Fourier Series Transform Proof Help

[evotunedscc]

Can someone fill in the blank between these two steps? I can't find Fourier series proof anywhere and my professor just left it out.

(1) y(t+nT)=y(t)

(2) y(t)=$$A_{0}$$ + $$\Sigma^{\infty}_{n=1}$$[$$A_{n}$$cos(n$$\omega$$t) + $$B_{n}$$sin(n$$\omega$$t)]

(The omega is going crazy on me... it's not supposed to be superscripted, just multiplied by n and t)

 

[HallsofIvy]

Can someone fill in the blank between these two steps? I can't find Fourier series proof anywhere and my professor just left it out.

(1) y(t+nT)=y(t)

(2) y(t)=$$A_{0}$$ + $$\Sigma^{\infty}_{n=1}$$[$$A_{n}$$cos(n$$\omega$$t) + $$B_{n}$$sin(n$$\omega$$t)]

(The omega is going crazy on me... it's not supposed to be superscripted, just multiplied by n and t)

What do you mean by "steps between them"? The first just says y is periodic with period T and the second is the general expression of a Fourier series of a function periodic with period $2\pi/\omega$- there is no mention of "T".

As for the LaTex, I would recommend putting the entire thing in [ t e x] not just individual parts:

$$y(t)=A_{0}+ \Sigma^{\infty}_{n=1}[A_{n}cos(n\omega t) + B_{n}sin(n\omega t)]$$

It looks better and is easier to type!

 

[dhris]

I would say that a general "Fourier expansion" is actually an integral. What (1) implies is that the modes are discrete and thus the integral becomes a sum, and therefore $\omega=2 \pi/T$, as Halls mentioned. Maybe this is the missing step you mean?

 

[Nima-k2]

have u find the gap between those two statements[evotunedscc]?

 

[blue_raver22]

I understand the importance of having a complete and accurate proof for any mathematical concept. In this case, the missing step between (1) and (2) in the Fourier Series Transform proof is the application of the Euler's formula. This formula states that e^ix = cos(x) + isin(x), where i is the imaginary unit.

Using this formula, we can rewrite the trigonometric functions in (2) as complex exponential functions. This leads to the final form of the Fourier series:

y(t) = A0 + Σ∞n=1 [An e^inωt + Bn e^-inωt]

I hope this helps in completing the proof and understanding the concept better. Additionally, there are many resources available online and in textbooks that provide a detailed explanation and proof of the Fourier series transform. I encourage you to explore these resources for a more comprehensive understanding.