Fourier transform of a function for a symbolic parameter :

Fourier transforms of trigonometric functions:

Evaluate the Fourier transform for a numerical value of the parameter :

TraditionalForm formatting:

Fourier transform of a power function:

Polynomial:

Fourier transform of rational functions:

Plot the transform:

Plot the real and imaginary parts:

Reciprocal of square root:

Plot the transform:

Expressions involving trigonometric functions:

Ratio of sine and linear function:

Plot the transform:

Composition of elementary functions:

Plot the transform:

Logarithmic function:

Plot the transform:

Sinc function:

Plot the transform:

Expressions involving Bessel functions:

Plot the transform:

SinIntegral function:

Laguerre polynomial:

Airy function:

Plot the magnitude of the Fourier transform for complex :

Fourier transform of a piecewise function:

Absolute value using Sign function:

Restriction of a sine function to a half-period:

Triangular function:

Ramp:

UnitStep:

Product of UnitStep and cosine functions:

Plot the magnitude and phase:

Fourier transform of SquareWave:

TriangleWave:

SawtoothWave:

Full-wave-rectified function with period :

Rectified wave:

Fourier transform involving HeavisideTheta:

Plot the magnitude and phase:

DiracDelta:

Derivative of DiracDelta:

HeavisideLambda:

HeavisidePi:

Bivariate Fourier transform of a constant:

Exponential function:

Trivariate cosine:

Product of power and exponential:

Fourier transform of a product of exponential and SquareWave functions:

Fourier transform of a first-order derivative:

Fourier transform of a second-order derivative:

Fourier transform threads itself over equations:

Calculate the Fourier transform at a single point:

Alternatively, calculate the Fourier transform symbolically:

Then evaluate it for the specific value of :

The option AccuracyGoal sets the number of digits of accuracy:

With default settings:

Specify the range of a variable using Assumptions:

Fourier transform for the unit box function with different parameters:

Create a nicely formatted table of the results:

Use GenerateConditionsTrue to get parameter conditions for when a result is valid:

The option PrecisionGoal sets the relative tolerance in the integration:

With default settings:

If WorkingPrecision is specified, the computation is done at that working precision:

With default settings:

Find the convolution of signals:

The product of their Fourier transforms:

Find the inverse transform:

Compare with Convolve:

Spectrum of the product of two signals, with one given in the frequency domain by:

The Fourier transform of the signal :

The Fourier transform of the product of with the original signal is the convolution of its transforms:

Its spectrum:

Frequency response of an LTI system defined by an ODE:

Apply the Fourier transform over the equation:

Solve for the Fourier transform of :

The frequency response of the LTI system is the ratio of the Fourier transforms of the output function over the input function :

Solve a differential equation using Fourier transforms:

Apply the Fourier transform over the equation:

Solve for the Fourier transform:

Find the inverse transform to get the solution:

Compare with DSolveValue:

Consider the heat equation: with initial condition :

Fourier transform with respect to :

With and , solve this ODE:

Compute the inverse Fourier transform:

And convolution to get the solution:

Consider the special case with initial condition and :

Compare with DSolveValue:

Plot the initial conditions and solutions for different values of :

Plot the solution over the - plane.

Calculate the following definite integral:

Compute the Fourier transform with respect to and interchange the order of transform and integration:

Integrate over :

Use the inverse Fourier transform to get the result:

Compare with Integrate:

The power spectrum of a damped sinusoid:

The Fourier transform of a radially symmetric function in the plane can be expressed as a Hankel transform. Verify this relation for the function defined by:

Plot the function:

Compute its Fourier transform:

Obtain the same result using HankelTransform:

Plot the Fourier transform:

Generate a gallery of Fourier transforms for a list of radially symmetric functions:

Compute the Hankel transforms for these functions:

Generate the gallery of Fourier transforms as required:

Calculate the power spectrum of a stationary OrnsteinUhlenbeckProcess:

A quick look at the Heisenberg uncertainty principle:

Consider a fixed-area box function as the position space wavefunction of a particle. Its Fourier transform gives the momentum space wavefunction of the particle:

When is small, the height of the fixed-area box is big, and the position of the particle is almost guaranteed. The momentum space wavefunction is approximately for values between its two roots closest to zero, which makes it almost impossible to find its momentum. Similarly, vice versa, as seen here: