orthogonal sets....trick question??

[Sarahisme]

Hey, i am little puzzled by this question.... i get that none of the sets (a,b or c) are orthogonal when taken over -infinity to infinity.

 

am i doing something wrong, i would have thought at least one of them would be orthogonal...???

 

this is the problem in question:

 

the inner product i think i am meant to be using is: <f|g> = ?f*g

 

where the thing on the left (f*) is complex conjugate of f.

 

 

the combinations i tried were:

 

(a) cos(x) & 3sin(x)

(b) 1 & (-x+1)

© x * 4x^{3}

 

i would have thought that cos(x) & 3sin(x) would be orthogonal...?? or is it that i am not meant to be integrating from -infinty to infinity? (that is should i just be going from 0 to 2pi?)

 

-a puzzled Sarah :S

[Sarahisme]

ok i have been told that they none are othorgoanl sets. but i know that (-x+1) & (x^2 - 4x +2)/2 is orthogonal. but i am can't seem to show it.... i just get inifinity terms...

[Tom Mattson]

i would have thought that cos(x) & 3sin(x) would be orthogonal...?? or is it that i am not meant to be integrating from -infinty to infinity? (that is should i just be going from 0 to 2pi?)

 

The inner product for those two functions on [imath](-\infty' date='\infty)[/imath'] isn't even defined, because the limit you obtain when you evaluate the resulting integral doesn't exist. Is this question from the same instructor as the Wronskian question?

 

but i know that (-x+1) & (x^2 - 4x +2)/2 is orthogonal. but i am can't seem to show it.... i just get inifinity terms...

 

Well then they aren't orthogonal! The inner product of two orthogonal functions is zero. I wouldn't expect them to be orthogonal anyway because their product has mixed symmetry (or no symmetry, if you prefer).

[Sarahisme]

phew, ok.

 

so are their any pairs in these sets (other than cos and sin if evaluated between 0 and 2pi) which are orthogonal, because i don't think there is, i just want to make sure i am not doing somethign wrong here

[Tom Mattson]

None of these sets are orthogonal.

 

Set (a): The inner product isn't defined on [imath](-\infty,\infty)[/imath] for any pair of these functions.

 

Set (b): All of the functions in this set have mixed symmetry (except the first one, which has even symmetry).

 

Set ©: The functions [imath]f(x)=x[/imath] and [imath]f(x)=4x^3[/imath] are both orthogonal to [imath]f(x)=e^{-ax^2}[/imath] on [imath](-\infty,\infty)[/imath], but they aren't orthogonal to each other.

[Sarahisme]

ok cool, thats what i was hoping for!

 

Thank you very much Tom!