I am trying to understand the various ways in which one can write the translation of a function by $a$ are related to each other:
- $\displaystyle f(x+a)=T_a [f(x)]=\lim_{N \rightarrow \infty} \left( T_{\frac{a}{N}}\right)^N [f(x)]$ where $T$ is the translation operator.
- $\displaystyle f(x+a)= \sum_k \frac{a^k \partial^k}{k!} f(x) = e^{a\partial} f(x) = \lim_{N \rightarrow \infty} \left(1 + \frac{a}{N} \partial\right)^N f(x)$, where $\partial=\partial_x$.
- $\displaystyle f(x+a) = f(x) + \int_x^{x+a} \partial_{x'}f dx' = f(x) + \lim_{N\rightarrow \infty} \sum_k \frac{a}{N} T^k_{\frac{a}{N}} [\partial f(x)]$
How can one show how 1,2 and 3 are related?