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Error functions

Error functions

Error function

The error function (also called the Gauss error function), often denoted by erf, is a function defined as

$$ \operatorname{erf} z = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}dt $$

Example

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use mathru::special::error;
use plotters::prelude::*;

fn main() {
    let x_start: f64 = -3.0;
    let x_end: f64 = 3.0;
    let length: usize = 2000;

    let mut graph_1: Vec<(f64, f64)> = Vec::with_capacity(length);

    for i in 0..length {
        let x: f64 = (x_end - x_start) / (length as f64) * (i as f64) + x_start;
        graph_1.push((x, error::erf(x)));
    }

    let root_area = BitMapBackend::new("./figures/erf.png", (600, 400)).into_drawing_area();
    root_area.fill(&WHITE).unwrap();

    let mut ctx = ChartBuilder::on(&root_area)
        .margin(20)
        .set_label_area_size(LabelAreaPosition::Left, 40)
        .set_label_area_size(LabelAreaPosition::Bottom, 40)
        .build_cartesian_2d(x_start..x_end, -1.1f64..1.1f64)
        .unwrap();

    ctx.configure_mesh()
        .x_desc("x")
        .y_desc("erf(y)")
        .axis_desc_style(("sans-serif", 15).into_font())
        .draw()
        .unwrap();

    ctx.draw_series(LineSeries::new(
        graph_1,
        Into::<ShapeStyle>::into(&BLUE).stroke_width(2),
    ))
    .unwrap();
}

Error function

Complementary error function

$$ \operatorname{erfc}(z) = 1 - \operatorname{erf} z, $$