I had a hunch that since the derivative of the inverse of the function is the reciprocated derivative in the original function as a function of the inverse that, if you had a function that you didn't know the inverse precisely, you could calculate the taylor series for its inverse using that derivative property. Upon researching this, the only conclusive thing I found was something called the Lagrange Inversion theorem. Coincidentally, the way it's described makes it seem like it does exactly what I want it to do, finding the inverse of a function which can only be described analytically. However, I'm not good at reading mathematics from scratch and whenever I look at the wiki page for it, it still looks like a mess of variables that aren't defined in much detail. Could someone work through an example with something simple showing how I use the theorem?

Edited March 2, 201510 yr by MWresearch

Since no one else will bother to answer, I will make the following observation, though I don't really feel inclined after your response to my last attempt to help.

 

You will find limited reference to the theorem you refer to since it has largely been superceded by Cauchy contour integration.

The technique is sound, but the problem is of determining the coefficients of the series representing the inverse function.

 

References are

 

Titchmarsh : The Theory of Functions

 

Copson : The Theory of Functions of a Complex Variable

 

Fort : Infinite Series

 

All from Oxford University Press

Edited March 2, 201510 yr by studiot

I guess I don't see how that Cauchy integration is used in place of the Lagrange Inversion theorem. Also, which technique are you referring to when you say it's hard to determine the coefficients?