# Compute with Integral Transforms

Properties of integral and other formal operators are applied to their inactive forms.

Integral operators include LaplaceTransform, FourierTransform, and Convolve.

 In[1]:= Xlt = Inactive[LaplaceTransform][a t^2, t, s]; ft = Inactive[FourierTransform][a f[t], t, \[Omega]]; conv = Inactive[Convolve][a f[t], g[t], t, s];

The derivatives of all of these with respect to the last argument can be expressed in terms of the integral operator itself.

 In[2]:= XD[lt, s]
 Out[2]=
 In[3]:= XD[ft, \[Omega]]
 Out[3]=
 In[4]:= XD[conv, s]
 Out[4]=

In all of these, is a dummy variable, so derivatives with respect to it are zero.

 In[5]:= XD[lt, t]
 Out[5]=
 In[6]:= XD[ft, t]
 Out[6]=
 In[7]:= XD[conv, t]
 Out[7]=

All of the transforms are linear, so derivatives with respect to parameters can simply be performed.

 In[8]:= XD[lt, a]
 Out[8]=
 In[9]:= XD[ft, a]
 Out[9]=
 In[10]:= XD[conv, a]
 Out[10]=

## Mathematica

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