How to Prove the Lagrange Inversion Theorem?

Click For Summary
The Lagrange Inversion Theorem relates to the Taylor series of the inverse of an analytical function. A user expressed difficulty in proving the theorem using basic algebra and calculus, without resorting to real analysis. The theorem asserts that if f is analytical at x=a, then the inverse function f-1 has a specific Taylor series expansion. The discussion highlights a lack of clear methods or resources for proving this theorem. Overall, the inquiry emphasizes the challenge of proving the theorem without advanced mathematical tools.
AdrianZ
Messages
318
Reaction score
0
I encountered this beautiful theorem and then I tried hard to prove it using ordinary algebraic methods and my understanding of calculus without involving real analysis in it but I didn't succeed. The theorem states that if f is an analytical function at some point x=a then f-1 has the following Taylor series:

c31894bd772bb55bd1f98d0e0dd770f2.png


How can I prove this formula?
 
Physics news on Phys.org
No one knows?
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
View full post »

Similar threads

Undergrad Usage of inverse function theorem in Folland
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
3K
High School Why do I not understand what seems to be basic Calculus?
  • · Replies 11 ·
Replies
11
Views
2K
The error caused by the lagrange inversion
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad On (real) entire functions and the identity theorem
  • · Replies 2 ·
Replies
2
Views
3K
Inverse function theorem over matrices
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Proving limit theorems when limit tends to infinity
  • · Replies 7 ·
Replies
7
Views
2K