Extract real part of a complex function

Question

I have a problem for extracting the real part of a complex number. The problem is the following: Suppose $f(z)=\frac{1}{z+\frac{\Delta_{1}}{z+\frac{\Delta_{2}}{z+\frac{\Delta_{3}}{z+...}}}}$, in which z is a complex number $(z= a+bi)$ and $\Delta_{1}$, $\Delta_{2}$ ... are number and $f(z)$ is a continued fraction. And I want to compute $\phi(b)$ since I would like to plot $\phi(b)$ in the end

$$\phi(b)=\lim_{a\rightarrow0}Ref(a+bi)$$

But Mathematica doesn't allow me to do that. I use "Limit" in Mathematica and set all $\Delta_{n} = 0 , n > 1$ and $\Delta_{1}=1$ , I get 0. Even I let some $\Delta_{n} $ to be nonzero, I still end up with 0.

Here is my code. I set $\Delta_{1}=1$ and $\Delta_{2}=0.5$, rest are zero.

Limit[ComplexExpand[Re[1/(a + b*I + (1/(a + b*I + (0.5/(a + b*I)))))]], a -> 0]

Can someone help me with this ? I am guessing the "Limit" function in Mathematica can't not handle this problem.

Thank you very much !!!

Answer 1

Are you sure 0 is not the right answer? The problem is not with the limit. Look at your function without limits. I changed your 0.5 to 1/2.

ComplexExpand[Re[1/(a + b*I + (1/(a + b*I + (1/2/(a + b*I)))))]] // Simplify
(*(a (4 a^4 + 8 a^2 (b^2 + 1) + 4 b^4 + 3))/((a^2 + b^2) (4 a^4 + 4 a^2 (2 b^2 + 3) + (3 - 2 b^2)^2))*)

with a as a factor in the numerator you don't need the limit.

% /. a -> 0
(*0*)