# Integral equations of convolution type

• catcherintherye
In summary, the conversation discusses two questions involving finding the transform and inverse transform of functions using convolution theorem and inversion formula. The first question involves finding the function f(x) given the integral equation and using differentiation and integration to derive the solution. The second question also uses the convolution theorem and contour integration to derive the transform of the function and it is shown that the inverse transform is equal to (e^-1)/(2pi)^1/2. The question concludes with the inquiry about the inverse of an operator Uφ for a given function φ ∈ L_2(−∞,∞).

#### catcherintherye

i am asked to find f(x) s.t. exp(-xsqd/2) =1/2the integral (-inf to +inf) of exp[-|x-u|f(u)du. I have got as far as to show that the transform f(k)=(1+ksqd)exp[-ksqd/2) and my notes show that this implies the next line which is f(x)=exp[-xsqd/2] -[exp(-xsqd/2)]'' {'' denotes twice differentiation w.r.t x} and this in turn is equal to (2-xsqd)exp[-xsqd/2]. I am not sure how these last two lines where reached, was the inversion formula applied to f(k) and then some integration done? or was some other result used?

I have a simliar question where i must use the convolution theorem to show that the solution to the following integral equation

integral(-inf to + inf) of {f(u)/1+(x-u)^2}du=1/(xsqd+4) is

f(x)=1/2pi.1/(1+xsqd)

i have got as far as to show transform of 1/(xsqd+4)=(2pi)^1/2.transform of1/(1+xsqd).transf and i have used contour integration to show transform of 1/(4+xsqd)=(pi/4).exp[-2]. I have also shown that trans 1/(1+xsqd) =(pi/2)exp[-1]

so I have derived that transform f =(e^-1)/(2pi)^1/2

is this correct up to this point? and where do I go from here, I'm stuck at the same point as the first question...

Last edited:
what is the inverse of the operator

Uφ ≡ φ(x) − 2e^{-x}\int_{-\infty}^{x} e^{t}φ(t)dt

where φ ∈ L_2(−∞,∞)?