Help: Commutator Expansion Proof

Hello, I was reading through a proof in the differential equations course that started with the equation above, saying that with some effort it could be shown that this expansion was true. Though I am incredibly curious about where this comes from.

I don't know much about commutators, but if someone could show me why this statement is true or send me some links that will help me with this proof that would be very much appreciated!

UPDATE: I managed to find a very nice proof on Youtube: https://www.youtube.com/watch?v=ndEBKIAJ_mg

#Calculus

Easy Math Editor

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Comments

I am working on your problem, but my first hint would be to use the Taylor series of $e^x$ .

Aruna Yumlembam - 11 months, 2 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$