For example, how do you find the inverse function of Y = X^X ? Solving for X in terms of Y can't be done using basic algebra. How do I calculate the power series expansion of the inverse function?

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A Taylor series doesn't help at all for the function

you mentioned.

Also, for even value of x between 0 and 1 , there are

two y values for every x so an inverse would not be a function.

so a Taylor series will be like the following

So with all different exponent

he Taylor

doesn't help find the inverse.

y = c + (x-a)*b + (x-a)^2*c + (x-a)^3*d + …