I need a numerical approximation around some functions at $x = 0^{-}$ from the left side, where $x = 0$ is unfortunately the right end of the domain (in the reals) so the functions are not differentiable there.

Nonetheless, all I what is an approximation so I actually just need a one-sided limit along the reals and don't need it to be differentiable. I don't care about the fact that different "series" will be obtained when one approaches $x=0$ from different directions in the complex plane.

That is, I'll take whatever works.

The two methods shown below of doing series expansion yield results that differ by a complex $i$. There are two examples, each being a Column object aligning the two ways of coding that seem mathematically the same to me.

ClearAll[x];
Column[{  (* example 1 *)
Series[ -1 + Sqrt[-x (2 + x)]
 , {x, 0, 4}, Assumptions -> x < 0] 
, Series[-1 + Sqrt[(2 - x) x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

Column[{  (* example 2 *)
Series[  ArcCos[1 + x]
 , {x, 0, 4}, Assumptions ->  x < 0] 
, Series[ ArcCos[1 - x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

The discrepancy most likely involve something like $\arg(x)$ and this has to do with the expansion being done near the branch cut.

However, why are there two different outcomes in terms of coding? To me the two ways of coding seem mathematically equivalent. If one of them is more appropriate, which one?

The expansion is meant to be, for instance more clearly seen in the 2nd example, $x \to 0^{-}$ coming from the left along the real line (inside the proper domain in reals). In both examples the function goes "vertical" at $x = 0$.

Thank you.

I need a numerical approximation around some functions at $x = 0^{-}$ from the left side, where $x = 0$ is unfortunately the right end of the domain (in the reals) so the functions are not differentiable there.

Nonetheless, all I want is an approximation so I actually just need a one-sided limit along the reals and don't need it to be differentiable. I don't care about the fact that different "series" will be obtained when one approaches $x=0$ from different directions in the complex plane.

That is, I'll take whatever works.

The two methods shown below of doing series expansion yield results that differ by a complex $i$. There are two examples, each being a Column object aligning the two ways of coding that seem mathematically the same to me.

ClearAll[x];
Column[{  (* example 1 *)
Series[ -1 + Sqrt[-x (2 + x)]
 , {x, 0, 4}, Assumptions -> x < 0] 
, Series[-1 + Sqrt[(2 - x) x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

Column[{  (* example 2 *)
Series[  ArcCos[1 + x]
 , {x, 0, 4}, Assumptions ->  x < 0] 
, Series[ ArcCos[1 - x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

The discrepancy most likely involve something like $\arg(x)$ and this has to do with the expansion being done near the branch cut.

However, why are there two different outcomes in terms of coding? To me the two ways of coding seem mathematically equivalent. If one of them is more appropriate, which one?

The expansion is meant to be, for instance more clearly seen in the 2nd example, $x \to 0^{-}$ coming from the left along the real line (inside the proper domain in reals). In both examples the function goes "vertical" at $x = 0$.

Thank you.

I need a numerical approximation around some functions at $x = 0^{-}$ from the left side, where $x = 0$ is unfortunately the right end of the domain (in the reals) so the functions are not differentiable there.

Nonetheless, all I what is an approximation so I actually just need a one-sided limit along the reals and don't need it to be differentiable. I don't care about the fact that different "series" will be obtained when one approaches $x=0$ from different directions in the complex plane.

That is, I'll take whatever works.

The two methods shown below of doing series expansion yield results that differ by a complex $i$. There are two examples, each being a Column object aligning the two ways of coding that seem mathematically the same to me.

ClearAll[x];
Column[{  (* example 1 *)
Series[ -1 + Sqrt[-x (2 + x)]
 , {x, 0, 4}, Assumptions -> x < 0] 
, Series[-1 + Sqrt[(2 - x) x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

Column[{  (* example 2 *)
Series[  ArcCos[1 + x]
 , {x, 0, 4}, Assumptions ->  x < 0] 
, Series[ ArcCos[1 - x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

The discrepancy most likely involve something like $Arg(x)$ and this has to do with the expansion being done near the branch cut.

However, why are there two different outcomes in terms of coding? To me the two ways of coding seem mathematically equivalent. If one of them is more appropriate, which one?

The expansion is meant to be, for instance more clearly seen in the 2nd example, $x \to 0^{-}$ coming from the left along the real line (inside the proper domain in reals). In both examples the function goes "vertical" at $x = 0$.

Thank you.

I need a numerical approximation around some functions at $x = 0^{-}$ from the left side, where $x = 0$ is unfortunately the right end of the domain (in the reals) so the functions are not differentiable there.

Nonetheless, all I what is an approximation so I actually just need a one-sided limit along the reals and don't need it to be differentiable. I don't care about the fact that different "series" will be obtained when one approaches $x=0$ from different directions in the complex plane.

That is, I'll take whatever works.

The two methods shown below of doing series expansion yield results that differ by a complex $i$. There are two examples, each being a Column object aligning the two ways of coding that seem mathematically the same to me.

ClearAll[x];
Column[{  (* example 1 *)
Series[ -1 + Sqrt[-x (2 + x)]
 , {x, 0, 4}, Assumptions -> x < 0] 
, Series[-1 + Sqrt[(2 - x) x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

Column[{  (* example 2 *)
Series[  ArcCos[1 + x]
 , {x, 0, 4}, Assumptions ->  x < 0] 
, Series[ ArcCos[1 - x]
 , {x, 0, 4}, Assumptions -> x > 0] }]

The discrepancy most likely involve something like $\arg(x)$ and this has to do with the expansion being done near the branch cut.

However, why are there two different outcomes in terms of coding? To me the two ways of coding seem mathematically equivalent. If one of them is more appropriate, which one?

The expansion is meant to be, for instance more clearly seen in the 2nd example, $x \to 0^{-}$ coming from the left along the real line (inside the proper domain in reals). In both examples the function goes "vertical" at $x = 0$.

Thank you.