1. The problem statement, all variables and given/known data
Greetings! I am reading section 2.8 of Jackson and trying to understand how completeness relation was derived.
It starts with the orthonormality condition:
[itex]∫U_N *(ε) U(ε) dε =δ_{nm}[/itex]
We can represent a function as a sum of orthonormal functions if N is finite:
[itex] f(ε) ⇔ \sum_{n=1}^N a_n U_n (ε) [/itex]
We can get the best coefficients a by minimizing the error MN:
[itex]M_N = \int_a ^b |f(ε) - \sum_{n=1}^N a_n U_n(ε)|^2 dε[/itex]
Jackson says that "it is easy to show that the coefficients are given by
[itex]a_n=\int_a ^b U_n*(ε) f(ε) dε [/itex]."
Question: How do I show this?
2. Relevant equations
Same as above.
3. The attempt at a solution
I guess minimizing MN means setting it to zero. And then I'm not sure what to do after. How do I extract the buried coefficients ? I am very sorry. :confused:
Greetings! I am reading section 2.8 of Jackson and trying to understand how completeness relation was derived.
It starts with the orthonormality condition:
[itex]∫U_N *(ε) U(ε) dε =δ_{nm}[/itex]
We can represent a function as a sum of orthonormal functions if N is finite:
[itex] f(ε) ⇔ \sum_{n=1}^N a_n U_n (ε) [/itex]
We can get the best coefficients a by minimizing the error MN:
[itex]M_N = \int_a ^b |f(ε) - \sum_{n=1}^N a_n U_n(ε)|^2 dε[/itex]
Jackson says that "it is easy to show that the coefficients are given by
[itex]a_n=\int_a ^b U_n*(ε) f(ε) dε [/itex]."
Question: How do I show this?
2. Relevant equations
Same as above.
3. The attempt at a solution
I guess minimizing MN means setting it to zero. And then I'm not sure what to do after. How do I extract the buried coefficients ? I am very sorry. :confused:
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