X3 Mathematical Methods 2000/2001
 | Look though the sections indicated. Where you seethe "Course Work" lables like these to the left and right do whatever it tells you. You should, of course, read all the material on these pages. You may be expected to know it!
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These pages provide material for part of the Mathematical Methods course at City University.
It is assumed that various facilities are pressent on the computers being used.
Contents:
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Here we are interested in the Laplace transform defined by
along with the inverse transform
Computers and Laplace Transforms
The following links have instructions and exercises for using computer algebra packages to manipulate Laplace transforms:
Here we use the definition of the Fourier transform
along with the inverse transform
The function f(x) must satisfy certain conditions, like decaying as x tends to + or - infinity.
Exercise 1: Make sure you can find the Fourier transforms of the following (done in lectures)
Warning: you may have to do some contour integration to find the solutions to the last two!
Exercise 2: Here are a couple more to try, these have not been done in lectures.
Exercise 3: Derive does not have inbuilt facilities to do Fourier transforms, while Mathematica can only do them numerically. This means that if you want to do things explicitly on the computer you have do Fourier transforms using Maple. But don't get your expectations too high ...
Exercise 4: If you have any time left try proving that the Fourier transform of an even function is real, and the Fourier transform of an odd function is imaginary.
Exercise: What are the possible branch cuts for the following, giving your reasons for your answers:
You can try and inspect the Riemann surfaces for
using Virtual Reality! The red and blue surfaces represent the differnt branches, the green lines the locations of the branch cuts. To find out how to move around in this virtual world use the help button on the VR browser (the small one with the ? near the bottom right corner). In my opinion these look better if you turn the headlights off.
Exercise: What would the Riemann surface for
look like? (Answer)
The conformal mappings page looks at the images of some grids under complex mappings. Check that you understand what is going on!
The example on the touching circles page has six touching circles between the inner and outer circles, as well as an example where the image circles are not nested. There is also an example where the circles fail to touch all the way around.
Here we will attempt to solve some more realistic problems. Firstly we will look at the heat conduction problem described in the lectures. Then we will look at the flow past an flat plate inclined to the oncoming flow.
Here is some material on using different definitions of inner products on functions to derive orthogonal polynomials. There is also a bit material on finding orthogonal bases and quadratic forms
So you now think you can solve systems of linear equations. Here is a problem to do with finding the price of beer. Note: there are no errors, and there is one sensible solution!
Here are a couple of links to some pages produced ny Douglas N. Arnold with "Graphics for complex analysis" with
I hope that they are not too slow in loading! You could always set up an extra browser and look at something else while you are waiting!
