Derivation of imaginary exponential function

[vinh pham]

I would like to find derivations of exp(-ik₀r) respect to k in order to calculate exp(-ik₁r) by using Taylor expansion:

exp(-ik₁r) = (exp(-ik₀r))⁽⁰⁾ +(k₁ -k₀)(exp(-ik₀r))⁽¹⁾/1! + (k₁ -k₀)²(exp(-ik₀r))⁽²⁾/2! + ...

This expansion converges when the value of r is relative low (0.3 - 1.2). However, when r grows with larger value (100-1000), the expansion does not converge any more.

k₀ = 21
k₁ = 27

Is there any solution to find exp(ik₁r) by using Taylor expansion for larger r?

Thank you very much

 

[Whovian]

How do you know that for large r, it doesn't converge anymore? Early terms start to get larger, but how do you know that later ones also do?

Anyways, all you need for derivation of imaginary exponential function is proving that e^{\theta\cdot i}=cis\left(\theta\right), none of these nasty k's, I think. Quite easily provable by substituting into the taylor expansion for exp, and similarly provable that it converges. I originally proved it, actually, with differential equations, noting that \dfrac{\exp'}{\exp}=\dfrac{cis'}{cis}=i and proving that they differ by a constant, and plugging in 0 gives us the desired.