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]:=
Click for copyable input
X

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

In[2]:=
Click for copyable input
X
Out[2]=
In[3]:=
Click for copyable input
X
Out[3]=
In[4]:=
Click for copyable input
X
Out[4]=

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

In[5]:=
Click for copyable input
X
Out[5]=
In[6]:=
Click for copyable input
X
Out[6]=
In[7]:=
Click for copyable input
X
Out[7]=

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

In[8]:=
Click for copyable input
X
Out[8]=
In[9]:=
Click for copyable input
X
Out[9]=
In[10]:=
Click for copyable input
X
Out[10]=
de es ja pt-br zh