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The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.
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More precisely, we know that for any base b, exponentiation satisfies:
  • .
  • .
And we also know that for in particular that we satisfy the exponential function differential equation and so:
One interesting fact is that the only thing we use from the exponential function differential equation is the value around , which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.
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Now suppose that we want to calculate . The idea is to start from and then then to use the first order of the Taylor series to extend the known value of to .
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E.g., if we split into 2 parts, we know that:
or in three parts:
so we can just use arbitrarily many parts that are arbitrarily close to :
and more generally for any we have:
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Let's see what happens with the Taylor series. We have near in little-o notation:
Therefore, for , which is near for any fixed :
and therefore:
which is basically the formula tha we wanted. We just have to convince ourselves that at , the disappears, i.e.:
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To do that, let's multiply by itself once:
and multiplying a third time:
TODO conclude.
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