Student Lectures - All lectures - Integral Transforms Lecture 5: The Laplace Transform & Differential Equations. 2nd Year Lecture.

This short course from Sam Howison, all 9 lectures of which we are making available (this is lecture 5), introduces two vital ideas. First, we look at distributions (or generalised functions) and in particular the mathematical representation of a 'point mass' as the Dirac delta function. Then, we see how to represent ordinary functions as weighted integrals of complex exponentials (the Fourier Transform, generalising Fourier series) or real exponentials (the Laplace Transform). The two parts are joined up, for example in the extraordinary result that the Fourier transform of 1 is a delta function. Along the way, we see how to use these techniques, individually or in combination, to solve a range of problems elegantly and economically. Deep and far-reaching, these ideas take linear mathematics to the next level; you will return to them again and again throughout your mathematical life. You can find the full Integral Transforms course here: https://www.youtube.com/playlist?list=PL4d5ZtfQonW0z1ZnUap34658VmfWgpnos You can watch many other student lectures via our main Student Lectures playlist (also check out specific student lectures playlists): https://www.youtube.com/playlist?list=PL4d5ZtfQonW0A4VHeiY0gSkX1QEraaacE All first and second year lectures are followed by tutorials where students meet their tutor to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.