itself and then try to figure out the taylor series expansion by differentiating etc but for example for the lambda w function it's very hard to get yourself the taylor series expansion without making use of the la crosse inversion theorem or the lagrange burman theorem now we can also use the lagrange inversion theorem to solve simple or harder polynomial equations but before we get into this we are going to write out what the lagrange inversion theorem actually says i'm not going to prove or derive it here the derivation is rather simple using just complex numbers so function theory or complex analysis just making use of the residue theorem and then going through the steps of expanding the integrand into a series expansion basically and so on so it's rather simple to prove this theorem i think it's been covered in stein and shakashi i'm not certain about it and shouts outlines of complex numbers so maybe take a look at that if you want to see a derivation but what does it state in general now suppose we have a function f which is analytic so let f be a function which is analytic and more importantly as for the taylor series expansion of a function what we want in a normal case is that we want to evaluate the series expansion at a certain point x naught so for the maclaurin series expansion it's at x not being equal to zero for the taylor series expansion it can be at any other point as long as the function is analytic their analytic meaning by the way that we can expand this function into a corresponding power series expansion a former series expansion now what we want is you want f to be analytic and more importantly analytic at a certain point x naught and since we are dealing with complex numbers let's say that x naught is element of the complex numbers for example now we want it to be analytic at x naught element of the complex numbers due to the residue theorem being at play we need one other condition which also needs to hold namely we need that the derivative of the of the function f which is analytic at the point x naught is non-zero this is very important because otherwise it would diverge in the series expansion so this is what we need for now and if what we have right here holds then
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