form was also required in the analytic case.
Week 07: OHPs used, 2x2 version to print.
A post-record version of the week 7 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: Recap of key points in chapter 3 about complex differentiability and the analytic property for a function w=f(z)=u(x, y)+iv(x, y).
The start of chapter 4 about the elementary functions of a complex variable. Representations of polynomials by a finite Maclaurin series, a finite Taylor series about any point and in terms of its factors determined by the roots of the polynomial. Simple roots and multiple roots of polynomials with a polynomial p(z) having a root or multiplicity r at z_0 if and only if p(z_0)=0 and the 1st, 2nd, ..., (r-1)th derivatives of p(z) are also zero at z_0 with the rth derivative being non-zero.
Rational functions R(z)=p(z)/q(z), p(z) and q(z) being polynomials, and the representation using partial fractions or a representation which involves a polynomial part and partial fraction part. The multiplicity of the zeros of q(z) are needed to give the representation. Examples with deg(p) less than deg(q) and with deg(p) greater than or equal to deg(q) and in cases when q(z) has simple poles and cases when q(z) has a double pole and a triple pole. The triple pole example was obtained from the previous double pole case which meant that not too much work was needed to get one of the residues.
In many cases a technique to get the coefficients using limits is given with L'Hopital's rule used to determine the limit value.
The definition of the residue as the coefficient of a simple pole term which appears in the representations.
Exercises session: Q13 of the exercises on chapter 3 material.
Week 08: OHPs used, 2x2 version to print.
A post-record version of the week 8 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: Reminders about polynomials, rational functions, pole singularities and partial fraction representations of at least part of the expression. Comments about how to get the residue at a pole.
Reminders of the techniques to get the representations of f_1, f_2, f_3 and f_4 done in the previous week. The f_3 function has a double pole as well as 2 simple poles. f_4(z) is in terms of f_3(z) and as f_3(z) had been done part of the work to get f_4(z) was already done.
The special case of one multiple pole and the use of a Taylor expansion of the numerator about the position of the pole to get the representation.
A brief indication as to why the representation using partial fractions is always possible for a rational function R(z)=p(z)/q(z) when deg(p) is less than deg(q) when there are poles of any order. This was just done using the slides.
The exponential function of z. A proof of a few results in the complex case, i.e. exp(-z)=1/exp(z), exp(z_1+z_2)=exp(z_1)exp(z_2) which all correspond to the real case. exp(z)=1 if and only if z is an integer multiple of 2\pi i. w=exp(z) is one-to-one on the strip with y in (-\pi, \pi] with the principal valued logarithm z=Log(w) being the inverse.
The definitions of cos, sin, cosh and sinh of a complex variable. The periods of exp, cos, sin, cosh and sinh. Verifying the expressions for the derivatives (corresponding to the real case).
From the definition showing the trig addition formula for sin(z_1+z_2) holds for all z_1 and z_2.
Exercises session:
Q1 and Q2 of the exercises on chapter 4 material. These questions involved rational functions, partial fractions and/or residues.
Week 09: OHPs used, 2x2 version to print.
A post-record version of the week 9 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: The lecture started with a review of a few of the main points from chapter 3 and what has been done in chapter 4 so far.
The new material started with tan(z) and cot(z) and the residues of the simple poles of these functions was covered. These function have period of \pi and have simple poles. The poles of cot(z) are at k\pi and those of tan(z) are at\pi/2+k\pi, k being an integer. All the residues of cot(z) are +1.
The principal value of a complex power and the multi-value version were considered next.
Expressions for z to the power \alpha for a general z and a general \alpha.
How many values in the multi-value case for different cases of \alpha was discussed. The cases when \alpha is an integer generates only has 1 value. The case when alpha=1/2 generates 2 value. The case when \alpha=1/n (n=integer, n is 2 or more ) gives n values. Thew multi-value z^{1/n} means all values of \zeta such that \zeta^n=z. More generally when \alpha is an irrational number or a non-real number the multi-value case is infinitely many values. The principal value of i^i and the corresponding multi-value case was one of the examples.
Qu 18 of the exercise sheet was done in the lecture hour.
The mapping properties of some standard functions, exp(z) and sin(z). In the case of sin(z) we have that it maps the strip |x|\le \pi/2, y\ge 0 onto the upper half plane.
Exercises session: These were all from the exercise sheet on chapter 4 material. Q3 (about partial fractions and residues), Q8 (about logs and complex powers) and Q10 (about properties of tan()) were the questions covered.
Week 10: OHPs used, 2x2 version to print.
A post-record version of the week 10 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: The slides started with comments about where some of the chapter 4 material on elementary functions of a complex variable will be seen again.
Chapter 5 about contour integrals started with review of what the integral of a function of a real variable means as the area under the curve with the idea of an appropriate limit of a sum.
The extension to the case of the integral on (a, b) when f=u+iv with an example of integrating exp(kt), k=p+iq.
The definition of a directed smooth arc using a parametrization z(t) to define the arc. A formula for the length of an arc. The integral of f(z) on an arc as a limit of a sum and also as an integral on (a, b) of an integrand in terms of f(z(t))z'(t).
The ML inequality to bound the contour integral with M being a bound for |f(z)| of the contour and L as the length of the contour.
Examples of evaluating contour integrals involving circles, straight line segments when the function is an integer power of z.
The independence of a contour integral on the path when f has an anti-derivative F. Only the case of a single arc was described at this point.
Exercises session: Q9, Q16, Q13 of the exercises on chapter 4 material. on elementary functions of a complex variable.
Week 11: OHPs used, 2x2 version to print.
A post-record version of the week 11 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: The independence of a contour integral on the path when f has an anti-derivative F. This extends the case of just a single arc done in the previous week.
The cases of z^n when n is an integer different from -1 considered using the anti-derivative (z^(n+1))/(n+1). The case when n=-1 by taking a path starting from one side of the branch cut of Log(z) and finishing at the corresponding point on the other side of the branch cut to deduce the value of 2\pi i.
Discussion of the equivalence of the following 3 statements:
All loop integrals of f are 0.
The value of the contour integral of f only depends on the end points.
There exists an anti-derivative.
The integral of a rational function explained by the use of partial fractions for part of the relation. This gives the residue theorem in the case of a rational function. This is a result which is geralised later in the module to any function which is analytic except for isolated singularities.
A trig integral converted to loop integral with a rational function by using the substitution z=exp(i \theta).
Exercises session: Q11 and Q6 of the exercises on chapter 5 material about contour integrals.
Week 15: There were no OHPs used in week 15, everything was done on the whiteboards. The questions done were from the following extra exercise sheet click here.
A post-record version of the week 11 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: In all the sessions it was questions from the exercise sheet indicated above which has 16 questions. At the Monday session it was questions 1, 2, 3, 8 and 5 (in this order in the panopto recording). At the Tuesday session it was questions 6, 10, 12, 13, 11 (in this order in the panopto recording for Tue 15:00) and 14, 15, 7 and 4 (in this order in the panopto recording for Tue 16:00). Question 4 is just beyond what was done on campus. The solution to questions 9 and 16 are just available in the printed version of the solutions. The printed solution to all 16 questions is available click here.
Week 18: OHPs used, 2x2 version to print.
A post-record version of the week 18 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
There were more slides this week compared with other weeks as they included a recap of some material up to the end of week 11 (the last previous week with new material) and an overview of some proofs which are not examinable. The reasons proofs are often shown is that it sometimes helps to understand why a given result is true.
The main proofs briefly outlined were of Cauchy theorem which is often also referred to as the Cauchy-Goursat theorem.
In one proof this involved writing the integrand f(z(t))z'(t) of a contour integral in cartesian form to indicate that both parts (i.e. the real part and the imag part) are line integrals in 2D space. Green's theorem could be applied to both line integral. The Cauchy Riemann equations being satisfied by analytic functions gave the result that loop integrals of analytic functions are 0 when f(z) is analytic on the loop and at all points inside the loop.
In the second proof this involved continuously deforming a contour to a point and the loop integral having the same value for each loop. It was assumed here that everything was sufficiently smooth that any partial derivatives needed could always be done.
The final proof mentioned involved showing why the result is true for a triangle of any size within the domain where f(z) is analytic. The technique involved repeatedly dividing a triangle into 4 similar triangles until we reached a stage where the triangle being considered is in a neighbourhood of an analytic point.
With Cauchy's theorem available a corollary of this is that it implies the existence of an anti-derivative F such that F'=f when we have a simply connected domain.
A further corollary of Cauchy's theorem about deforming the loop where f(z) is analytic without changing the value of the integral. This proof was mainly indicated using the slides as this helped with the diagrams. The details of this proof are also not examinable.
Consideration again of the loop integral with any loop and an integrand of 1/(z-z_0). with z_0 being inside the loop. The method involved deforming to a circle centered at z_0 where the integral can be evaluated directly. There was no need to mention the logarithm in this version.
The residue theorem result in the case of rational functions was mentioned again. This was used to evaluate trig. integrals. The general idea was given and two examples of trig. integrals were done in the Monday 16:00--17:00 hour.
In the Tuesday lecture it was the start of the chapter 6. This involved a proof of the Cauchy integral formula (CIF) which gives an integral represntation of f(z). In the lecture a similar integral represntation of f'(z) was also proved. Partially differentiating the CIF representation for f(z) does give the representation for f'(z).
Week 19: OHPs used, 2x2 version to print.
A post-record version of the week 19 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: It was first shown how partially differentiating with respect to z the integrand in the Cauchy integral formula representation leads to the generalised Cauchy integral formula.
Two examples using the generalised Cauchy integral formula to evaluate loop integrals when there is one isolated singularity inside the loop.
Mention that we can define analytic functions by loop integrals.
A harmonic function u has a harmonic conjugate v such that u+iv is analytic is guaranteed when the domain is simply connected. This was shown by first showing that if \phi is harmonic then g=\phi_x-i*\phi_y is analytic. It is guaranteed an anti-derivative throughout the domain D when D is simply connected.
The example of ln(r) being harmonic in 2D space with 0 removed, a doubly-connected domain, with no harmonic conjugate which works for this domain. Log(z)=ln(r)+i Arg(z) is however analytic in the simply-connected domain obtained when the negative real axis is removed.
The generalised Cauchy integral formula when the loop is a circle and bounds on the nth derivative at a point. Liouville's theorem that the only bounded entire functions are constants. Outline, using the projector, of the fundamental theorem of algebra that a polynomial of degree at least 1 must have a zero in the complex plane.
Statement of the maximum modulus theorem but no proof was done.
Exercises session: The Monday 16:00 hour in semester 2. In this session it was Qu 10 and Qu 16 of the exercise sheet on chapter 5 material. Qu 10 involved contour integrals involving 2z+z^2 and 1/z In each case the question wanted the integrals on each of 3 parts of a loop. The value when a loop is considered gave a way to check the consistency of the values and/or a way to determine one of the values when the other two had already been found. Qu 16 was a trig integral with the final part to show them sin(\theta) is replaced by \sin(m\theta), m an integer, the value does not change.
Week 20: OHPs used, 2x2 version to print.
A post-record version of the week 20 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: The start of chapter 7 about series. A quick run through of definitions and results about sequences and series which are the same in the complex case as they are in the real case. This is because all these definitions and results just use the absolute value. A new result about series of analytic functions which converge uniformly have a limit which is also analytic. This used contour integrals and Morera's theorem done in chapter 6 to deduce that the limit is an analytic function.
Most of the proof of the Taylor series representation in a disk with centre z_0 and radius R when f(z) is analytic in the disk. Comment that the largest R is such that there is a non-analytic point on the circle |z-z_0|=R when f(z) is not entire.
The example of both tan(z). to determine R=\pi/2 by observing the isolated singularity at \pm \pi/2 for tan(z). tanh(z) similarly has poles at i\pm \pi/2.
Other examples done were 1/(1+exp(z)) has radius of \pi and the functions f_1(z), f_2(z) and f_3(z) from the May 2024 exam (these are on the exercise sheet about series).
Exercises session: Q1 of the exercises on chapter 6 material about the Cauchy integral formula. and Q9 of the chapter 5 material involving contour integrals.
Week 21: OHPs used, 2x2 version to print.
A post-record version of the week 21 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2024_2025.
Content: Examples to determine the radius of convergence R by observing where the function has singularities. The circle of convergence was also mentioned.
Some comments about Maclaurin series.
Real valued functions have real coefficients.
Expressing the coefficients in the Maclaurin series as integrals involving a circle about 0. Explanation as to why even functions only have even powers and odd functions only have odd powers.
The analytic function \conj{f(\conj{z})} and its Maclaurin series.
Some standard standard Maclaurin series you should know, exp(z), exp(-z), cos(z), sin(z), cosh(z), sinh(z). These were deduced either from directly from the series or from one of the other series once that had been obtained techniques such as term-by-term differentiation or term-by-term integration or add or subtracting series.
The series for (1+z)^\alpha which usually has a non-analytic point at -1.
Series derived from 1/(1-z) and the series for Log(1-z) and 1/(1-z)^2 obtained by term-by-term integration or differentiation. The Koebe function z/( (1-z)^2 ).
The Cauchy product expression when multiplying two series. Several examples using this technique including the example of tan(z) obtained after first writing tan(z)*cos(z)=sin(z). In each case just the first few terms are obtained.
L'Hopitals rule with multiple zeros of f and g in the limit of f/g. This is explained using the Taylor seriesof each function about a zero.
Analytic functions which are not the zero function have isolated zeros.
Exercises session: Q5a and Q5b of the exercises on chapter 6 material about the Cauchy integral formula. These involved just evaluating certain loop integrals and representations of finite difference approximations of f'(0).
Also Q11b of the exercises on chapter 7 material about series involving the first few terms of 3/(2+exp(z)).
Brief mention of Qu 7 on chapter 6 material which related the 2-point Gauss Legendre quadrature rule on (-1, 1). It involved representations of f(1)-f(-1), of the approximation and of the difference between the two.
Week 22: OHPs used, 2x2 version to print.
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Week 24: OHPs used, 2x2 version to print.
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Week 25: OHPs used, 2x2 version to print.
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Week 26: OHPs used, 2x2 version to print.
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Week 27: OHPs used, 2x2 version to print.
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