# line_integral: Line integral (in the complex plane) In pracma: Practical Numerical Math Functions

## Description

Provides complex line integrals.

## Usage

 `1` ```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.

`integral`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## 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 ```

### Example output

```[1] 0
[1] 0
[1] 5.015201e-17
```

pracma documentation built on Dec. 11, 2021, 9:57 a.m.