Suggest a simple example of change of variables.

JuliaMath / Cubature.jl

One- and multi-dimensional adaptive integration routines for the Julia language

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Although there is a link introducing how to apply change of variables method to compute integrals over infinite or semi-infinite domains, I think taking an example would be more friendly way to explain how to use it.

If f(x)=1/(x^2), we could calculate the integral of f(x) over [a, ∞] by hands in advance.
$\int_{a}^{\infty}f(x)dx = \int_{a}^{\infty}\frac{1}{x^2}dx = \frac{1}{a}$

To compute an integral over a semi-infinite interval, you have to perform the change of variables x=a+t/(1-t):

$\int_{a}^{\infty}f(x)dx = \int_{0}^{1}f(a+\frac{t}{1-t})\frac{1}{(1-t)^2}dt = \int_{0}^{1}g(a, t)dt$

Now we could turn all of them into Julia script.

a = 1 # assume that the lower bound of integral is 1.

function f(x)
    return 1/(x^2)
end

function g(a, t)
    return f(a+(t/(1-t)))*(1/(1-t)^2)
end

hquadrature(g, 0, 1)

returning (1.0, 1.1102230246251565e-14) which are the same as our calculation by hands.

Another example about infinite interval integral.

If f(x)=e^(-x^2), we could calculate the integral of f(x) over [-∞, ∞] by hands in advance.
$\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}\approx 1.772453850905516\ \ (by\ Gaussian\ integral)$

To compute an integral over a infinite interval, you have to perform the change of variables x=t/(1-t²):

$\int_{-\infty}^{\infty} f(x)dx=\int_{-1}^{1} f\bigg(\frac{t}{1-t^2}\bigg)\frac{1+t^2}{(1-t^2)^2}dt=\int_{-1}^{1}g(t)dt$

Now we could turn all of them into Julia script.

function f(x)
    return exp(-x^2)
end

function g(t)
    return f(t/(1-t^2))*(1+t^2)/((1-t^2)^2)
end

hquadrature(g, -1, 1)

returning (1.772453850905516, 2.95326836403382e-9) which are the same as our calculation by hands.

JuliaMath / Cubature.jl

Suggest a simple example of change of variables. #42