Complex exponential and sine-cosine Fourier series

In summary, the sine-cosine (SC) Fourier series and the complex exponential (CE) Fourier series are related by the coefficients ##A_j## and ##B_j##, which can be expressed in terms of the CE coefficients ##C_n## using Euler's formula. This relationship can be proven by expanding the CE series and applying the evenness and oddness of cosine and sine.
  • #1
sikrut
49
1
The sine-cosine (SC) Fourier series: $$f(x) = \frac{A_0}{2} + \sum_{j=1}^{+\infty} A_j cos(jx) + \sum_{j=1}^{+\infty} B_jsin(jx) $$

This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$

and vice versa. Here we define, for convenience: $$ B_n := 0 $$

(a) Prove that the SC Fourier coefficients ##A_j## and ##B_j## for non-negative integer ## j = 0,1,2,3...## are related to the CE Fourier coefficients ##C_n## by: $$ A_j = C_j + C_{-j} ; \\ B_j = i(C_j - C_{-j}) ; \\ for: j = 0,1,2,3... $$

Hint: All you need here is Euler: ## e^{i\gamma} = cos(\gamma) + isin(\gamma) ## or ## cos(\gamma) = (e^{i\gamma} + e^{-i\gamma})/2 ## or ## sin(\gamma) = (e^{i\gamma} - e^{-i\gamma})/(2i) ##


I plugged in the Euler formula to the CE Fourier series and evaluated the definite integral from -pi to pi, which equaled zero, soooo I'm not too sure what to do...
 
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  • #2
You don't need to integrate. Just expand the (CE) series using Euler's formula and group the sines and cosines into their own sums. Then judiciously apply the evenness and oddness of cosine and sine respectively to get your coefficient identities.
 

Related to Complex exponential and sine-cosine Fourier series

1. What is a complex exponential Fourier series?

A complex exponential Fourier series is a mathematical tool used to represent periodic functions as a sum of complex exponential functions. It is a type of Fourier series that is commonly used in electrical engineering and signal processing.

2. How is a complex exponential Fourier series different from a sine-cosine Fourier series?

A complex exponential Fourier series uses complex exponential functions (e.g. eix) as the basis functions, while a sine-cosine Fourier series uses sine and cosine functions as the basis functions. Both types of Fourier series can be used to represent periodic functions, but the choice of basis functions depends on the application.

3. Why are complex exponential and sine-cosine Fourier series important?

Complex exponential and sine-cosine Fourier series are important because they allow us to represent a wide variety of periodic functions in terms of simpler components. This makes it easier to analyze and manipulate these functions, and can be useful in fields such as mathematics, physics, and engineering.

4. What is the relationship between complex exponential and sine-cosine Fourier series?

The complex exponential Fourier series is a special case of the sine-cosine Fourier series, where the coefficients of the sine terms are equal to the imaginary parts of the coefficients of the complex exponential terms, and the coefficients of the cosine terms are equal to the real parts. In other words, the complex exponential Fourier series can be thought of as a simplified version of the sine-cosine Fourier series.

5. Can any periodic function be represented by a complex exponential or sine-cosine Fourier series?

Yes, any periodic function can be represented by both a complex exponential Fourier series and a sine-cosine Fourier series. However, in some cases, one type of Fourier series may be more useful or efficient than the other in terms of convergence or ease of calculation.

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