- #1

julian

Gold Member

- 809

- 315

- 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.

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: