how to get the Fourier series coefficients (fourier series engineering mathematics)

Learn how to derive the Fourier series coefficients formulas. Remember, a Fourier series is a series representation of a function with sin(nx) and cos(nx) as its building blocks. Meanwhile, a Taylor series is a series representation of a function with x^n as its building blocks. These are two must-know series in your calculus and engineering math classes.

@blackpenredpen

[arequina]

Every time I watch one of your videos, my love for mathematics just keeps increasing. Fourier series was never explained like this in any of my classes. We are just told to accept it.

[DasIllu]

Points at Fourier and says it's just a name.
Engineers feel a cold shiver runing down their spines without knowing why.

[davidkwon]

Thank you.
Finally I understand what the Fourier is all about.
Comparing it with Taylor is awesome.
You are a great teacher.

[elenaclaramaria]

I love you, you’re such a great and humble person man. All the best from Italy, watching you to prepare my Calculus Exam!

[andrewhaar]

I learned about these recently in my partial differential equations class and I think I can shed some light on why you would multiply by cos(nx) or sin(nx). When Fourier series come up in a PDE the sin and cos terms are eigenfunctions after separating variables. There is a theorem in PDE about Sturm-Liouville differential equations that says that if the DE for the eigenfunctions is in Sturm-Liouville form then the eigenfunctions are orthogonal to each other with a specific weight function, which comes from the form of the DE (and form a complete set). Knowing about orthogonality of functions it would seem only natural to multiply by orthogonal functions. It is just like in linear algebra when you have an orthogonal set you can easily calculate each coefficient in a linear combination using the fact that the dot product is 0 when vectors are orthogonal. One other thing I think is interesting to note is that the coefficient term for a_0 is also the average value of f(x) on the interval. Something interesting and fun to consider is why it would be the average. There are good intuitive reasons...

[connoratkinson]

You're a blessing to calc students everywhere thank you so much

[calvinjackson]

Great presentation.
Have a better understanding of Fourier series now. How this man came up with the series and transform is beyond me.

[rybaplcaki]

please Fourier transform derivation, complex Fourier or other Fourierr-ish stuff

[boogychan]

You are so good Mr. Blackpenredpen . Never learned Fourier series in this way. The way you showed the derivation of the formulas made much sense. Thanks a lot.🥺👍

[aaronwong]

i like how each time i don't understand a topic and i saw your videos i heave a sigh of

[Gold]

I figured out that part about m being the same as n! The integrands can be written as (cos((n-m)x)+/-cos((n+m)x)/2, and since we've shown twice that the integral from -pi to pi of cos(kx) is zero when k is nonzero, the only thing that matters is when n-m=0, so we can integrate 1/2 over (-pi, pi).
I know you said you're making a follow-up, I'm just really proud of myself for some reason :)

[muazzamhmaiyali]

This happened to be the first time for me to understand this furrier series. Thank you abundantly, sir.

[whatidoknow]

Finally thank you sooo much!! After being pushed around all over Youtube and different materials, I actually found something that relates to what I'm doing and it's then broken down and explained gradually in a way our lecturer didn't bother to do. Thank you soo much. I finally understand have a full grasp of what I'm doing. While watching the video, I was also proving the cases so I can defend everything you've taught. Thank you.

[emirbakunic]

Great videos! You're making student lives so much easier. Best math teacher ever! All the best from Bosnia!

[kristianfella-glanville]

Great video. You explained it in a really nice (completely perfect I'd say) way and I enjoyed filling in the gaps with the double angle formulas and stuff.

[carlmcgrath]

Love this guy's channel. never stop holding that mic and saving my degree. big love from England

[TheCeava]

wow ! you are amazing ! 2 weeks lecture of my prof isnt even comparable to your video ! Thanks so much for this video !! helped me A LOT. also your attitude is amazing :) keep going

[evazhang]

IT IS SO SATISFYING. THANK YOU!!! I really appreciate you list the first two lines shows the taylor and fourier and proceed everything afterwards. I was always confused that how does the summation of cos and sin come from at the first place. MATH IS BEAUTIFUL! Would you please explain how to get 0 and pi when n not equal and equal to m? orthogonality ? do you have videos on that? Thank you so much!

[albert]

Bro you look so happy teaching math, i wish professors and teachers would also have the same passion and love for teaching and solving math problems

Your energy altered my mood too, i was so stressed
Thank you so much for everything

With Love from syria❤

[qracy-kun]

this guy is literally became my math teacher
appreciate your teaching