R: Line integral (in the complex plane)

line_integral {pracma} | R Documentation

## Line integral (in the complex plane)

### Description

Provides complex line integrals.

### Usage

    line_integral(fun, waypoints, method = NULL, reltol = 1e-8, ...)

### Arguments

`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.

### Details

`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.

### Value

Returns the integral, no error terms given.

### See Also

`integral`

### Examples

    ##  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

[Package pracma version 2.0.4 Index]