# Tagged Questions

**6**

votes

**4**answers

698 views

### Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$

In the book, "Pi and the AGM" by Borwein and Borwein, it is mentioned that Gauss computed the following integral to the eleventh decimal palce.
$\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How did he do it? ...

**0**

votes

**0**answers

61 views

### Characterisation of completely monotone functions in terms of Mellin transform

A smooth function $f(x)$ of variable $x=(x_1,\ldots,x_n)>0$ is called completely monotone if for any multiindex $\alpha \in \mathbb{N}^n_0$ the equality holds:
$$
(-1)^{|\alpha|} \frac{\partial ...

**1**

vote

**3**answers

715 views

### A definite integral

Hello,
I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...

**5**

votes

**1**answer

345 views

### Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise.
On ...