# How to interpolate/extrapolate a complex function?

Let us assume that we have $$f:\mathbb{R} \to \mathbb{R}$$. Also let us assume that $$x_1\in \mathbb{R}$$ and $$x_2\in \mathbb{R}$$ are given too. With this we can calculate $$y_1 = f(x_1)$$ and $$y_2 = f(x_2)$$.

Now if we want to interpolate/extrapolate an in-between value we could take a value like e.g. $$\alpha \in \mathbb{R}$$ (but for interpolation $$\alpha$$ would be in range $$[0,1]$$).

With this we could calculate the interpolated values $$x_{irp},y_{irp} \in \mathbb{R}$$ like:

$$x_{irp}=\alpha \cdot x_1 + (1- \alpha) \cdot x_2$$ $$y_{irp}=\alpha \cdot y_1 + (1- \alpha) \cdot y_2$$

Now my question:

Does the same approach also works for complex numbers too? In this case instead of $$\mathbb{R}$$ we would have $$\mathbb{C}$$.

(I know indeed that for multidimensional input and output this approach works, but for complex numbers too?)

## 1 Answer

Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).

• Is there a better approach too? I mean like for the $\mathbb{R}$ numbers with a square line approximation? How would this one work for $\mathbb{C}$? – PiMathCLanguage Nov 26 '18 at 13:51