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 1-5) 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,

Exercises involving contour integrals (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.
Chap 6: Cauchy's integral formula, consequences and bounds
Chap 7: Taylor series and Laurent series
Chap 8: Residue theory

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.
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