I am working on the heat equation in a semi-inf domain. I am doing an even extension:
$$u(x,t)= \int_{-\infty}^0 G(x-y,t)f(-y)dy + \int_{0}^\infty G(x-y,t)f(y)dy $$ Now in the next step they do:
$$u(x,t)= \int_{0}^\infty G(x+y,t)f(y)dy + \int_{0}^\infty G(x-y,t)f(y)dy $$
How did they do that? I have tried all of the properties of definite integrals and haven't found anything.