MA3614

Any slides used in the lectures

2x2 version of slides suitable for the printer

Summary of what was done each week -- brief detail

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  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_2023_2024.

  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 in |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 and 9 of the introductory exercises were done.

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  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_2023_2024.

  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.

  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 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, 6 and 12 from the introduction exercises were done.

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  Week 03: OHPs used, 2x2 version to print.

  A post-record version of the week 3 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  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.
  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: Q7, Q10, Q11 and Q22 from the first exercise sheet were done.

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  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_2023_2024.

  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).
  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 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. With the polar form we more quickly show that the Cauchy Riemann equations are satisfied for Log(z). The derivative of Log(z) is 1/z.

  Exercises session: Q3 (from Dec 2022) 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_2(z) in terms of z only.

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  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_2023_2024.

  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 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 it was indicated that the level curves of u and v are at right angles at points for which f'(z_0) is not zero.

  Exercises session: Q1, Q13(b)(iv) of the exercises on chapter 3 material. The post record also has Q7.

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  Week 06: OHPs used, 2x2 version to print.

  A post-record version of the week 6 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  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 in a case when q(z) has a double pole.
  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: Q11 of the exercises on chapter 3 material.

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  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_2023_2024.

  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 and f_3 done in the previous week. The f_3 function has a double pole as well as 2 simple poles. 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.

  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: Part of Q4 of the exercoses on chapter 3 material. This involved showing that a function is analytic using the Caucht Riemann equations. The function is a polynomial. A technique for expressing in terms of z only.
  Q1 and f_3 of Q2 of the exercises on chapter 4 material. These questions involved rational functions, partial fractions and/or residues.

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  Week 08:

  A reading week when students can spend all their time on their final year project as they get ready for the viva in week 9. No new material is covered in this week with no event on Monday and with Tuesday 15:00--17:00 running as a drop-in sessions in the usual rooms.

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  Week 09: OHPs used, 2x2 version to print.

  A post-record version of the week 09 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: This was the first session for nearly 2 weeks due to the so-called reading week in week 8 for year 3 modules. Partlty as a consequence 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.

  When new material started tan(z) and cot(z) and the residues of the simple poles of these functions was covered. All the residues of cot(z) are +1.

  The principal value of a complex power and the multi-value version were considered next.

  How many values in the multi-value case for different cases of \alpha was discussed. The cases when \alpha is an integer which only has 1 value. The case when alpha=1/2 which has 2 values. The case when \alpha=1/n (n=integer, n is 2 or more ) has n values. For generally when \alpha is an irrational number or a non-real number. The principal value of i^i was one of the examples.

  Expressions for z to the power \alpha for a general z and a general \alpha.

  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: Q3(b) on chapter 3 material about an analytic function created from a harmonic function. Q4, Q16, Q17(b) of the exercises on chapter 4 material on elementary functions of a complex variable. Q4 involves rational functions, partial fractions and residues. Q16 and Q17(b) involve Logarithms and complex powers.

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  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_2023_2024.

  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.
  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 an example involving 1/z^2 and a circle of radius R.
  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: Q6, Q8, Q15, Q19 of the exercises on chapter 4 material. on elementary functions of a complex variable.

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  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_2023_2024.

  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.

  A trig integral converted to loop integral with a rational function by using the substitution z=exp(i \theta).

  Exercises session: Q6 and Q9 of the exercises on chapter 5 material about contour integrals.

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  Week 12: There were no OHPs used in week 12, everything was done on the whiteboards.

  A post-record version of the week 12 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: In all the sessions it was question from the exercise sheet called ``Extra exercises before the class test''. All 16 questions were just about done in the session and all are in the post record version.

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  Week 13: The winter exam week when the MA3614 class test took place.

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  Week 17: The last winter exam week which had exams in other modules. There were no events for MA3614.

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  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_2023_2024.

  Some proofs of the Cauchy theorem were described. Non-examinable proofs of the Cauchy theorem (Cauchy-Goursat theorem) were briefly considered.

  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.

  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 theorame 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.

  Consideration again of the loop integral with any loop and an integrand of 1/(z-z_0). Here z_0 is inside the loop by deforming the loop. The method involved deforming to a circle centered at z_0 where the integral can be 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 17:00--18:00 hour.

  Exercises session: Only some of the trig integrals towards the end of the chapter 5 notes were done in the Mon 17:00 session as mentioned above. There were no questions from the exercise sheets covered.

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  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_2023_2024.

  Content: Examples using the generalised Cauchy integral formula to evaluate loop integrals when there is one isolated singularity inside the loop.

  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. 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: Q10 (May 2022), Q17 (May 2023) of the exercises on chapter 5 material about contour integrals an trig integrals were done.

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  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_2023_2024.

  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. 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.

  A 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).

  Exercises session: Q1 and Q2 of the exercises on chapter 6 material about the Cauchy integral formula.

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  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_2023_2024.

  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).

  Various techniques to derive one series from another, i.e. term-by-term differentiation or term-by-term integration or add or subtracting series.

  Series derived from 1/(1-z). The series for (1+z)^\alpha.

  The Cauchy product expression when multiplying two series. Several examples using this technique and the example of tan(z).
  L'Hopitals rule with multiple zeros of f and g in the limit of f/g.

  Exercises session: Q6a and Q4 of the exercises on chapter 6 material about the Cauchy integral formula. Also Q10b of the exercises on chapter 7 material about series. The series question was about deducing the radius of convergence from the nearest non-analytic points to the centre and by using the Cauchy product technique to get the first few coefficients in the Maclaurin series of a given function.

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  Week 22: OHPs used, 2x2 version to print.

  A post-record version of the week 22 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: Power series. The ratio test and the root test to get the radius of convergence. Examples using the ratio test and also examples using the root test. A formula for the radius which works in all cases.
  Laurent series. Examples indicating techniques to get the Laurent series. A sketch of the proof as to why a function analytic in an annulus has a Laurent series representation. Classifying isolated singularities using the Laurent series. The zeros of non-zero analytic functions are isolated and examples where this can be used.

  Exercises session: Q10(a) and (c), Q11(b) of the exercises on chapter 7 material about series.

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  Week 23: OHPs used, 2x2 version to print.

  A post-record version of the week 23 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: Zeros of non-zero analytic functions are isolated and examples where this can be used to explain why real identities also hold in the complex case.

  Removaeble singularities and classifying isolated singularities using the Laurent series more generally. Defining the residue as the coefficient of (z-z_0)^{-1} in the Laurent series about z_0.

  Integrating a Laurent series.

  A start of the last chapter about residue theory with an overview of loop integral results covered so far.

  The principal value meaning of an integral from -\infty to \infty. Determining real integrals of this type by using a path which is [-R, R] union a half circle. The details involve the residue theory and the ML inequality.

  The examples 1/(x^2+2x+2) and 1/( (x^4+1), There is 1 simple pole in the upper half plane in the first case and there is 2 somple poles in the upper half plane in the second case.

  With rational functions the degree of the denominator needs to be at least 2 more than the numerator for the upper half circle contribution to tend to 0 as R tends to \infty.

  Exercises session: Q13 of the exercises on chapter 7 material about series.

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  Week 24: OHPs used, 2x2 version to print.

  A post-record version of the Mon and Tue and Thu week 24 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: Integrals involving cos(ax) and sin(ax) are obtained by taking the real and imaginary of exp(iax).

  The example involving exp(iax)/(1+x^2). and an example involving (x sin(x))/(1+x^2). The second case needs Jordan's lemma to explain why the upper half circle contribution tends to 0 as R tends to infty.

  Jordan's lemma indicating that the degree of the denominator only needs to be 1 more than the numerator for the contribution due to the half circle to tend to 0 as R tends to \infty. Briefly the reasoning for this was as follows. By parametrising and considering the integral wrt \theta in a bit more detail we have |exp(iaz)|=exp(-ay)=exp(-aR\sin(\theta)). The rhs is 1 when \theta is 0 or \pi but it is small at \theta=\pi/2, and for most \theta, and the area under the curve of this function tends to 0 as R tends to \infty.

  Singularities on the real line and the examples of integrating sin(x)/x and sin^2(x)/x^2 by using suitable functions f(z). It is exp(iz)/z in the first case and it is (1-exp(2iz)/(2z^2) in the second case. These functions have a simple poles` at z=0. The loop used in each case is an indented contour involving [-R, -r], [r, R] with a half circle in the upper half plane of radius r and centre 0 considered clockwise (the indent) and the usual half circle in the upper half plane of radius R and centre 0 considered anti-clockwise. By considering the Laurent series representation of f(z) about z=0 the integral around C_r^+ (considered anti-clockwise) tends to \pi i \Res(f, 0).

  The simple poles of f'/f for which the residues are integers. At a zero of f(z) it is the multiplicity. At a pole of f(z) it is -n where n=order of the pole. Statement of Rouche's theorem about the number of zeros of f+g inside a loop when |g(z)| is less than |f(z)| on the loop. Use of Rouche's theorem to prove the fundamental theorem of algebra. An example of a polynomial of degree 5 to show that all roots are in an annulus by applying Rouche's theorem to both the polynomial and the reverse polynomial.

  Exercises session: Q5 and Q6 of the exercises on chapter 8 material about residue theory.

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  Week 25: OHPs used, 2x2 version to print.

  A post-record version of the week 25 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: It was just questions from the exercise sheet on residue theory in week 25.

  Exercises session: Q8, Q9, Q12 and Q1 of the exercises on chapter 8 material (residue theory). Q1 is a trig integral. Q9 involves an indented contour as the function f(z) has a simple pole on the real line at z=0. Q12 involves a quarter circle as part of the loop. Q8 has a loop which involves an upper half circle as was the case in the exercises done in week 24.

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  Week 31: OHPs used, 2x2 version to print.

  A post-record version of the Mon and Tue week 31 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: The slides indicate briefly the main topics of the module and some aspects were briefly discussed before exercises were done. Some of the exercises had past exam questions or were similar to past exam questions. Q1 involved using the Cauchy Riemann equations to determine whether or not the given functions were analytic. There are also parts of the question involving getting a harmonic conjugate, expressing a polynomial in z-only form. Q1 also had a part about the function \conj{f(\conj{z})} which is analytic when f(z) is analytic. The first part of Q2 involved using an anti-deriv to compute certain contour integrals. In one case care is needed when having an appropriate anti-derivative for 1/z. The last parts of Q2 involved using the generalised Cauchy integral formula. Q3 is about series. To answer the first parts just needs the use of the ratio test to determine the values of z where the given series converges. There is a part where you need to identify the nearest non-analytic point to 0 to determine the radius of convergence of the Maclaurin series. In that part the Cauchy product technique is needed to get the first few coefficients. The last part involves the Laurent series representation valid for large |z| of a given function. Th first part of Q4 involves evaluating a trig integral after first making the substitution z=exp(i\theta). This gives a loop integral involving the unit circle with a rational function as the integrand. The last part of Q4 is a variation on what has been done in the lectures and exercises involving the function 1/(z^4+16). All of Q4 needs the residue theorem and in the last part the ML inequality is needed.

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  Week 32:

  A post-record version of the week 32 sessions are available on panopto and you can get to panopto from Brightspace. The name of the folder on panopto is called post_records_2023_2024.

  Content: The post record has the solutions to questions 5--11 of the revision exercises. Many of these questions were done in the on-campus session.

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Brunel Matlab portal, Online Matlab version.

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