Intereting Posts

Linear independence of the numbers $\{1,e,e^2,e^3\}$
Questions about Fubini's theorem
Basic problem on topology $( James Dugundji)$
Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$
Quotient topology by identifying the boundary of a circle as one point
If coprime elements generate coprime ideals, does it imply for any $a,b\in R$ that $\langle a\rangle+\langle b\rangle=\langle \gcd (a,b)\rangle$?
Number of conjugacy classes of the reflection in $D_n$.
Conjecture regarding integrals of the form $\int_0^\infty \frac{(\log{x})^n}{1+x^2}\,\mathrm{d}x$.
Calculate the expectation of Inverse Bessel Process
Cyclic modules over a polynomial ring
Ring where irreducibles are primes which is not an UFD
An example of a function uniformly continuous on $\mathbb{R}$ but not Lipschitz continuous
Find $\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^x\right)}{\ln\left(1+2^x\right)}}$
Problem similar to folland chapter 2 problem 51.
How to prove that $\sum _{k=0}^{2n-1} \frac{(-2n)^k}{k!}<0 $

The complete elliptic integral of the first kind is defined as $$K(k) = \int_0^{\pi/2} \frac{d x}{\sqrt{1 – k^2 \sin^2 x}}.$$ I would like to derive (at least the first term of) the asymptotic expansion for $k = 1 -\epsilon$. This is certainly not trivial due to lack of uniform convergence which implies that I can’t use the Taylor expansion of the integrand. What is the best way to proceed?

- To find area of the curves that are extension of ellipse
- Reformulation of Goldbach's Conjecture as optimization problem correct?
- Equivalence to the prime number theorem
- Closed form for the integral $\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$
- How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?
- Effect of differentiation on function growth rate
- Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$
- Singular asymptotics of Gaussian integrals with periodic perturbations
- Asymptotic behavior of $|f'(x)|^n e^{-f(x)}$
- Explanation of the binomial theorem and the associated Big O notation

You can employ the substitution $y=1- k^2 \sin^2x$ such that the integral can be written as

$$ K(k) = \int_{1-k^2}^1\!dy\,\frac{1}{2 \sqrt{y(1-y)(y-1 +k^2)}}.$$

Now, we can expand the integrand in $\epsilon = 1-k$ and obtain

$$ K(k) = \int_{1-k^2}^1\!dy\frac{1}{2 y \sqrt{1-y}} + O(1). \tag{1}$$

The estimate of the error term follows from the fact that expanding in $\eta =1-k^2$, we have

$$K(k) = \sum_{n=0}^\infty c_n \eta^n \int_{\eta}^1\!dy\, y^{-1-n} (1-y)^{-1/2}$$

with $c_n$ some constants. Now, we have for $n>0$ that

$$ \int_{\eta}^1\!dy\, y^{-1-n} (1-y)^{-1/2} = O(\eta^{-n})$$

such that only the $n=0$ term diverges for $\eta \to 0$ (which corresponds to $\epsilon \to 0$).

It thus remains to estimate the first term in (1) for $k \to 1$ which is not that difficult:

In fact due to the $1/y$ behavior close to $y=0$, the integral is logarithmically divergent and we have that

$$K(k) = \frac12 \log|1-k^2| + O(1) = \frac12 \log|\epsilon| + O(1).$$

Write $k’^2 = 1 – k^2$. Make the substitution $v = k’ \cot x$ to obtain

$$K(k) = \int_0^{+\infty} (1 + v^2)^{-1/2} (1 + k’^2 v^2)^{-1/2} \, dv.$$

Now it is easy to see that this integral is $O(1)$ on any bounded interval $[0,A]$. Since we’re only interested in the leading term, we can look at the integral only on $[A + \infty]$. On that interval, the $(1 + v^2)^{-1/2}$ term is very close to $1/v$ (to within a factor of $(1 + 1/A^2)^{1/2}$, which can be as close to $1$ as we like, if we take $A$ large enough). Therefore we can write

$$K(k) \sim \int_A^{+\infty} v^{-1} (1 + k’^2 v^2)^{-1/2} \, dv.$$

Making the substitution $w = k’v$, we find

$$K(k) \sim \int_{k’A}^{+\infty} \frac{dw}{w(1+w^2)^{1/2}} \sim \int_{k’A}^{B} \frac{dw}{w(1+w^2)^{1/2}},$$

where $B$ is small and fixed, since the integral on $[B,+\infty]$ is clearly $O(1)$. Since $B$ is small, the factor $(1 + w^2)^{1/2}$ can be assumed close to $1$, so as a result

$$K(k) \sim \int_{k’A}^B \frac{dw}{w} \sim – \ln k’ = -\frac{1}{2}\ln(2\epsilon – \epsilon^2) \sim -\frac{1}{2} \ln \epsilon.$$

- fair and unfair games
- Is the axiom of choice needed to show that $a^2=a$?
- If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$?
- Representation theory of the additive group of the rationals?
- Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
- Why does $\mathbb{C}$ have transcendence degree $\mathfrak{c}$ over $\mathbb{Q}$?
- Creating a bijection from $(a,b)$ to $\mathbb R$ that is visually compelling
- Is it possible for a quadratic equation to have one rational root and one irrational root.
- Calculating volume of convex polytopes generated by inequalities
- Presentation of group equal to trivial group
- Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
- The dense topology
- Can someone give me the spherical equation for a 26 point star?
- Asymptotic of a sum involving binomial coefficients
- Dual space of $H^1(\Omega)$