Line integral (in the complex plane)
Provides complex line integrals.
line_integral(fun, waypoints, method = NULL, reltol = 1e-8, ...)
fun |
integrand, complex (vectorized) function. |
method |
integration procedure, see below. |
waypoints |
complex integration: points on the integration curve. |
reltol |
relative tolerance. |
... |
additional parameters to be passed to the function. |
line_integral realizes complex line integration, in this case straight
lines between the waypoints. By passing discrete points densely along the
curve, arbitrary line integrals can be approximated.
line_integral will accept the same methods as integral;
default is integrate from Base R.
Returns the integral, no error terms given.
## Complex integration examples points <- c(0, 1+1i, 1-1i, 0) # direction mathematically negative f <- function(z) 1 / (2*z -1) I <- line_integral(f, points) abs(I - (0-pi*1i)) # 0 ; residuum 2 pi 1i * 1/2 f <- function(z) 1/z points <- c(-1i, 1, 1i, -1, -1i) I <- line_integral(f, points) # along a rectangle around 0+0i abs(I - 2*pi*1i) #=> 0 ; residuum: 2 pi i * 1 N <- 100 x <- linspace(0, 2*pi, N) y <- cos(x) + sin(x)*1i J <- line_integral(f, waypoints = y) # along a circle around 0+0i abs(I - J) #=> 5.015201e-17; same residuum
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