Chapter 19: Problem 1
A function \(f: G \rightarrow \mathbb{R}\) is said to be an aralytic function on \(G\) if it can be expanded as a Taylor serics at any point \(g \in G\). Show that if \(X\) is a left-invariant vector ficld and \(f\) is an analytic function on \(G\) then $$ f(g \exp t X)=\left(c^{t x} f\right)(g) $$ where, for any vector ficld \(Y\), we define $$ \mathrm{e}^{\gamma} f=\boldsymbol{f}+Y \boldsymbol{f}+\frac{1}{21} Y^{2} f+\frac{1}{3 !} Y^{3} f+\cdots=\sum_{i=0}^{\infty} \frac{Y^{n}}{n !} f $$ The operator \(Y^{\pi}\) is defined inductively by \(Y^{n} f=Y\left(Y^{n-1} f\right)\).
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