Taylor's series for Lie groups Let $G_1$ and $G_2$ be two (matrix) Lie groups, with $L(G_1)$ and $L(G_2)$ their respective Lie algebras.
I am interested to know if there is a well developed theory to approximate a (sufficiently) smooth function $f:G_1 \rightarrow G_2$ using a "Taylor's series" expansion. 
That is, I'd like to know how I can compute the functions $a_i: L(G_1) \rightarrow L(G_2)$, $i = 1,2, \dots$ such that the following identity holds
$
f(g \exp( \varepsilon \zeta)) = f(g) \exp( \varepsilon a_1(\zeta) + \frac{\varepsilon^2}{2!} a_2(\zeta) + \frac{\varepsilon^3}{3!} a_3(\zeta) + \dots)
$
with $\varepsilon \in \mathbb{R}$ and $\zeta \in L(G_1)$.  
Clearly, $a_1(\zeta) = f(g)^{-1} Df(g)\cdot g\zeta$... 
Thanks.
 A: Let me sketch a solution as a three step process: 
For smooth function $f:M\to G_2$ consider its left logarithmic differential
$\delta^l f\in \Omega^1(M,\mathfrak g_2)$ which satisfies the right Maurer-Cartan equation
$d(\delta^l f) +\frac12 [\delta^l f,\delta^lf]=0$. It can be reconstructed on simply connected domains in $M$ uniquely up to constant right translation from $\delta^l f$. This called the Cartan development. 
See 4.2 of here, e.g., for a detailed proof.   
Thus for $f:G_1\to G_2$ we have $\delta^l f\in \Omega^1(G_1,\mathfrak g_2)$.  By left trivializing $TG_1$ we can view $\delta^l f$ as an element of 
$C^\infty (G_1, L(\mathfrak g_1,\mathfrak g_2))$.
Thm 2.6 in the following paper is the Taylor theorem with remainder term for functions on a Lie group $G_1$ (with values in a vector space, here $L(\mathfrak g_1,\mathfrak g_2))$.  The infinite Taylor series factors to a linear functional on the universal enveloping algebra of the Lie algebra $\mathfrak g_1$. 


*

*Peter W. Michor: The cohomology of the diffeomorphism group is a Gelfand-Fuks cohomology. Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 14 (1987), 235-- 246 (pdf)
Putting all together again, we get a Taylor series with remainder term for a smooth mapping $G_1\to G_2$. 
