# How is Lagrange's inversion theorem derived?

I am interested in the complex-analysis version of deriving Lagrange's inversion theorem:

If $y=f(x)$ with $f(a)=b$ and $f'(a)\neq 0$, then $$x(y)=a+\sum_{n=1}^{\infty} \left(\lim_{x\to a}\frac{d^{n-1}}{dx^{n-1}}\left(\frac{x-a}{f(x)-b}\right)^n \frac{(y-b)^n}{n!}\right).$$

The derivative expression immediately suggests some type of residue calculation but I'm not able to derive it or find any reference on line that goes over it.

Can someone here help me with this?

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gallica.bnf.fr/ark:/12148/bpt6k229222d/f6 is the paper cited by wikipedia (en francais), en.wikipedia.org/wiki/… is a formal power series argument, people.math.carleton.ca/~zgao/MATH5819/Notes/NoteT6.pdf is another explanation – yoyo Dec 13 '11 at 14:42
Those references are worst than nothing as it gives the illusion the question is being answered. It's an english forum but assumes the reader knows french. How bout Russian? The other two do not address this specific quesion and one is so cryptic I have doubts anyone but the writer understands the proof given. This is a perfect example of why the question has been asked several times on the internet in the past several years without answers. Nevermind guys. I'll get it on my own eventually. – Jack Dec 13 '11 at 19:41
This shows the more general Lagrange-Bürmann theorem via Cauchy. This gives an algebraic proof. Which of these are you more comfortable with? – J. M. Dec 14 '11 at 3:21
@Jack: That seems to me an overreaction. It is unlikely to inspire anyone to want to help you with your question. Note that yoyo's helpful comment is just that: a comment, with no claim that it is supposed to be a final answer to the question. – Jonas Meyer Jan 23 '12 at 5:05
@Jack - by the way, you haven't been entirely accurate in your problem statement - although I agree that most people familiar with the theorem will know that it applies only where f is analytic at the value a! – user71551 Apr 21 '13 at 10:08