Complex Sinusoids
The complex function: facilitates many kinds of mathematical operations involving, due in large part to Euler's simplification:
This very useful form is often referred to as a complex sinusoid, and it preserves the distinction between positive and negative .
- For positive values, it is also called the analytic representation of .
The Fourier transform of produces a non-zero response only at frequency .
- The transform of has responses at both and, which reflects the fact that is insufficient to determine the sign of .
- An alternative, and surprisingly useful, viewpoint is that both frequencies are present, as implied by the inverse of Euler's formula: .
Read more about this topic: Negative Frequency
Famous quotes containing the word complex:
“In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
—Georg Wilhelm Friedrich Hegel (1770–1831)