# Tagged Questions

**1**

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**1**answer

102 views

### Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly
$$d_{n,m}(x) \le d_{n,m}(1) = ...

**1**

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**0**answers

80 views

### Simplify this expression with modified Bessel functions of the second kind

I'm interested in the function
$$\frac{K^{(0,2)}(0,x)}{K(0,x)}$$
where the numerator is the modified Bessel function of the second kind twice differentiated with respect to $\alpha$ and taken at ...

**4**

votes

**1**answer

190 views

### How to get an expression for this integral(Numerically/Analytically)

I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...

**2**

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**2**answers

223 views

### Computing hypergeometric function of matrix argument

In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...

**5**

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**0**answers

416 views

### Parabolic cylinder functions - explicit estimates?

I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker).
Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in ...

**2**

votes

**2**answers

2k views

### Numerical Computation of arcsin and arctan for real numbers [closed]

I'm coding some numerical methods and I do not know what the correct analysis would be for choosing the implementation for $arcsin$ and $arctan$ for real numbers. Here's what I know:
Both functions ...

**2**

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**0**answers

658 views

### Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...

**3**

votes

**3**answers

3k views

### Finding the length of a cubic B-spline

Can I find an analytical solution to the the length of an 2-dimensional cubic B-spline? All I can find are chorded approximations and the opinion that the analytic solution is "unbearably gruesome". ...