The Fourier transform computation toolkit
Build a new transform from a known one -- six properties and five standard pairs, side by side, so almost nothing needs a fresh integral.
Before setting up \int f(t)e^{-i\omega t}\,dt from scratch, check whether the function in front of you is a shifted/scaled/multiplied version of something already in the pairs table below -- one of the six properties usually gets you there in one line.
The properties -- turn a known transform into a new one
Section
Linearity
\mathcal{F}[af+bg]=a\hat f+b\hat g -- combine known transforms directly.
Section
Time shifting
Delaying f multiplies \\hat f(\\omega) by a pure phase factor -- magnitude unchanged.
Section
Frequency shifting (modulation)
Multiplying f by e^{i\\omega_0 t} slides the whole spectrum to \\omega_0 -- the duality partner of time shifting.
Section
Scaling
Squeeze in time, spread out in frequency -- never both at once.
Section
Differentiation
A derivative in time becomes multiplication by i\\omega in frequency -- turns a differential equation into algebra.
Section
Duality
If f\\leftrightarrow\\hat f, then \\hat f's own transform lands back near f, time-reversed -- every pair below is really two pairs at once.
The pairs -- recognize the shape, skip the integral
Example
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Section
The Gaussian
The one function whose transform has the same shape it started with.
Section
The one-sided exponential
Example