Ooura Fourier Integrals

Computes Fourier sine and cosine integrals using Ooura's robust double exponential method (1999). These methods are designed to handle oscillatory integrals that are problematic for standard quadrature.

Ooura Fourier Sine: Computes the integral: $$\int_0^\infty f(t) \sin(\omega t) dt$$

Ooura Fourier Cosine: Computes the integral: $$\int_0^\infty f(t) \cos(\omega t) dt$$

Usage

ooura_fourier_sin(
  f,
  omega = 1,
  relative_error_tolerance = sqrt(.Machine$double.eps),
  levels = 8
)

ooura_fourier_cos(
  f,
  omega = 1,
  relative_error_tolerance = sqrt(.Machine$double.eps),
  levels = 8
)

Arguments

f

A function to integrate. It should accept a single numeric value and return a single numeric value.

omega

The frequency parameter \(\omega\) for the sine or cosine term.

relative_error_tolerance

The relative error tolerance for the approximation. Default is sqrt(.Machine$double.eps).

levels

The number of levels of refinement to apply. Default is 8.

Value

A single numeric value with the computed Fourier sine or cosine integral.

Details

The method precomputes nodes and weights for efficiency. Convergence depends on the position of the poles of the integrand in the complex plane. If poles are too close to the real axis, convergence may be slow.

References

Ooura, Takuya, and Masatake Mori. "A robust double exponential formula for Fourier-type integrals." Journal of computational and applied mathematics 112.1-2 (1999): 229-241.

See also

double_exponential_quadrature

Examples

# Fourier sine integral of 1/x -> integral convergent to pi/2 approx
# sin(x)/x from 0 to Inf is pi/2. Here we integrate 1/x * sin(1*x).
ooura_fourier_sin(function(x) { 1 / x }, omega = 1)
#> [1] 1.570796
#> attr(,"relative_error")
#> [1] 1.265522e-11

# Fourier cosine integral of 1/(x^2 + 1) * cos(x)
# Expected value is pi/(2*e) approx 0.57786
ooura_fourier_cos(function(x) { 1/ (x * x + 1) }, omega = 1)
#> [1] 0.5778637
#> attr(,"relative_error")
#> [1] 6.417739e-09