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Fourier Analysis 1: Definition of the Fourier Series
Uploaded by: donylee
Video Description:
We begin our study on the work of Joseph Fourier (1768-1830) with the definition of the Fourier Series - a way of expressing functions as infinite sums or integrals or trigonometry functions.
Please check out www.gaussianmath.com for a deeper look into this or other mathematics topics.
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Physics needs Fourier Analysis too! As for me, I ALWAYS need a beer once in a while.
I did look it up in a Real Analysis book. Interchanging of integrals with summation is a generalization of something called Tonelli's theorem. Conditions are that either the function is a positive function or if the integral of the sum (or vice versa) is finite.
bmxtra211, you are indeed correct.
The 1/2 is brought outside the calculation of a0.
Also, I find your pace to be just right and your videos to be very informative and inspiring. An excellent resource; keep doing what you're doing.
Just go slower please!
Thank you for sharing your knowledge.
but at one point it seems, that you have deleted too many terms at once:
When it came to the part, that the Integral over the summation with the coefficients a_{n} collapses except to k, i can clearly see, that the Term "a_{n}*Pi" is left there.
But i see no reason, why you deleted the part with the coefficients b_{n}, since, for the same reasons, there should be left "b_{n}*Pi", and not zero, as it seems.
Please explain this lack.
Regards,
Ismail
I see now, that the integral with the integrand "cos(x)sin(y), x=-Pi..Pi" is zero for every real number, and not just only for integers...
Problem solved.
Thanks sir! :-)