# Complex Fourier Series & Full Fourier Series

• kingwinner
In summary, the author is attempting to find a proof that the full Fourier series is equivalent to the complex Fourier series, but is having difficulty.
kingwinner

## Homework Statement

Claim: If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series.

This is a claim stated in my textbook, but without any proof. I also searched some other textbooks, but still I have no luck of finding the proof.
I've already spent an hour thinking about how to show that this is true, but still I am not having much progress. Here is what I've got so far:

Full Fourier series is:

where

Complex Fourier series is:

where

And now I am having trouble with this...how can I use the last part to show that if f(x) is REAL-valued, the complex Fourier series can be reduced to the full Fourier series. Can someone please show me how to continue from here? I also don't see how a sum from negative infinity to infinity (for complex Fourier series) can possibly be reduced to a sum from 0 to infinity (for full Fourier series). It seems like I have no hope...

As shown above

## The Attempt at a Solution

As shown above

I am really frustrated now and any help is very much appreciated! :)

#### Attachments

• fourier_series.gif
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Last edited:
An obvious first step is to use the fact that $e^{in\pi x/L}= cos(n\pi x/L)+ i sin(n\pi x/L)$. Multiply it out and use the fact that cos(-x)= cos(x), sin(-x)= sin(x).

HallsofIvy said:
An obvious first step is to use the fact that $e^{in\pi x/L}= cos(n\pi x/L)+ i sin(n\pi x/L)$. Multiply it out and use the fact that cos(-x)= cos(x), sin(-x)= sin(x).
OK, the following is what I got.
(for simplicity I am taking the interval to be from -pi to pi)

Is this a correct proof??

Thanks!

The claim is
"If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series."

But nowhere in the proof have I assumed f(x) is real-valued. Is it absolutely necessary for f(x) to be REAL-valued in order to prove that the full Fourier series is exactly equivalent to the complex Fourier series??

That's because the equality of those sums does not depend upon real or complex numbers. We require that F be real valued in order to have the coefficients real numbers.

HallsofIvy said:
That's because the equality of those sums does not depend upon real or complex numbers. We require that F be real valued in order to have the coefficients real numbers.
But looking at my proof above, I believe that the full Fourier series and the complex Fourier series are equivalent in general, even when f(x) is complex-valued. Right??

## 1. What is a complex Fourier series?

A complex Fourier series is a mathematical representation of a periodic function using a combination of sine and cosine waves with different amplitudes and frequencies.

## 2. What is the difference between a complex Fourier series and a full Fourier series?

A complex Fourier series includes only sine and cosine waves, while a full Fourier series can also include other types of waves, such as square waves or triangular waves.

## 3. How is a complex Fourier series calculated?

A complex Fourier series is calculated by finding the coefficients of the sine and cosine waves that make up the periodic function using integration and trigonometric identities.

## 4. What is the purpose of using a complex Fourier series?

A complex Fourier series allows us to represent complex periodic functions in a simpler form, making it easier to analyze and manipulate them in mathematical equations.

## 5. Can a complex Fourier series be used for non-periodic functions?

No, a complex Fourier series can only be used for functions that are periodic, meaning they repeat themselves over a certain interval. For non-periodic functions, a different type of Fourier series, such as a discrete Fourier transform, may be used.

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