Derivative of the Legendre function of the second kind

Usage

Qndk(n, x, k = 1L)

Arguments

n: Numeric vector. Degree (order) of the Legendre function.
x: Numeric vector. Real argument(s) at which to evaluate the derivative. Avoid \(x = \pm 1\).
k: Integer. Order of the derivative (\(k \geq 0\)). Default is 1.

Value

A complex matrix of dimension length(n) by length(x), where element [i, j] contains \(\frac{d^k}{dx^k} Q_{n_i}(x_j)\).

Note

Derivatives are computed via finite differences with step size \(h = 10^{-6}\). Accuracy is typically 4-6 significant digits for first derivatives, less for higher orders.

This function calls underlying \(C++\) code via Rcpp for computational efficiency and to support different cases for both order and argument that are not readily available in R.

References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover Publications. Chapter 8: Legendre Functions.

Examples

# First derivative of Q_1(x) at x = 0.5
Qndk(1, 0.5, 1)
#>             [,1]
#> [1,] 1.215973+0i

# Compare with numerical derivative
h <- 1e-6
(Qn(1, 0.5 + h) - Qn(1, 0.5 - h)) / (2 * h)
#>             [,1]
#> [1,] 1.215973+0i

# Second derivative
Qndk(2, 0.5, 2)
#>            [,1]
#> [1,] 5.42566+0i

# Multiple orders
Qndk(c(1, 2, 3), 0.5, 1)
#>               [,1]
#> [1,]  1.2159728+0i
#> [2,] -0.8427075+0i
#> [3,] -2.8773435+0i

# Complex result for |x| > 1
Qndk(1, 2.0, 1)
#>                      [,1]
#> [1,] -0.1173605-1.570796i