---
title: Jump condition for the Greens function
tags: Applied_math_method
GA: G-77TT93X4N1
---

# Jump condition for the Greens function

Consider Green's function that satisfies
$$
a_2(x) g'' + a_1(x)g' + a_0(x) g = \delta(x-\xi),
$$
where $a_2$, $a_1$ and $a_0$ are continuous functions.

To derive the jump condition, we integrate the equation on a small region $(\xi^-, \xi^+)$. The left hand side gives
$$
\int^{\xi^+}_{\xi^-}a_2(x) g'' + a_1(x)g' + a_0(x) g\,dx.
$$
In the following we evaluate this quantity term-by-term.

Since $a_2$ is continuous,
$$
\int^{\xi^+}_{\xi^-}a_2(x) g''\,dx = a_2(\xi)\int^{\xi^+}_{\xi^-} g''\,dx = a_2(\xi)(g'(\xi^+) - g'(\xi^-)).
$$

$a_1$ and $g$ are continuous, so we have
$$
\int^{\xi^+}_{\xi^-}a_1(x) g'\,dx = a_1(\xi)\int^{\xi^+}_{\xi^-} g'\,dx = a_1(\xi)(g(\xi^+) - g(\xi^-))=0,
$$
and use the fact that $a_0$ and $g$ are both continuous to have
$$
\int^{\xi^+}_{\xi^-}a_0(x) g\,dx = 0.
$$

The right hand side gives
$$
\int^{\xi^+}_{\xi^-} \delta(x-\xi)\,dx = 1.
$$

In summary, we have
$$
g'(\xi^+) - g'(\xi^-) = \frac{1}{a_2(\xi)}.
$$