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1.5.11 Integral Transforms

Laplace transforms.
This computes a Laplace transform.
In[1]:= LaplaceTransform[t^3 Exp[a t], t, s]
Out[1]= 
Here is the inverse transform.
In[2]:= InverseLaplaceTransform[%, s, t]
Out[2]= 

Fourier transforms.
This computes a Fourier transform.
In[3]:= FourierTransform[t^4 Exp[-t^2], t, w]
Out[3]= 
Here is the inverse transform.
In[4]:= InverseFourierTransform[%, w, t]
Out[4]= 
Note that in the scientific and technical literature many different conventions are used for defining Fourier transforms. Section 3.8.3 describes the setup in Mathematica.
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