Legendre Function of the Second Kind, $Q_\nu(x)$

Computes the Legendre function of the second kind, $Q_\nu(x)$, for real order $\nu$ (integer or fractional) and real argument $x$. Returns complex values when $|x| > 1$.

Usage

Qn(n, x)

Arguments

n: Numeric vector. Degree (order) of the Legendre function. Can be integer or fractional (e.g., 0, 1, 2.5, 3.7).
x: Numeric vector. Real argument(s) at which to evaluate the function. Valid for all real values. Note that $x = \pm 1$ are singularities.

Value

A complex matrix of dimension length(n) by length(x), where element [i, j] contains $Q_{n_i}(x_j)$. For $|x| < 1$, the imaginary part is zero.

Details

The Legendre function of the second kind satisfies the same differential equation as $P_\nu(x)$: $$(1 - x^2) \frac{d^2 Q_\nu}{dx^2} - 2x \frac{dQ_\nu}{dx} + \nu(\nu + 1) Q_\nu = 0$$

but represents the linearly independent second solution.

For $|x| < 1$ (real result):

For integer order $n$: Uses the C++ Boost library implementation
For fractional order $\nu$: Uses the Ferrers identity $$Q_\nu(x) = \frac{\pi}{2 \sin(\pi \nu)} \left[ \cos(\pi \nu) P_\nu(x) - P_\nu(-x) \right]$$

For $x = \pm 1$: Returns infinity (singularity).

For $|x| > 1$ (complex result):

Real part: Computed via the integral representation $$\text{Re}[Q_\nu(x)] = \int_0^\infty \frac{dt}{(x + \sqrt{x^2-1} \cosh t)^{\nu+1}}$$
Imaginary part: $$\text{Im}[Q_\nu(x)] = -\frac{\pi}{2} P_\nu(x)$$

Note

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 with Formulas, Graphs, and Mathematical Tables. Dover Publications. Chapter 8: Legendre Functions.

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/14

Examples

# Single values
Qn(1, 0.5)
#>               [,1]
#> [1,] -0.7253469+0i

# Multiple orders, single argument
Qn(c(1, 2, 3), 0.5)
#>               [,1]
#> [1,] -0.7253469+0i
#> [2,] -0.8186633+0i
#> [3,] -0.1986548+0i

# Single order, multiple arguments
Qn(1, c(-0.5, 0, 0.5))
#>               [,1]  [,2]          [,3]
#> [1,] -0.7253469+0i -1+0i -0.7253469+0i

# Multiple orders and arguments (returns a matrix)
Qn(c(1, 2, 3), c(0.25, 0.5, 0.75))
#>               [,1]          [,2]          [,3]
#> [1,] -0.9361468+0i -0.7253469+0i -0.2702837+0i
#> [2,] -0.4787615+0i -0.8186633+0i -0.7905467+0i
#> [3,]  0.4246139+0i -0.1986548+0i -0.8079942+0i

# Fractional orders
Qn(c(0.5, 1.5, 2.7), c(0.5, 0.9))
#>               [,1]          [,2]
#> [1,] -0.2655964+0i  0.7873898+0i
#> [2,] -0.8959028+0i -0.0245780+0i
#> [3,] -0.4202294+0i -0.5761958+0i

# Arguments |x| > 1 return complex values
Qn(c(1, 2, 3.5), c(2, 3.5))
#>                        [,1]                     [,2]
#> [1,] 0.098612289- 3.141593i 0.0286266636-  5.497787i
#> [2,] 0.021183794- 8.639380i 0.0033433176- 28.077984i
#> [3,] 0.002370082-47.971060i 0.0001501177-390.254185i

# Singularity at x = 1
Qn(1, 1)
#>          [,1]
#> [1,] Inf+Infi

Legendre Function of the Second Kind, \(Q_\nu(x)\)