Integral equations of convolution type

[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...

 

[skybsk]

what is the inverse of the operator

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

where φ ∈ L_2(−∞,∞)?