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]:=

X lt = 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]:=

X D[lt, s]

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In[3]:=

X D[ft, \[Omega]]

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In[4]:=

X D[conv, s]

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

In[5]:=

X D[lt, t]

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In[6]:=

X D[ft, t]

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In[7]:=

X D[conv, t]

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

In[8]:=

X D[lt, a]

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In[9]:=

X D[ft, a]

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In[10]:=

X D[conv, a]

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