How can we expand a function near a singularity x=0?

[eljose]

let,s suppose we have a function f so the limit when $$\epsilon\rightarrow{0}$$ is infinite..now i would like to know how could i make an expansion of the function f near the singularity x=0 so we have..
$$f(x)=\frac{a0(x)}{(x-\epsilon)}+\frac{a1(x)}{(x-\epsilon)^{2}...$$
i say a series that is valid near the point x=0+e

 

[Zurtex]

let,s suppose we have a function f so the limit when $$\epsilon\rightarrow{0}$$ is infinite..now i would like to know how could i make an expansion of the function f near the singularity x=0 so we have..
$$f(x)=\frac{a0(x)}{(x-\epsilon)}+\frac{a1(x)}{(x-\epsilon)^{2}...$$
i say a series that is valid near the point x=0+e

Perhaps you meant:

$$f(x)=\frac{a0(x)}{(x-\epsilon)}+\frac{a1(x)}{(x-\epsilon)^{2}}\ldots$$

 

[Edwin]

consider a function of two of more variables.

Here can take the jacobian, and if the absolute value of the jacobian at point in the domain is greater than 1, expansion is accuring at that point in the domain, if the absolute value of the jacobian is less than 1, then contraction is occurring at that point in the domain. If the absolute value of the jacobian is equal to one, then there is neither contraction, nor expansion occurring at the point in the domain. Mind you, in this case, I believe that contraction/expansion is defined as occurring on the domain of the given function, and we are assuming that the function is continuous and differentiable over it's entire domain.

Does this help? Do you know how to take the jacobian of the a function of n variables? If not I can explain pretty quickly in my next post. It's pretty easy thing to do for functions of 2 or 3 variables.

Best Regards,

Edwin