Exponential Formula

$f(x)=a\_1\; x+\{a\_2\; \backslash over\; 2\}x^2+\{a\_3\; \backslash over\; 6\}x^3+\backslash cdots+\{a\_n\; \backslash over\; n!\}x^n+\backslash cdots$

$\backslash exp\; f(x)=e^\{f(x)\}=1+\backslash sum\_\{n=1\}^\backslash infty\; \{b\_n\; \backslash over\; n!\}x^n$

$b\_n=\backslash sum\_\{\backslash pi=\backslash left\backslash \{\backslash ,B\_1,\backslash ,\backslash dots,\backslash ,B\_k\backslash ,\backslash right\backslash \}\}\; a\_\{\backslash left|B\_1\backslash right|\}\backslash cdots\; a\_\{\backslash left|B\_k\backslash right|\},$

$b\_3=a\_3+3a\_2\; a\_1\; +\; a\_1^3$

References

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