Derivation of imaginary exponential function

In summary, the conversation discusses the use of Taylor expansion to find derivations of exp(-ik0r) with respect to k in order to calculate exp(-ik1r). It is noted that the expansion converges for low values of r (0.3 - 1.2) but does not converge for larger values (100-1000). The question is raised if there is a solution to find exp(ik1r) using Taylor expansion for larger r. The speaker also mentions an alternative method of proving e^{\theta\cdot i}=cis\left(\theta\right) using differential equations.
  • #1
vinh pham
1
0
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
 
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  • #2
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 [tex]e^{\theta\cdot i}=cis\left(\theta\right)[/tex], 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 [tex]\dfrac{\exp'}{\exp}=\dfrac{cis'}{cis}=i[/tex] and proving that they differ by a constant, and plugging in 0 gives us the desired.
 

FAQ: Derivation of imaginary exponential function

1. What is the definition of an imaginary exponential function?

An imaginary exponential function is a mathematical expression of the form f(x) = e^ix, where i is the imaginary unit (√-1) and e is the base of the natural logarithm. It is also known as a complex exponential function.

2. How is an imaginary exponential function derived?

An imaginary exponential function can be derived using the Taylor series expansion of the exponential function. This involves expressing e^ix as a sum of terms with increasing powers of i. The result is a series that converges to the imaginary exponential function.

3. What are the properties of an imaginary exponential function?

Some key properties of an imaginary exponential function include: it has a period of 2π, it is an entire function (meaning it is analytic everywhere in the complex plane), and it can be used to represent circular motion in the complex plane.

4. How is an imaginary exponential function used in physics and engineering?

Imaginary exponential functions are commonly used to describe oscillatory processes in physics and engineering. They can also be used to simplify complex calculations involving trigonometric functions.

5. What is the relationship between an imaginary exponential function and the unit circle?

An imaginary exponential function can be represented as a point on the unit circle in the complex plane. This is because the magnitude of e^ix is always equal to 1, and the argument (or angle) is equal to the value of x.

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