I need to prove a transformation theorem $T(ψ(x)) = (e^{hD})*ψ(x)$ and use Taylor series to do this task. It is known that $T(ψ(x)) = ψ(x + h)$ and $D$ is a derivatation.
I have no idea, how to start this issue. What function do I have to approximate to get this? I tried to do this with both sides of equations:
Series[f[x+h], {x, 0, 4}]
Series[f[x]*Exp[D[h, x]], {x, 0, 4}]
But it didn't work. I got output like this: output1 output2
You are my last hope :(
I assume (based on the context that you had previously decided to remove from the question) that $\psi: \Bbb R \to \Bbb R$, and $D$ is the operator $D(\psi) = \frac{d \psi}{dx}$. With that said, note the following:
Using the Taylor series defintion of $e^{hD}$, we have $$ e^{hD} \psi = \left( \sum_{k=0}^\infty \frac 1{k!}(hD)^k\right)\psi = \sum_{k=0}^\infty \frac 1{k!}h^k D^k\psi = \sum_{k=0}^\infty \frac 1{k!}\frac{d^k\psi}{dx^k} h^k $$ Compare this to the Taylor series of $f(x+h)$ centered at $x$.
Alternatively: use Mathematica expand over $h$ rather than over $x$. In other words, use the following modifications to your commands