SUMMARY
The Fourier transform of the function f(t) = sin(t) for the interval -π < t < π is correctly computed by integrating from -π to π, as the function is zero outside this range. The initial confusion arose from the standard Fourier transform limits of -∞ to ∞, which include the Dirac delta function. However, since the function is defined to be zero outside the specified interval, the integration limits can be adjusted accordingly. This approach eliminates the need for the Dirac delta function in this specific case.
PREREQUISITES
- Understanding of Fourier transforms and their properties
- Familiarity with the Dirac delta function
- Knowledge of piecewise functions
- Basic calculus skills for integration
NEXT STEPS
- Study the properties of the Fourier transform, particularly for piecewise functions
- Learn about the Dirac delta function and its applications in Fourier analysis
- Explore the implications of changing integration limits in Fourier transforms
- Practice computing Fourier transforms of other trigonometric functions
USEFUL FOR
Students in mathematics or engineering, particularly those studying signal processing or Fourier analysis, will benefit from this discussion.
