Professor MathTheBeautiful, thank you for showing How to determine the Fourier Series Coefficients in classical Partial Differential Equations. This is an error free video/lecture on TH-cam TV.
Sir, in 11:39 you mention you read a book about Fourier series, do you remember which book it is? I and Thanks for your clear explanation . I study for many days with 3 books. and you just solve me the answer about A_n and b_n in 12 minis !
08:15 Hmm... Doesn't it mean that we could also use sines and cosines of semisums and semidifferences as our basis functions? Then we could decompose any function into even and odd functions :q 09:10 Couldn't we use a substitution `u = cos(5·x)` and integrate `u²` first, and then substitute back for `u`? 11:17 Hmm... but did we though? :q We only showed that we can decompose a function that was already composed from those sines of cosines added together with different weights. But it doesn't show yet that this will work with any other function written in a different way (e.g. `x³ + 2·x² + 4·x + 5`). If we had to be rigorous, we would have to show that *correlating* a function `f(x)` (whatever it is) with sines and cosines will separate out the particular sinusoidal component from that function, or in other words, how much that function resembles the sinusoid of that particular frequency (how often they are "in phase", so to speak, and how often in "opposite phase", and how often they are uncorrelated).
Professor MathTheBeautiful, thank you for showing How to determine the Fourier Series Coefficients in classical Partial Differential Equations. This is an error free video/lecture on TH-cam TV.
brilliant lessons!
Also, the "twice the angle" was great :P .
Thank you, that means a lot!
Sir, in 11:39 you mention you read a book about Fourier series, do you remember which book it is? I
and Thanks for your clear explanation . I study for many days with 3 books. and you just solve me the answer about A_n and b_n in 12 minis !
amazing
08:15 Hmm... Doesn't it mean that we could also use sines and cosines of semisums and semidifferences as our basis functions? Then we could decompose any function into even and odd functions :q
09:10 Couldn't we use a substitution `u = cos(5·x)` and integrate `u²` first, and then substitute back for `u`?
11:17 Hmm... but did we though? :q We only showed that we can decompose a function that was already composed from those sines of cosines added together with different weights. But it doesn't show yet that this will work with any other function written in a different way (e.g. `x³ + 2·x² + 4·x + 5`). If we had to be rigorous, we would have to show that *correlating* a function `f(x)` (whatever it is) with sines and cosines will separate out the particular sinusoidal component from that function, or in other words, how much that function resembles the sinusoid of that particular frequency (how often they are "in phase", so to speak, and how often in "opposite phase", and how often they are uncorrelated).