Hi,(adsbygoogle = window.adsbygoogle || []).push({});

Let me describe the question. Suppose you have two distinct solutions, say, [itex]G_{1}(z)[/itex] and [itex]G_{2}(z)[/itex], to such a linear differential equation, [itex](\partial^2_z-q^2)G(z)=\delta(z)[/itex], where [itex]\delta(z)[/itex] denotes the Dirac function and [itex]q^2[/itex] is a constant. Now I'd like to evaluate this integral: [itex]\int^{\infty}_0\int^{\infty}_0dz_1dz_2G_1(z-z_1)G_2(z'-z_2)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)[/itex]. I will use this property: [itex]f(z)\delta^{n}(z)=(-1)^{n}f^{n}(z)\delta(z)[/itex]. Thus, if I in the first place integrate over [itex]z_1[/itex], I should then find

[tex]\begin{eqnarray*}

\int^{\infty}_0\int^{\infty}_0dz_1dz_2G_1(z-z_1)G_2(z'-z_2)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)

& = &\int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1G_1(z-z_1)(\partial^2_{z_1}-q^2)\delta(z_1-z_2)\\

& = & \int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1(\partial^2_{z_1}-q^2)G_1(z-z_1)\delta(z_1-z_2)\\

& = &\int^{\infty}_0dz_2G_2(z'-z_2)\int^{\infty}_0dz_1\delta(z-z_1)\delta(z_1-z_2)\\

& = & G_2(z-z')

\end{eqnarray*}[/tex] On other hand, if it is first integrated over [itex]z_2[/itex], one would instead find [itex]G_1(z-z')[/itex], which differs from previous result. I don't know how to make sense of such ambiguity. Can anybody come to help ?!

Thank you in advance.

hiyok

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# What is wrong with this integral?

**Physics Forums | Science Articles, Homework Help, Discussion**