Exponential-kernel half-line eigenfunction
= Exponential-kernel half-line eigenfunction
{title2=$f(x)=A(\cos x+\sin x)$}
The equation $\int_0^\infty e^{-|x-t|}f(t)\,dt=f(x)$ has this solution family in the ordinary bounded-growth transform class. Differentiating twice gives $f''=-f$, and the integral's right derivative at zero gives $f'(0)=f(0)$. Alternatively its positive-support <Fourier transform> is $A(ik-1)/(k^2-1)$; inversion above both real poles gives the same answer.