Complex Fourier Series

Fourier Analysis · Axiom Academy

Expressing Fourier series using complex exponentials Euler's formula connects complex exponentials with trigonometric functions. It states that for any real number x: This remarkable identity shows that complex exponentials trace out circular motion in the complex plane. 2. Expressing cos and sin in Terms of Complex Exponentials From Euler's formula, we can derive expressions for cosine and sine. Using both e ix and e -ix : These identities allow us to replace trigonometric functions with complex exponentials. 3. Transforming the Trigonometric Series Starting with the trigonometric Fourier series: We substitute the complex exponential forms of cos and sin, then collect terms to obtain the complex form. 4. The Complex Fourier Series Formula After collecting terms, we arrive at the elegant complex Fourier series: The sum now runs from -∞ to +∞, with complex coefficients c n . The fundamental angular frequency is ω = 2π/T. 5. The Complex Coefficient Formula The complex Fourier coefficients are computed using: This single formula replaces the separate formulas for a n and b n . The integration is over one period T. 6. Relationship Between c n and a n , b n The complex and trigonometric coefficients are related by:

This is the written version of the interactive lesson above. See the full Fourier Analysis course.