This is how I learned about Green's functions:(adsbygoogle = window.adsbygoogle || []).push({});

For the 1-D problem with the linear operator L and the inner product,

[itex](\cdot,\cdot)[/itex],

[itex]Lu(x) = f(x) \rightarrow u=(f(x),G(\xi,x))[/itex]

if the Green's function G is defined such that

[itex]L^*G(\xi,x) = \delta(\xi-x)[/itex]

I understand how to arrive at this algebraically. However, most articles I read define the Green's function backwards (?) like this:

[itex]LG(x,\xi)=\delta(x-\xi)[/itex]

How do I arrive at this definition? As in, how do I work through the algebra to show that the green's function can be defined like this? I am assuming that the swapping of the variables indicates an equivalence between the two definitions, but I do not immediately see it, and it has been confusing me for quite a bit. Does it have something to do with whether we're on the interval [a,b] in [itex]\xi[/itex] vs [itex]x[/itex]? Could someone walk me through the steps? No, this is not a homework or coursework question. I'm just confused with the definition.

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# Confusion over the definition of a Green's function

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