Function-type arguments in function definition
Carsten Reckord
I'm pretty new to Mathematica so please excuse me if this is kind of a silly
question (though I couldn't find any answer after a full day of searching).
I'm trying to define functions that take other functions as arguments and
need those functions' arguments in their own definition. An example would be
the definition of convolution:
h(x)=f(x)*g(x) is defined as Integral over f(y)g(x-y) with respect to y.
As you can see it is important in the definition of convolution to treat the
arguments f and g as functions because the definition makes use of their
arguments.
I've seen this done in Mathematica as
convolute[f_,g_,x_]:=Integrate[f[y]*g[x-y],{y,-inf,inf}]
but that's not exactly what I'm looking for because I can only use function
names as arguments to convolute[...], not arbitrary expressions in x. So I
can't for example use it for the convolution f(s(x))*g(t(x)) without
defining intermediate functions for f(s(x)) and g(t(x))...
So, my question is if there is any way to define such a function that can
make use of its arguments being functions and yet supports arbitrary
expressions as its arguments?
Thanks,
Carsten
Jens-Peer Kuska
and
conv[f_, g_][x_] := Module[{y}, Integrate[f[y]*g[x - y], {y, a, b}]]
conv[#^2 &, Exp[-#] &][x]
does not what you whant ?
Regards
Jens
Paul Abbott
"Carsten Reckord" <ne...@reckord.de> wrote:
I would do this as follows. Define your convolution operation for pure
functions f and g with variable x as
Star[f_Function, g_Function][x_] :=
Integrate[f[y] g[x - y], {y, -Infinity, Infinity}]
If you now add a rule that converts f and g to pure functions (assumed
to be functions of x),
Star[f_,g_][x_] := Star[Function[x,f], Function[x,g]][x]
then you can compute convolutions of the type you require as
Star[f[s[x]], g[t[x]]][x]
(An aside: Mathematica already has an infix notation suitable for
convolution -- the Star operator. Above I've used the InputForm, Star,
which adds rules to Star, so that you can enter input expressions as
(f[s[x]] \[Star] g[t[x]])[x]
where \[Star] is converted to the appropriate special character on
input.)
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
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Jean-Claude Poujade
one needs an extra argument telling the name
of the variable used in f and g (it's not necessarily 'x').
Example :
In[1]:=ClearAll[convolute];
convolute[f_,g_,x_]:=
Integrate[f[y]*g[x-y],{y,-Infinity,Infinity}];
(* 'u' is the variable used in expressions f and g *)
convolute[f_,g_,u_,x_]:=
Integrate[(f /. u -> y)*(g /. u ->(x-y)),{y,-Infinity,Infinity}];
In[2]:=f1[x_]:=E^(-x^2);
f2[x_]:=E^(-x^2);
convolute[f1,f2,x]
Out[2]=Sqrt[Pi/2]/E^(x^2/2)
In[3]:=exp1=E^(-u^2);
exp2=E^(-u^2);
convolute[exp1,exp2,u,x]
Out[3]=Sqrt[Pi/2]/E^(x^2/2)
---
jcp