- #1
the_dialogue
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I remember some of my linear algebra from my studies but can't wrap my head around this one.
Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So my solution is:
f(x)= A*J(d*x) + B*Y(d*x)
I can find a general solution for f(x) by imposing 2 boundary conditions f(x1)=f(x2)=0. That would give me an equation for f_n(x).
First question: The author calls this f_n(x) the "eigenfunctions" and the "orthogonal basis". Why is this given these names? I'm not sure why these solutions form an orthogonal basis.
Second question:
The author then states that an arbitrary vector F(x) "can be expanded in this orthogonal basis" via:
F(x)= sum{from n=1 to inf} [ a_n*f_n(x) ]
where
a_n = [ (f_n(x) , F(x)) ] / [ (f_n(x) , f_n(x) ]
What in the world is this on about? Any guidance would be helpful!
Homework Statement
Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So my solution is:
f(x)= A*J(d*x) + B*Y(d*x)
I can find a general solution for f(x) by imposing 2 boundary conditions f(x1)=f(x2)=0. That would give me an equation for f_n(x).
First question: The author calls this f_n(x) the "eigenfunctions" and the "orthogonal basis". Why is this given these names? I'm not sure why these solutions form an orthogonal basis.
Second question:
The author then states that an arbitrary vector F(x) "can be expanded in this orthogonal basis" via:
F(x)= sum{from n=1 to inf} [ a_n*f_n(x) ]
where
a_n = [ (f_n(x) , F(x)) ] / [ (f_n(x) , f_n(x) ]
What in the world is this on about? Any guidance would be helpful!