In mathematics, the Fourier series is a way to represent a function as a sum of periodic components, and it is useful in many fields, including engineering, physics, and economics. Given a function f(x) defined on some interval [a,b], the Fourier series of f is the infinite sum
f(x) = a0 + a1cos(x2pi/T) + b1sin(x2pi/T) + a2cos(2x2pi/T) + b2sin(2x * 2*pi/T) + …
where T is the period of f (that is, T = b-a), and the coefficients an and bn are defined by
a0 = 1/T * integral from a to b of f(x)*dx
an = 1/T * integral from a to b of f(x)*cos(n * x * 2 * pi / T)*dx
bn = 1/T * integral from a to b of f(x)*sin(n * x * 2 * pi / T)*dx
The terms in the Fourier series are often referred to as “harmonics”, because they are periodic with period T, and each harmonic has its own regular frequency given by n/T.
The Fourier Series
In mathematics, the Fourier series is a tool used to represent a function as the sum of a series of sinusoidal functions. It is named after French mathematician Joseph Fourier, who showed that any continuous function could be represented as such a sum.
The Fourier series has many applications in physics and engineering because it can be used to model periodic waveforms. For example, the Fourier series can be used to represent the sound of a musical instrument, the light emitted by stars, or the shape of an object.
The Fourier series of a function is typically written as:
f(x) = a0 + a1cos(x) + b1sin(x) + a2cos(2x) + b2sin(2x) + …
where x is the independent variable, and a0, a1, b1, a2, b2, … are coefficients that can be determined from the function f(x). In practice, only a finite number of terms are usually needed to get an accurate representation of f(x).
The Fourier Transform
In mathematics, the Fourier transform is a tool used to change a function from its original form (often called the “time domain”) to a new form (the “frequency domain”). This transformation can be used to analyze periodic functions, such as those that occur in music or in signals from communication devices. The Fourier transform can also be used to solve certain partial differential equations, such as the heat equation.
The Inverse Fourier Transform
The inverse Fourier transform is used to convert a signal from the frequency domain back into the time domain. This is the opposite of the Fourier transform, which converts a signal from the time domain into the frequency domain.
The inverse Fourier transform is represented by:
g(t) = ∫G(f)e^(2πift) df
where G(f) is the Fourier transform of g(t).
The inverse Fourier transform can be used to reconstruct a signal from its Fourier transform. This is done by using the inversetransform to convert the signal back into the time domain.
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