I have to expand $\exp[-(x-t)]$ as product of series of $f(x)$ and $f(t)$, where $f(x)$ and $f(t)$ are polynomials.

Question

I am trying to solve homogenous fredholm equation of second kind. The kernel is given by $\exp[-(x-t) ]$. I have to separate the kernel as product of two function $f(x)$ and $f(t)$.

Answer 1

Note that

$e^{-(x - t)} = e^{-x} e^{t}$

Using the Taylor series formulas, we can expand as:

$e^{-(x - t)} = \left( 1 - x + {x^2 \over 2!} - {x^3 \over 3!} + \cdots \right) \left( 1 + t + {t^2 \over 2!} + {t^3 \over 3!} + \cdots \right)$