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Hankel Functions

Source: R/hankel_functions.R
hankel_functions.Rd

Functions to compute cylindrical and spherical Hankel functions of the first and second kinds.

Cyclic Hankel Functions

  • cyl_hankel_1(v, x): The Hankel function of the first kind: \(H_v^{(1)}(x) = J_v(x) + iY_v(x)\)

  • cyl_hankel_2(v, x): The Hankel function of the second kind: \(H_v^{(2)}(x) = J_v(x) - iY_v(x)\)

Where \(J_v(x)\) is the Bessel function of the first kind and \(Y_v(x)\) is the Bessel function of the second kind.

Spherical Hankel Functions:

  • sph_hankel_1(v, x): The spherical Hankel function of the first kind: \(h_v^{(1)}(x) = \sqrt{\frac{\pi}{2}}\frac{1}{\sqrt{\pi}}H_{v + \frac{1}{2}}^{(1)}(x)\)

  • sph_hankel_2(v, x): The spherical Hankel function of the second kind: \(h_v^{(2)}(x) = \sqrt{\frac{\pi}{2}}\frac{1}{\sqrt{\pi}}H_{v + \frac{1}{2}}^{(2)}(x)\)

Usage

cyl_hankel_1(v, x)

cyl_hankel_2(v, x)

sph_hankel_1(v, x)

sph_hankel_2(v, x)

Arguments

v

Order of the Hankel function (can be any real number)

x

Argument of the Hankel function (can be any real number)

Value

A single complex value with the computed Hankel function.

See also

Boost Documentation for more details on the mathematical background.

Examples

cyl_hankel_1(2, 0.5)
#> [1] 0.03060402-5.441371i
cyl_hankel_2(2, 0.5)
#> [1] 0.03060402+5.441371i
sph_hankel_1(2, 0.5)
#> [1] 0.01637111-25.05992i
sph_hankel_2(2, 0.5)
#> [1] 0.01637111+25.05992i