He didn't ask for the Fourier transform of F(ω). He asked for the spectrum, i.e. a graph in the frequency domain, which is what BvU said, except it's not sufficient to say " ... spikes which go up to to infinity ..." of course. What of the ω coefficient?George Jones said:xoxmae already has a function ##F\left( \omega\right)## that has two "spikes", one at ##\omega = \omega_0## and one at ##\omega = -\omega_0##. The Fourier transform of this will not have spikes.
The Dirac Delta Function, also known as the unit impulse function, is a mathematical function that is defined as zero everywhere except at the origin, where it is infinite. It is often used to represent a point mass or point charge in physics and engineering.
Mathematically, the Dirac Delta Function is represented as a spike at the origin, with an area of 1. It is often denoted by the symbol δ(x) or δ(x-a), where a is the location of the spike. In integral form, it is represented as ∫δ(x-a)dx = 1.
The Dirac Delta Function has several physical interpretations, depending on the context in which it is used. In physics, it is often used to represent a point mass or charge, where the mass or charge is concentrated at a single point. In signal processing, it is used to represent an impulse or instantaneous change in a system.
The Dirac Delta Function is a discontinuous function, as it is defined as being infinite at a single point and zero everywhere else. However, it is often treated as a continuous function in mathematical calculations, as it is often integrated over a range of values.
The Dirac Delta Function has a wide range of applications in physics, engineering, and mathematics. Some examples include representing point masses or charges in physics, modeling impulse responses in signal processing, and solving differential equations in mathematics. It is also used in Fourier analysis and as a basis for other distributions in functional analysis.