What is the process for determining Fourier coefficients?

[member 428835]

hey pf!

can someone explain to me what to do if presented with an equation like this: $$\sum_{i=1}^{n}A_i=i$$
is this identical to stating $A_i=i$? either way, can you please explain.

thanks!

josh

 

[Mark44]

hey pf!

can someone explain to me what to do if presented with an equation like this: $$\sum_{i=1}^{n}A_i=i$$
is this identical to stating $A_i=i$? either way, can you please explain.

thanks!

josh

It doesn't make much sense to me. On the left side, i is an index variable that takes on the values 1, 2, 3, ..., n, so I have no idea what it means on the right side.

Where did you see this? If it's from a textbook, can you post a picture?

 

[AlephZero]

Just taking it at face value, it means
$A_1 + A_2 + \cdots + A_n = i$

The $i$ in the sum is a "bound variable" or "dummy variable". You could replace it by anything else (except $n$) without changing the meaning. The $i$ on the right hand side means $i$.

But using $i$ twice in one equation like that is horrible, as Mark44 said. It would have been more literate to write something like
$$\sum_{k=1}^n A_k = i$$

 

[member 428835]

yea, i was trying to generalize for determining the Fourier coefficients. i have a link here on pf: https://www.physicsforums.com/showthread.php?t=733877
my question there has been answered, but if you're curious to this type of problem, there's the link.

 

[CellCoree]

The process for determining Fourier coefficients involves breaking down a periodic function into a series of sine and cosine functions. This is done by using the Fourier series formula, which involves integrating the function over one period and multiplying it by the sine or cosine function with the appropriate frequency. This process is repeated for each frequency, resulting in a series of coefficients that represent the amplitude of each sine and cosine function.

In regards to the equation \sum_{i=1}^{n}A_i=i, this is not identical to stating A_i=i. The summation symbol (\sum) indicates that the values of A_i are being added together, starting at i=1 and ending at i=n. This equation represents a relationship between the coefficients (A_i) and the index (i). To solve for the coefficients, you would need more information about the function or the values of i and n. I hope this helps clarify the process for determining Fourier coefficients.