Take $\mathfrak{g}$ a simple Lie algebra and $k$ the Killing form of $\mathfrak{g}$. Now I found two definitions of affine Kac Moody algebra associated to $\mathfrak{g}$.
The first: a one-dimensional central extension of $\mathfrak{g}\otimes\mathbb{C}((t))$
The second: a one-dimensional central extension of $\mathfrak{g}\otimes\mathbb{C}(t)$
In both of them, if $A$ is a central element, the bracket is defined by $[x \otimes f, y \otimes h]=[x,y]\otimes fh + k(x,y)Res(fh')A.$
I'd like to know if them are isomorphic lie algebras and in case some reference to study them on.