copula (version 0.999-13)

math-fun: Sinc, Zolotarev's, and Other Mathematical Utility Functions

Description

sinc(x) computes the sinc function $s(x)=\sin(x)/x$ for $x\ne 0$ and $s(0) = 1$, such that $s()$ is continuous, also at $x = 0$.

A..Z(x, a) computes Zolotarev's function to the power 1-a.

Usage

sinc(x)
A..Z(x, alpha, I.alpha = 1 - alpha)

Arguments

x

numeric argument in $[0,\pi]$, typically a vector.

alpha

parameter in (0,1].

I.alpha

must be = 1 - alpha, maybe more accurately when alpha is very close to 1.

Value

• A..Z(x,alpha) is $\tilde A_{Z}(x,\alpha)$, defined as $$\frac{\sin (\alpha x{)}^{\alpha }\sin ((1-\alpha )x{)}^{1-\alpha }}{\sin (x)},x\in [0,\pi ],$$

  where $\alpha\in(0,1]$ is alpha.

Details

For more details about Zolotarev's function, see, for example, Devroye (2009).

References

Devroye, L. (2009) Random variate generation for exponentially and polynomially tilted stable distributions, ACM Transactions on Modeling and Computer Simulation 19, 18, 1--20.

See Also

retstable internally makes use of these functions.

Examples

curve(sinc, -15,25); abline(h=0,v=0, lty=2)
curve(A..Z(x, 0.25), xlim = c(-4,4),
      main = "Zolotarev's function A(x) ^ 1-alpha")