Uses of eulers equation in fourier series

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SUMMARY

The discussion centers on the application of Euler's equation, e^(jωt) = cos(ωt) + i sin(ωt), in Fourier series analysis. It highlights that the complex Fourier series provides a streamlined approach to representing real periodic functions by combining sine and cosine components into a single series. This method simplifies the process of determining coefficients, as it allows for the use of a single integral equation rather than separate equations for sine and cosine series.

PREREQUISITES
  • Understanding of Fourier series and their components
  • Familiarity with Euler's formula and complex numbers
  • Knowledge of integral calculus for coefficient determination
  • Basic concepts of periodic functions in mathematics
NEXT STEPS
  • Study the derivation of Fourier series from periodic functions
  • Explore the application of Euler's formula in signal processing
  • Learn about the computation of Fourier coefficients using integrals
  • Investigate the differences between complex and real Fourier series
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Mathematicians, engineers, and students in fields such as signal processing and applied mathematics who are interested in the theoretical and practical applications of Fourier series and Euler's equation.

amaresh92
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greetings,

why do we use Euler equation that is e ^(jωt)=cos(ωt)+i sin(ωt) in Fourier series and what does it represent?
advanced thanks.
 
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In general for a real periodic function, the Fourier decomposition is composed of both a Sine series and a Cosine series, both having real coefficients. The complex Fourier series (of a real function) that you describe is a convenient way of combining both of these into a single series solution.

It also has the advantage that often both sets of coefficients can be found using just the one integral equation, as opposed to a separate one for each of the Sine and Cosine series.
 

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