# The asymptotic behaviour of Elliptic integral near k=1

• I
• julian
In summary, the conversation is about understanding a step in the proof of the asymptotic expression for the Elliptic function of the first kind. The step involves an integral and the goal is to show that it is less than or equal to another integral. The first step in the proof is shown, but the next steps are still being worked on.
julian
Gold Member
TL;DR Summary
I'm looking at a proof of the asymptotic expression for the Elliptic function of the first kind and I'm having trouble understanding a step in the proof.
I'm looking at a proof of the asymptotic expression for the Elliptic function of the first kind

https://math.stackexchange.com/ques...ptotic-behavior-of-elliptic-integral-near-k-1

and I'm having trouble understanding this step in the proof:
\begin{align*} \frac{1}{2} \int_0^k \dfrac{dx}{1 - x} \dfrac{ \sqrt{1 - x^2} }{ \sqrt{1 - k^2 x^2} } + \mathcal{O} (1) = \frac{1}{2} \int_0^k \dfrac{dx}{1 - x} + \mathcal{O} \left( \int_0^k \dfrac{1-k}{1-x} dx \right) + \mathcal{O} (1) \end{align*}
I've written
\begin{align*} \frac{1}{2} \int_0^k \dfrac{dx}{1 - x} \dfrac{ \sqrt{1 - x^2} }{ \sqrt{1 - k^2 x^2} } &= \frac{1}{2} \int_0^k \dfrac{dx}{1 - x} + \frac{1}{2} \int_0^k \frac{dx}{1 - x} \left( \dfrac{ \sqrt{1 - x^2} }{ \sqrt{1 - k^2 x^2} } - 1 \right) \nonumber \\ &= \frac{1}{2} \int_0^k \dfrac{dx}{1 - x} - \frac{1}{2} \int_0^k \dfrac{1-k}{1-x} dx + \frac{1}{2} \int_0^k \frac{dx}{1 - x} \left( \dfrac{ \sqrt{1 - x^2} }{ \sqrt{1 - k^2 x^2} } - k \right) \end{align*}
But not sure where to go from here.

Last edited:
Delta2
Work towards proving that the last integral is less or equal than ##\int_0^k \frac{1-k}{1-x}dx##. I think it is sufficient to show that for "proper" values of ##x## and ##k## it is $$\frac{\sqrt{1-x^2}}{\sqrt{1-k^2x^2}}\leq 1$$

julian
So for ##0 \leq k < 1## we have for ##0 \leq x \leq k## that

\begin{align*}
\dfrac{ \sqrt{1 - x^2} }{ \sqrt{1 - k^2 x^2} } - k < 1 - k .
\end{align*}

Huzzah.

Delta2

## 1. What is the significance of the asymptotic behavior of Elliptic integrals near k=1?

The asymptotic behavior of Elliptic integrals near k=1 is important because it helps us understand the behavior of these integrals as the parameter k approaches 1. This is particularly useful in applications where k is close to 1, as it allows us to approximate the value of the integral without having to compute it directly.

## 2. How does the asymptotic behavior of Elliptic integrals change as k approaches 1?

As k approaches 1, the value of the Elliptic integral becomes increasingly sensitive to small changes in k. This means that the integral will approach infinity as k approaches 1, and its behavior can be described using a series expansion.

## 3. Can the asymptotic behavior of Elliptic integrals near k=1 be expressed mathematically?

Yes, the asymptotic behavior of Elliptic integrals near k=1 can be expressed using a series expansion known as the Laurent series. This series provides a mathematical representation of how the integral behaves as k approaches 1.

## 4. What are some practical applications of understanding the asymptotic behavior of Elliptic integrals near k=1?

The asymptotic behavior of Elliptic integrals near k=1 has many practical applications, particularly in physics and engineering. For example, it can be used in the calculation of the period of a pendulum, the motion of a particle in a gravitational field, and the calculation of the electric potential of a charged ring.

## 5. Are there any limitations to using the asymptotic behavior of Elliptic integrals near k=1?

While the asymptotic behavior of Elliptic integrals near k=1 can be useful in certain applications, it is not always accurate. This is because the series expansion used to approximate the integral only holds true for values of k close to 1. Therefore, it is important to consider the limitations of this approach and use it carefully in practical calculations.

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